research article optimal investment-consumption strategy under...

Research ArticleOptimal Investment-Consumption Strategy underInflation in a Markovian Regime-Switching Market

Huiling Wu

China Institute for Actuarial Science Central University of Finance and Economics Beijing 100081 China

Correspondence should be addressed to Huiling Wu sunnyling168hotmailcom

Received 30 December 2015 Accepted 9 June 2016

Academic Editor Silvia Romanelli

Copyright copy 2016 Huiling Wu This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies an investment-consumption problem under inflation The consumption price level the prices of the availableassets and the coefficient of the power utility are assumed to be sensitive to the states of underlying economy modulated bya continuous-time Markovian chain The definition of admissible strategies and the verification theory corresponding to thisstochastic control problem are presented The analytical expression of the optimal investment strategy is derived The existenceboundedness and feasibility of the optimal consumption are proven Finally we analyze in detail by mathematical and numericalanalysis how the risk aversion the correlation coefficient between the inflation and the stock price the inflation parameters andthe coefficient of utility affect the optimal investment and consumption strategy

1 Introduction

The investment-consumption optimization problemhas beenone of the time-honored topics in which the decision-makerseeks to maximize the expected utility of intertemporal con-sumption plus the terminal wealth The classical investment-consumption model can be traced back to the foundationalwork of Samuelson [1] Hakansson [2] Fama [3] andMerton[4] Their foundational work has inspired various extensionsand applications from different aspects in the past forty yearsincluding Zariphopoulou [5] Akian et al [6] Liu [7] Zhaoand Nie [8] and Dai et al [9] with transaction costs Taksarand Sethi [10] and Zariphopoulou [11] with bankruptcyMunk and Soslashrensen [12] andWang andYi [13] with stochasticinterest rate or stochastic return rate Munk [14] and Dybvigand Liu [15] with a stochastic income and Pliska and Ye [16]and Kwak et al [17] with a life insurance purchase

No matter what the models are the existing researchpapers mentioned above share the common setting that theinvestor makes the investment-consumption decision underthe uncertainty of assetsrsquo prices However the uncertainty ofthe assetsrsquo prices also comes from the uncertainty of underly-ing economy To be more specific the market mode in a realworld usually has a finite number of states such as ldquobullishrdquo

and ldquobearishrdquo and could switch among them It is calledldquoregime switchingrdquo The empirical analysis shows that thereturns of the assets such as the stocksrsquo appreciation ratesand volatility rates are sensitive to the states of underlyingeconomy and are quite different in different states For exam-ple Hardy [18] used monthly data from the Toronto StockExchange 300 indices and the Standard and Poorrsquos 500 tofit a regime-switching log-normal model and found that theregime-switching model is better than all the models consid-eredUsually themovement of themarket states is depicted bya continuous-timefinite-stateMarkovian chain and the assetrsquosreturn at time 119905 is assumed to be a function of the currentmarket state In the past years there has been great interestof using regime-switching models in finance and actuarialscience Here we review the related literature with regimeswitching according to the research topic Zhou and Yin[19] Cakmak and Ozekici [20] Celikyurt and Ozekici [21]Wei and Ye [22] Wu and Li [23 24] and Wu and Zeng [25]considered the optimal investment strategy under the mean-variance criterion while Chen et al [26] investigated anasset-liability management problem Cheung and Yang [27]and Canakoglu and Ozekici [28 29] studied the investmentpolicy under the power utility but the latter ones assumedthat the utilities have the regime-dependent parameters

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 9606497 17 pageshttpdxdoiorg10115520169606497

2 Discrete Dynamics in Nature and Society

that is the utility is being of the form 119880(119894 119909) = 119870(119894) +

119862(119894)(119909 minus 120573)120574120574 As for the optimal investment-consumption

problem Cheung and Yang [30] considered a multiperiodmodel where the return of the risky assets depends onthe economic environments with an absorbing state whichrepresents the bankruptcy state Li et al [31] and Zeng etal [32] investigated a discrete-time investment-consumptionproblem with regime switching and uncertain time horizonGassiat et al [33] studied this problem in an illiquid financialmarket where the asset trading has time restriction Pirvuand Zhang [34] considered a continuous-time investment-consumption problem with regime-switching discount rateand asset returns In this paper enlightened by the existingliterature we assume that the utility function is of this form119880(119909) = 120577(119894)119909

120574120574 where 120577(119894) is dependent on the current

market state 119894 Under this assumption the utility functionis changed according to the market states over time In thissense our paper has adopted the similar assumption aboutthe movement of the financial market and the utility param-eters as some existing literature For example Canakogluand Ozekici [28 29] also assumed that the parametersof the power utility are state-dependent However thereare some differences between our paper and the existingliterature Firstly the optimization problem considered isdifferent Canakoglu and Ozekici [28 29] investigated anoptimal portfolio selection problem while our paper studiesan investment-consumption strategy Secondly the above-mentioned research papers do not analyze the effects of theinflation In contrast in addition to the financial risk theinflation risk is also considered in our paper

In recent years the phenomenon of inflation has beencausing grave concern in developed and developing coun-tries When the continuous increase in price level exceedsa tolerable limit the inflation can cause many distortionsin investment behavior and effect greatly on daily life ofthe people The persistence of inflation can diminish theinvestment enthusiasm on the normal financial productssince the investors are not really earning money They preferacquisition of land and other assets which yield quick capitalgains When inflation continues over a period of time italso erodes the motivation for saving due to the fact thatthe money is worth more presently than in the future Forexample if the return of the bank account was 4 and theinflation was 5 then the real return on investment wouldbe minus 1 In addition when commodity price is raisedthe consumers cannot buy as much as they could previouslyand hence they have opted for major cuts in their dailybudget Nowadays the problem of inflation is quite commonfor the people all over the world Therefore we think thatit is theoretically and practically important to consider theinvestment-consumption problem under inflation

However none of the above papers allow for stochasticinflationHerewe introduce some existing literature allowingfor inflation For the optimal portfolio selection problemunder inflation Brennan and Xia [35] Munk et al [36] andChiarella et al [37] aimed to maximize the expected powerutility from terminal real wealth and obtained the closed-form investment strategies for the investors Menoncin [38]studied an optimal portfolio selection problem for a HARA

utility investor under stochastic inflation and wage incomeHe obtained a quasiexplicit solution for this problemMamunand Visaltanachoti [39] analyzed numerically how the antic-ipated rate of inflation affected the investment strategy ofUS investors under the assumption that the assets availableincluded treasury inflation protected securities equity realestate treasury bonds and corporate bonds Their studyindicated that when the anticipated rate of inflation is higherthe investor should allocate more wealth to the treasuryinflation protected securities For the defined contribu-tion management problem under inflation Battocchio andMenoncin [40] considered an optimal pension managementunder stochastic interest rate wage income and inflationThey wanted to maximize the expected exponential utilityfrom terminal wealth and found a closed-form solutionfor this problem Zhang and Ewald [41] assumed that thefinancial market consists of a money account a stock and aninflation linked bondThey wanted tomaximize the expectedpower utility from the terminal wealth and obtained theoptimal investment strategy using the martingale methodHan and Hung [42] assumed that the retired individualreceived a guarantee as a downside protection The closed-form solution is obtained under the power utility functionFor more information refer to Battocchio and Menoncin[43] Zhang et al [44] and de Jong [45] For the optimalconsumption problem under inflation Brennan and Xia [35]investigated a problem for the interim consumption underthe power utility and obtained the explicit expression ofthe consumption Menoncin [46] generalizedMenoncin [38]to the case with intertemporal consumption He aimed tomaximize the expected HARA utility of the intertemporalconsumption plus the terminal wealth under the stochasticincome and inflation and computed a quasiexplicit solutionfor both optimal consumption and investment Chou et al[47] considered an optimal portfolio-consumption problemunder stochastic inflation with nominal and indexed bondsThey studied respectively an optimization problem that aimsto maximize the expected terminal wealth at a fixed terminaltime 119879 and an optimization problem that maximizes theintertemporal consumption utility with infinite time horizonParadiso et al [48] studied the existence and stability ofthe consumption function in the United States of Americasince the 1950s They introduced inflation as an additionalexplanatory variable to analyze the life-cycle consumptionfunction

We can see that literature on optimal investment-con-sumption under inflation is so limitedMoreover the existingliterature has not studied how the commodity price levelaffects the optimal investment-consumption decision of theinvestors in a Markovian regime-switching market as men-tioned above This paper aims to bridge the gap Referring toKorn et al [49] we assume that the instantaneous expectedrate and volatility rate of inflation are also dependent on themarket states

The rest of our paper is organized as followsThe problemformulation and the verification theory are presented inSection 2 The explicit expressions of the investment strategyand consumption are obtained in Section 3The properties ofinvestment strategy are analyzedmathematically in Section 4

Discrete Dynamics in Nature and Society 3

The properties of the optimal consumption proportion aredemonstrated in Section 5 by mathematical and numericalanalysis This paper is concluded in Section 6

2 Problem Formulation and Notations

In this paper there are a bank account and a stock traded con-tinuously within a time horizon [0 119879] whose price processesdepend on the states of an underlying economy Here theevolution of the market states is modulated by a continuous-time Markov chain 120585(119905) 119905 ge 0 taking discrete valuesin a finite space 119878 = 1 2 119871 and having a generator119876 = (119902119894119895)119894119895isin119878

The price process of the bank account satisfiesthe following differential equation

1198891198600 (119905) = 1198600 (119905) 119903 (119905 120585 (119905)) 119889119905 119860 (0) = 1198860 (1)

where 119903(119905 120585(119905)) is the instantaneous interest rate of the bankaccount corresponding to themarket state 120585(119905)The evolutionof the price process of the stock is governed by the followingMarkovian regime-switching geometric Brownian motion

1198891198601 (119905) = 1198601 (119905) [120583 (119905 120585 (119905)) 119889119905 + 120590 (119905 120585 (119905)) 119889119882 (119905)]

1198601 (0) = 1198861

(2)

where119882(119905) is a standard one-dimensional Brownian motionand 120583(119905 120585(119905)) and 120590(119905 120585(119905)) are respectively the appreciationrate and volatility rate of the stock corresponding to themarket state 120585(119905)

Let 119868(119905) denote the nominal price level per unit ofconsumption goods at time 119905 Then the evolution of 119868(119905) isassumed to follow the stochastic differential equation

119889119868 (119905) = 119868 (119905) [120583119868 (119905 120585 (119905)) 119889119905 + 120590119868 (119905 120585 (119905)) 119889119882119868 (119905)]

119868 (0) = 1198862 gt 0

(3)

where119882119868(119905) is a standard one-dimensional Brownianmotionand 120583119868(119905 120585(119905)) and 120590119868(119905 120585(119905)) are the expected inflation rateand volatility rate at time 119905 respectively Generally weassume that119882(119905) and119882119868(119905) are correlated with a correlationcoefficient 120588(119905) isin [minus1 1] Referring to Koo [50] (3) can beexpressed as

119889119868 (119905) = 119868 (119905) [120583119868 (119905 120585 (119905)) 119889119905 + 120590119868 (119905 120585 (119905)) 120588 (119905) 119889119882 (119905)

+ 120590119868 (119905 120585 (119905))radic1 minus 120588

2(119905)1198891198820 (119905)]

(4)

where1198820(119905) is a standard one-dimensional Brownianmotionindependent of 119882(119905) Furthermore we assume that 120585(119905) and(1198820(119905)119882(119905)) are independent of each other To describe

uncertainty we employ a complete filtered probabilityspace (ΩF 119875 F119905119905ge0) where F119905 is defined as F119905 =

120590(1198820(119904)119882(119904)) 120585(119904) 0 le 119904 le 119905We also assume throughoutthis paper that 119903(119905 119894) 120583(119905 119894) 120590(119905 119894) 120583119868(119905 119894) and 120590119868(119905 119894) aredeterministic and uniformly bounded in 119905 for any given state120585(119905) = 119894

Referring to Menoncin [46] the variable 119868minus1(119905) ldquorepre-

sents the purchasing power of a nominal monetary unitFurthermore if we identify the value of a monetary unit withthe number of goods that can be purchased against it then119868minus1(119905) can also be interpreted as the value of moneyrdquoAn investor joins the market at time 0 with initial wealth

1199090 and plans to invest and consume his wealth dynamicallyover a fixed time horizon 119879 Let 120579(119905) be the proportion ofthe wealth available invested in the stock at time 119905 and let119888(119905) ge 0 be the real consumption that is the ratio betweenthe nominal consumption and the price level 119868(119905) Then thenominal wealth process 119883120579119888(119905) 119905 isin [0 119879] satisfies thefollowing stochastic differential equation

119889119883120579119888

(119905) = 119883120579119888

(119905) (1 minus 120579 (119905)) 119903 (119905 119894) 119889119905

+ 119883120579119888

(119905) 120579 (119905) [120583 (119905 119894) 119889119905 + 120590 (119905 119894) 119889119882 (119905)]

minus 119888 (119905) 119868 (119905) 119889119905

= 119883120579119888

(119905) 119903 (119905 119894) 119889119905

+ 119883120579119888

(119905) 120579 (119905) [ (119905 119894) 119889119905 + 120590 (119905 119894) 119889119882 (119905)]

minus 119888 (119905) 119868 (119905) 119889119905

(5)

where (119905 119894) = 120583(119905 119894) minus 119903(119905 119894)

Denote by 119883120579119888

(119905) = 119883120579119888(119905)119868(119905) the real wealth level at

time 119905 after considering the inflation Then according to Itorsquosformula the dynamics of119883120579119888(119905) is

119889119883120579119888

(119905) = 119889(119883120579119888

(119905)

119868 (119905)) = 119883

120579119888

(119905) [119903 (119905 119894) minus 120583119868 (119905 119894)

+ (120590119868 (119905 119894))2

+ 120579 (119905) ( (119905 119894) minus 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894))] 119889119905

minus 119888 (119905) 119889119905 + 119883120579119888

(119905) 120579 (119905) 120590 (119905 119894) 119889119882 (119905) minus 119883120579119888

(119905)

sdot 120590119868 (119905 119894) (120588 (119905) 119889119882 (119905) + radic1 minus 1205882(119905)1198891198820 (119905))

(6)

with initial value119883(0) = 1199090 = 11990901198862The investorrsquos optimization problem could be described

by the following

maxA(0119879)

E011989401199090 [int119879

0

119890minus120575119905

119880 (120585 (119905) 119888 (119905)) 119889119905 + 119890minus120575119879

119880(120585 (119879) 119883120579119888

(119879))] (7)


where120575 is the discount rate and the set of admissible strategiesA(0 119879) is defined below In our paper the utility function119880(119894 119909) is defined as 119880(119894 119909) = 120577(119894)119909

120574120574 where 120574 lt 1 120574 = 0

and 120577(119894) gt 0

Definition 1 A strategy (120579(119905) 119888(119905) ge 0) 0 le 119905 le 119879 isadmissible if

(i) for any initial wealth 1199090 gt 0 the stochastic differen-tial equation (6) has a unique solution 119883

120579119888

(119905) corre-sponding to (120579(119905) 119888(119905))

(ii) the corresponding solution 119883120579119888

(sdot) satisfiesE(sup

119905isin[0119879]|119883120579119888

(119905)|2120574) lt +infin for all 120574 le 1

(iii) E(int1198790(120579(119905))2119889119905) lt +infin E(int119879

0(119888(119905))120574119889119905) lt +infin for all

120574 le 1

(iv) 119883120579119888(119879) gt 0 as

For convenience denote byA(119905 119879) the set of admissible strat-egies (120579(119904) 119888(119904)) 119905 le 119904 le 119879

We can write the value function in 119905 isin [0 119879) as

119881 (119905 119909 119894) = maxA(119905119879)

E119905119894119909 [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(8)

with terminal condition 119881(119879 119909 119894) = 119880(119894 119909)Then the optimal investment-consumption problem can

be formulated by the dynamic programming equation

minus 120575119881 (119905 119909 119894) + 119881119905 (119905 119909 119894) +

119871

sum

119895=1

119902119894119895119881 (119905 119909 119895) +1

2

sdot 1199092119881119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2+ sup120579(119905)119888(119905)ge0

119880 (119894 119888 (119905))

+ 119881119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092119881119909119909 (119905 119909 119894)

sdot [1205792(119905) 1205902(119905 119894) minus 2120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)] = 0

(9)

where 120578(119905 119894) = 119903(119905 119894)minus120583119868(119905 119894)+(120590119868(119905 119894))2 and 120581(119905 119894) = (119905 119894)minus

120588(119905)120590(119905 119894)120590119868(119905 119894)The optimality condition (9) is not sufficient if a verifica-

tion theorem is not provided so we present the verificationtheorem before we give the explicit solution to this problemLet 11986212([0 119879] times O times 119878) where O sube R denote the set of allcontinuous functions 119891(119905 119909 119894) [0 119879] times O times 119878 rarr R thatare continuously differentiable in 119905 and twice continuouslydifferentiable in 119909 for any 119894 isin 119878

Theorem 2 Let V(119905 119909 119894) isin 11986212([0 119879] times119874times 119878) where119874 sube R

be a solution to the HJB equation (9) with boundary condition119881(119879 119909 119894) = 119880(119894 119909) If for all (119905 119909 119894) isin [0 119879] times 119874 times 119878 and alladmissible controls there exists 120573 gt 1 such that

E119905119894119909 ( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

) lt +infin (10)

then we have

(a) V(119905 119909 119894) ge 119881(119905 119909 119894)

(b) if there exists an admissible strategy (120579lowast(sdot) 119888lowast(sdot)) thatis a maximizer of (9) then V(119905 119909 119894) = 119881(119905 119909 119894) for all119894 isin 119878 119909 isin 119874 and 119905 isin [0 119879] Furthermore (120579lowast(sdot) 119888lowast(sdot))is an optimal strategy

Proof (a) Applying Itorsquos formula to 119890120575(119879minus119905)V(119905 119909 119894) yields

119880(120585 (119879) 119883120579119888

(119879)) = V (119879119883120579119888

(119879) 120585 (119879))

= 119890120575(119879minus119905)V (119905 119909 119894)

+ int

119879

119905

minus120575119890120575(119879minus119904)V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890120575(119879minus119904)V119905 (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))

sdot [119883120579119888

(119904) (120578 (119904 120585 (119904)) + 120579 (119904) 120581 (119904 120585 (119904))) minus 119888 (119904)] 119889119904

+1

2int

119879

119905

119890120575(119879minus119904)

(119883120579119888

(119904))

2

V119909119909 (119904 119883120579119888

(119904) 120585 (119904))

times [(120579 (119904) 120590 (119904 120585 (119904)) minus 120590119868 (119904 120585 (119904)) 120588 (119904))2

+ (120590119868 (119904 120585 (119904)))2(1 minus 120588

2(119904))] 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot

119871

sum

119895=1

119902120585(119904)119895V (119904 119883120579119888

(119904) 119895) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904) minus int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(11)


Denote

A120579119888V (119905 119909 119894) = 119880 (119894 119888 (119905)) minus 120575V (119905 119909 119894) + V119905 (119905 119909 119894)

+

119871

sum

119895=1

119902119894119895V (119905 119909 119895) +1

21199092V119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2

+ V119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092V119909119909 (119905 119909 119894)

sdot [(120579 (119905))2(120590 (119905 119894))

2minus 2120579 (119905) 120588120590 (119905 119894) 120590119868 (119905 119894)]

(12)

Thus we have

int

119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904 + 119890minus120575(119879minus119905)

119880(120585 (119879)

119883120579119888

(119879)) = V (119905 119909 119894)

+ int

119879

119905

119890minus120575(119904minus119905)

119860120579119888V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904)

sdot [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

minus int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(13)

We first assume that O isin R is bounded When V(119905 119909 119894) isin11986212([0 119879]timesOtimes119878) and 120579(119905) and 119888(119905) are admissible according

to Definition 1 we know that

int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(14)

are martingales and E[int119879119905119890minus120575(119904minus119905)

119860120579119888V(119904 119883

120579119888

(119904) 120585(119904))119889119904] lt

+infin Since V(119905 119909 119894) solves HJB equation (9) taking expecta-tion on both sides of the above equality yields

E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))] le V (119905 119909 119894)

(15)

which immediately implies that 119881(119905 119909 119894) le V(119905 119909 119894)

In the general case when O isin R might not be boundedfor a relatively fixed time 119905 isin [0 119879) we define

O119901 = O

cap 119911 isin R |119911| lt 119901 dist (119911 120597O) gt 119901minus1 119901 isin N

119876119901 = [119905 119879 minus 119901minus1) timesO119901

(16)

where 119901 satisfies 119901minus1 lt 119879 and 119879 minus 119901minus1

gt 119905 Let 120591119901 be thefirst exit time of stochastic process (119904 119883

120579119888

(119904))119904ge119905 from 119876119901

and 120603119901 = min120591119901 119879 Then 120603119901 119901 isin N is a sequence ofstopping times Furthermore as 119901 rarr +infin 120603119901 increases to119879 with probability 1 Since now 119874119901 is bounded referring tothe analysis above we can derive

E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))] le V (119905 119909 119894)

(17)

Equation (10) implies uniform integrability of V(119905 119909 119894)There-fore we have

V (119905 119909 119894) ge lim119901rarr+infin

E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))]

= E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(18)

which implies that V(119905 119909 119894) ge 119881(119905 119909 119894)(b) When taking the strategy (120579

lowast(119905) 119888lowast(119905)) 0 le 119905 le

119879 the inequalities become equalities Hence conclusion (b)holds

3 Optimal Investment-Consumption Strategy

In this section we assume that the utility of the investor instate 119894 is given by the power utility function

119880 (119894 119909) = 120577 (119894)119909120574

120574 (19)

where 120577(119894) gt 0 for all 119894 isin 119878 119909 gt 0 120574 lt 1 and 120574 = 0Suppose that a solution toHJB equation (9) is of this form

V (119905 119909 119894) = 120577 (119905 119894)119909120574

120574

V (119879 119909 119894) = 120577 (119894)119909120574

120574

(20)


Then substituting (20) into (9) yields

minus 120575120577 (119905 119894)119909120574

120574+ 120577119905 (119905 119894)

119909120574

120574minus1

2120577 (119905 119894) (1 minus 120574)

sdot 119909120574(120590119868 (119905 119894))

2+ 120577 (119905 119894) 120578 (119905 119894) 119909

120574+119909120574

120574

119871

sum

119895=1

119902119894119895120577 (119905 119895)

+ sup120579(119905)119888(119905)ge0

120577 (119894)119888 (119905)120574

120574minus 120577 (119905 119894) 119909

120574minus1119888 (119905)

+ 120577 (119905 119894) 119909120574120579 (119905) 120581 (119905 119894)

minus1

2120577 (119905 119894) (1 minus 120574) 119909

1205741205792(119905) 1205902(119905 119894)

+ 120577 (119905 119894) (1 minus 120574) 119909120574120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894) = 0

(21)

where 120577119905(119905 119894) is the partial derivative to 119905If 120577(119905 119894) gt 0 and 119909 gt 0 differentiating with respect to 120579(119905)

and 119888(119905) in (21) respectively gives the maximizers as follows

120579lowast(119905 119894) =

120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

(22)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (23)

where 120577(119905 119894) solves the following equation

0 = 120577119905 (119905 119894) + (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

+

119871

sum

119895=1

119902119894119895120577 (119905 119895) + 120577 (119905 119894) (120574120578 (119905 119894) minus 120575

+1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2)

120577 (119879 119894) = 120577 (119894) gt 0

(24)

Next we shall show that 120577(119905 119894) gt 0 and the wealth process119883120579lowast119888lowast

(119905) gt 0 by the following lemmas step by step

Lemma 3 If 120577(119905 119894) solves (24) then

(a) 120577(119905 119894) gt 0 furthermore 120577(119905 119894) is uniformly boundedfrom below that is there exists a constant gt 0 suchthat 120577(119905 119894) ge

(b) 120577(119905 119894) is the only continuous solution of (24) and 120577(119905 119894)has an uniformly upper bound in [0 119879] times 119878

Proof (a) Denote

120601 (119905 119894) = 120574120578 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2

= 120574119903 (119905 119894) minus 120574120583119868 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

2(119905 119894)

1205902(119905 119894)

+1

2120574 (1 + 120574) (120590119868 (119905 119894))

2

minus1205742

1 minus 120574

120588 (119905) (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

+1

2

1205743

1 minus 1205741205882(119905) (120590119868 (119905 119894))

2

(25)

119870 (119905 119904) = exp [int119904

119905

120601 (119906 120585 (119906)) 119889119906] (26)

119872(119905 119904) = sum

119905leVle119904[120577 (V 120585 (V)) minus 120577 (V 120585 (Vminus))]

minus int

119904

119905

119871

sum

119895=1

119902120585(Vminus)119895120577 (V 119895) 119889V(27)

Then in view of (24) we have

119889 [119870 (119905 119904) 120577 (119904 120585 (119904))] = 120577 (119904 120585 (119904)) 119870119904 (119905 119904)

+ 119870 (119905 119904) 119889120577 (119904 120585 (119904)) = 119870 (119905 119904)

sdot [120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) 119889119904 + 119889120577 (119904 120585 (119904))] = 119870 (119905 119904)

sdot [

[

120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) + 120577119904 (119904 120585 (119904))

+

119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895)]

]

119889119904 + 119870 (119905 119904) [120577 (119904 120585 (119904))

minus 120577 (119904 120585 (119904minus))] minus 119870 (119905 119904)

119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895) 119889119904 = minus (1

minus 120574)119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119904 120585 (119904))

)

120574(1minus120574)

119889119904

+ 119870 (119905 119904) 119889119872 (119905 119904)

(28)


The solution of the above equation is of this form

119870 (119905 119879) 120577 (120585 (119879)) = 120577 (119905 119894) minus (1 minus 120574)

sdot int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119905 120585 (119904))

)

120574(1minus120574)

119889119904

+ int

119879

119905

119870 (119905 119904) 119889119872 (119905 119904)

(29)

It is well known that119872(119905 119904) is a martingale then we have

120577 (119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [

[

int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (119904 120585 (119904))

120577 (120585 (119904)))

120574(120574minus1)

119889119904]

]

(30)

To prove 120577(119905 119894) gt 0 we construct a Picard iterativesequence 120577

(119896)

(119905 119894) 119896 = 0 1 2 as follows

120577(0)

(119905 119894) = 120577 (119894)

120577(119896+1)

(119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [int119879

119905

119870 (119905 119904) [120577 (120585 (119904))]1(1minus120574)

sdot (120577(119896)

(119904 120585 (119904)))

120574(120574minus1)

119889119904]

(31)

Noting that 120577(119894) gt 0 and119870(119905 119904) gt 0 we have

120577(119896)

(119905 119894) ge E119905119894 [120577 (120585 (119879))119870 (119905 119879)] gt 0 119896 = 1 2 (32)

Since all the coefficients in our paper are uniformly bounded(32) indicates that 120577

(119896)

(119905 119894) gt gt 0 for 119896 = 1 2 Atthe same time it is well known that 120577(119905 119894) is the limit of thesequence 120577

(119896)

(119905 119894) 119896 = 0 1 2 as 119896 rarr +infinThus 120577(119905 119894) ge gt 0 119905 isin [0 119879]

(b) For 119894 = 1 2 119871 denote

119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

= minus (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

minus

119871

sum

119895=1

119902119894119895120577 (119905 119895)

minus 120577 (119905 119894) 120601 (119905 119894)

(33)

We have

120577119905 (119905 119894) = 119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

119894 = 1 2 119871

(34)

which is a system of the first-order ordinary differentialequations Since 120601(119905 119894) is uniformly bounded for 119894 isin 119878 119891119894satisfies that

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus (1 minus 120574) (120577 (119894))1(1minus120574)

sdot [(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)

] minus

119871

sum

119895=1

119902119894119895

sdot [120577 (119905 119895) minus 120577lowast

(119905 119895)] minus 120601 (119905 119894) [120577 (119905 119894) minus 120577lowast

(119905 119894)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198601

10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

+ 1198602

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(35)

for suitable constants 1198601 and 1198602 Moreover1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120597 (120577 (119905 119894))120574(120574minus1)

120597120577 (119905 119894)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

120577 (119905 119894)

)

1(1minus120574)

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574)

(36)

Then10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574) 100381610038161003816100381610038161003816120577 (119905 119894) minus 120577

lowast

(119905 119894)

100381610038161003816100381610038161003816

(37)

Therefore100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198603

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(38)

which leads to119871

sum

119894=1

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198604

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(39)

Now it obvious that 119891119894rsquos satisfy Lipschitz condition Conse-quently (24) has a unique continuous solution denoted by


120577(119905 119894) in [0 119879] A continuous function 120577(119905 119894) defined in aclose interval [0 119879] must have an upper bound 119872119894 If wedefine119872 = max11987211198722 119872119871 we know that 120577(119905 119894) has auniformly upper bound119872

The next step is to prove that the stochastic differentialequation (6) under 120579lowast(119905 119894) in (22) and 119888

lowast(119905 119909 119894) in (23) has a

unique and nonnegative solution 119883120579lowast119888lowast

(119905) The main resultsare presented in the following lemma

Lemma 4 For any initial wealth 1199090 gt 0 the stochastic differ-ential equation (6) under 120579lowast(119905 119894) and 119888

lowast(119905 119909 119894) has a unique

nonnegative solution119883120579lowast119888lowast

(119905) Furthermore

E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572

) lt +infin forall120572 isin R (40)

Proof Substituting (22) and (23) into (6) yields

119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)

minus 120590119868 (119905 119894)radic1 minus 120588

2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)

Since the coefficients of (41) are uniformly bounded it isobvious that there exists a unique solution to (41) such as

119883120579lowast119888lowast

(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)

Therefore119883120579lowast119888lowast

(119905) gt 0 for all 119905 isin [0 119879]Next we shall prove that E(sup

119905isin[0119879]|119883120579lowast119888lowast

(119905)|120572) lt +infin

for120572 isin R To this end define119885(119905) = expint1199050ℎ(119904 120585(119904))

1015840119889(119904)

where (119905) is an 119899-dimensional standard Brownian motionand ℎ(119905 119894) is an 119899 times 1 column vector whose components areuniformly bounded in [0 119879] for any 119894 isin 119878 For 119885(119905) we have

119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)

The stochastic differential equation of (119905) is of this form

119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)

The uniformly bounded ℎ(119905 119894) results in (119905)ℎ(119905 120585(119905))2le

1198672|(119905)|2 then according to Krylov [51 p 85] we have

E(sup119905isin[0119879]

|(119905)|) lt +infin It follows 119885(119905) le 1198671(119905) that

E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)

where ℎ(119905 119894) is any 119899 times 1 column vector whose componentsare uniformly bounded in [0 119879] for any 119894 isin 119878 In view of (43)for any given 120572 isin R we have

(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)

It follows (46) that E(sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))120572) lt +infin

Lemma 5 120579lowast(119905 119894) in (22) and 119888lowast(119905 119909 119894) in (23) are admissibleand then are optimal strategies for the power utility model

Proof By Lemma 4 we know that conditions (i) and (ii)in Definition 1 hold and 119883


(119905) gt 0 for all 119905 isin [0 119879]which guarantees (iv) in Definition 1 holds Since 120579

lowast(119905 119894)

and 120577(119894)120577(119905 119894) are time deterministic and uniformly bounded


functions for any given market state 119894 E(int1198790|120579lowast(119905 120585(119905))|

2) lt

+infin holds naturally By Lemma 4 we have

E(int119879

0

1003816100381610038161003816100381610038161003816119888lowast(119905 119883120579lowast119888lowast

(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)

Nowwe have verified that 119888lowast(119905 119909 119894) and 120579lowast(119905 119894) are admissibleand hence optimal for the power utility model

The next work is to prove that the candidate value func-tion V(119905 119909 119894) in (20) satisfies all the conditions in Theorem 2First of all it is obvious that V(119905 119909 119894) isin 119862

12 is a solution of (9)Moreover for any (119905 119909 119894) isin [0 119879]times[0 +infin)times119878 and admissiblecontrol (120579(119905) 119888(119905)) there exists a 120573 = 2 gt 1 such that

E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)

The detailed analysis above gives the main results of thispaper presented in the following theorem

Theorem 6 The optimal investment proportion and the opti-mal consumption for the power utility model are respectively

120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)

where 120577(119905 119894) solves (24) and the value function is

119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)

4 Analysis of the OptimalInvestment Proportion

First of all if there is no inflation by (50) the optimalinvestment proportion is

120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)

which clearly shows that when the market state has higherexpected return per unit risk or the investor has lower riskaversion the investor would like to invest higher proportionof his wealth on the stock which is a classical conclusion inthe existing literature if the investor does not need to face theinflation

However when there is inflation this conclusionmay nothold First we can prove that the higher expected return perunit risk does not result in a higher investment proportion By(50) the investment proportion is decreased by an amountof (120574(1 minus 120574))120588(119905)120590119868(119905 119894)120590(119905 119894) compared with the portfolioselection without inflation This amount is increased withrespect to the volatility rate of the inflation and the correlationcoefficient 120588(119905)When 120588(119905) equiv 1 that is the stock price and theinflation index are modulated by the same Brownian motionthe investment proportion is decreased by the largest amountThat means if the stock and the commodity price level havethe same volatility trend the inflation volatility will diminishthe investment proportion the most Therefore when theincreasing range of the expected return per unit is lower thanthat of the inflation volatility the investorwould not buymorestocks and could even short sell the stock because he worriesthe high volatility of the inflation would seriously damage hisinvestment return

Next we shall present the effects of the risk aversion onthe investment proportion

Lemma 7 When (119905 119894) gt 120588(119905)120590119868(119905 119894)120590(119905 119894) the optimalinvestment proportion is increased with respect to the risk tole-rance 1(1 minus 120574) when (119905 119894) lt 120588(119905)120590119868(119905 119894)120590(119905 119894) the optimalinvestment proportion is decreased with respect to the risk tole-rance when (119905 119894) = 120588(119905)120590119868(119905 119894)120590(119905 119894) the optimal investmentproportion is a constant 120588(119905)120590119868(119905 119894)120590(119905 119894)

Proof We rewrite (50) as

120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)

it is clear that the conclusions of Lemma 7 hold

Remark 8 When 120590119868(119905 119894) = 0 (119905 119894) gt 0 holds naturallyTherefore the investment proportion increases as the risktolerance increases which reduces to a classical conclusionin the model without inflation

Remark 9 When there is no inflation the investment pro-portion 120579

lowast(119905 119894) is a positive number if (119905 119894) gt 0 However

this conclusion does not hold in the case with inflation evenif (119905 119894) gt 120588(119905)120590119868(119905 119894)120590(119905 119894) When 0 lt 120574 lt 1 that is the risktolerance is greater than 1 (119905 119894) gt 120588(119905)120590119868(119905 119894)120590(119905 119894) leadsto (119905 119894) gt 120574120588(119905)120590119868(119905 119894)120590(119905 119894) By (50) 120579lowast(119905 119894) gt 0 When120574 lt 0 that is the risk tolerance is less than 1 (119905 119894) gt

120588(119905)120590119868(119905 119894)120590(119905 119894) cannot always guarantee a positive invest-ment proportion if 120588(119905) lt 0


Remark 10 If 0 lt (119905 119894) lt 120588(119905)120590(119905 119894)120590119868(119905 119894) the investmentproportion will decrease according to the risk toleranceMoreover if the risk tolerance is high enough the investorwill tend to short sell herhis stock and the short sellingproportion is increasing according to the risk tolerance

5 Analysis of the OptimalConsumption Proportion

Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)

the consumption proportion Next we shall analyze in detailthe effects of the risk aversion the correlation coefficient theexpected rate and volatility rate of the inflation index andthe utility coefficients on the consumption proportion Weassume that the risk-free return rate is a constant 119903 = 002

independent of time and market states and the appreciationrate 120583 and volatility rate 120590 of the stock depend on the marketstates only Let there be two market states and 120583(1) = 02120583(2) = 015120590(1) = 025120590(2) = 04 the discount rate 120575 = 08the time horizon 119879 = 5 and the generator

119902 = (

minus25 25

4 minus4

) (56)

51 Effects of the Risk Aversion In this subsection assumethat 120588 = 04 120583119868 = (005 005) 120590119868 = (015 015) and 120577 =

(1 1)We increase 120574 fromminus04 to 095with step size 01Thenthe effects of risk aversion on the consumption proportion areobtained as demonstrated in Figure 1

Figure 1 shows the following(i) As 119905 rarr 119879 consumption proportion approaches 1

which is consistent with the conclusion in Cheungand Yang [30]

(ii) As 120574 is increased from minus04 to some extent the con-sumption proportion is raised accordingly Howeverthere come changes when 120574 continues to increaseThe consumption proportions almost decrease to 0

as 120574 increases to 095 Actually since now 120581(119905 sdot) =

(119905 sdot)minus120588(119905)120590119868(119905 sdot)120590(119905 sdot) = (0165 0106) according toLemma 7 an investorwith higher risk tolerance 1(1minus120574) will invest more of herhis wealth in the stock andconsequently consume less of herhis wealth That iswhen 120574 is close to 1 the consumption proportion isalmost zero in most cases

(iii) When 120574 is relatively small the investor consumes alarger proportion of our wealth if it is closer to theend of the horizon When 120574 is close to 1 that is therisk tolerance is relatively high the consumption ratedecreases with time

52 Effects of the Correlation Coefficients

Lemma 11 When 120588(119905) is a constant 120588 in [0 119879] and 120574 lt 0the consumption proportion 119888119901(119905 119894) is increasing according to

the correlation coefficient 120588 if (119904 119895) gt 120574120588120590(119904 119895)120590119868(119904 119895) for all119895 isin 119878 and 119904 isin [119905 119879]

Proof By (25) we have

120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)

If (119904 119895) gt 120574120588120590(119904 119895)120590119868(119904 119895) for all 119895 isin 119878 and 119904 isin [119905 119879]we know that 120601(119904 119895) decreases with respect to 120588 in 119904 isin

[119905 119879] which has a consequence that 119870(119905 119904) in (26) decreasesaccordingly if 120588 increases for all 119904 isin [119905 119879] When 120574 lt 00 lt 120574(120574 minus 1) lt 1 Therefore (120577(119905 119894))120574(120574minus1) is an increasingfunction of 120577(119905 119894) This together with the Picard sequence(31) indicates that 120577

(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)

decreases Since 120577(119905 119894) is the limit of the Picard sequence weimmediately obtain that 120577(119905 119894) decreases as 120588 increases Nowit follows (55) that the conclusion in Lemma 11 holds

Let 120574 = minus08 and increase the correlation coefficient120588 from minus1 to 1 with step size 05 while keeping otherparameters unchangeable Since theminimal value of (119905 sdot)minus120574120588120590(119905 sdot)120590119868(119905 sdot) is (0150 0082) we can see clearly in Fig-ure 2 that the consumption proportion at state 1 increasesaccording to the increasing correlation coefficients Howeverif we assume that 119903 = 014 120583 = (016 015) and 120574 = minus4then (119905 119895) lt 120588120574120590(119905 119895)120590119868(119905 119895) given that 120588 = minus1 and minus05Therefore we obtain Figure 3 which shows that the higherthe 120588 is the lower the consumption proportion cp is

53 Effects of the Expected Inflation Rate

Lemma 12 The consumption proportion 119888119901(119905 119894) decreases ifthe expected inflation rate 120583119868(119904 119895) increases for all 119895 isin 119878 and119904 isin [119905 119879] when 120574 lt 0

Proof The proof of Lemma 12 is similar to that of Lemma 11so it is omitted here

Let 120574 = minus05 and 120588 = 04 and increase respectively120583119868(1) and 120583119868(2) from 005 to 015 with step size 002 whilekeeping other parameters unchangeable we obtain Figure 4But if we change 120574 to be 05 while keeping other parametersunchangeable we obtain Figures 5 and 6

Figures 4ndash6 show that if the risk aversion 1 minus 120574 is greaterthan 1 then the higher the expected inflation rate is the lowerthe consumption proportion is otherwise if the risk aversion1 minus 120574 is less than 1 the best decision for the investor is toconsume a high proportion of herhis wealth at the currenttime when the expected inflation rate in the future is high nomatter what the market state is

54 Effects of the Inflation Volatility Let 120574 = 08 120588 = 04120590119868(2) = 015 120583119868 = (005 005) and 120577 = (1 1) and increase120590119868(1) from 015 to 025 with step size 002 The effects ofthe volatility of inflation on the consumption proportion aredemonstrated in Figure 7 One can see that the higher thevolatility rate is the more the investor consumes A similar


0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05

120574 = minus04120574 = minus03120574 = minus02

120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)

Figure 1 Consumption proportion with respect to 120574

phenomenon happens when we increase 120590119868(2) from 015 to025with step size 002while keeping 120590119868(1) = 015 To explainthis we notice that 120574 gt 0 in Figure 7 which has a consequencethat the higher the volatility rate 120590119868(119905 119894) is the lower theinvestment proportion is by (50) Therefore more wealth isused for personal consumption

55 Effects of the Utility Coefficient In this subsection let120583119868 = (005 005) 120590119868 = (015 015) 120574 = 06 and 120588 = 04

and increase 120577(1) and 120577(2) from 02 to 1 with step size 02respectively Then we have Figures 8 and 9

Figures 8 and 9 present an interesting phenomenonthat the increasing 120577(119894) results in an increasing cp(119905 119894) and

a decreasing cp(119905 119895) 119895 = 119894 Actually we can regard 120577(119894) as theattention degree of the consumption at state 119894 Hence a larger120577(119894) indicates that the investor caresmore about the consump-tion utility at state 119894 and hence consumes a larger amount ofherhis wealth In contrast the consumption proportion atother market states will be diminished correspondingly

6 Conclusion

This paper considers a continuous-time investment-con-sumption problem under inflation where the stock pricethe commodity price level and the coefficient of the powerutility all dependon themarket statesThe admissible strategy


0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)


and verification theory corresponding to this problem areprovidedWe obtain the closed-form investment strategy andquasiexplicit consumption strategy by dynamic program-ming and stochastic control technique By mathematical andnumerical analysis we obtain some interesting properties ofthe optimal strategies

For the optimal strategy (a) we say that a market has abetter state if at this state the stock has a higher expectedexcess return per unit risk (the Sharpe ratio) Under theinfluence of the inflation the investorwould not always investmore wealth in the stock even if the market state is better Ifthe increasing range of the inflation volatility is higher than


0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)

Figure 4 Consumption proportion with respect to 120583119868(1)

0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


that of the Sharpe ratio of the stock the investor would notinvest more of his wealth on this stock since the high inflationerodes greatly the investment enthusiasm of the investor evenif he is at a better market state (b) if there is no inflationthen when the Sharpe ratio is greater than 0 an investorwith higher risk aversion would invest less of his wealth inthe stock But if there exists inflation the positive Sharpe ratiocannot guarantee this conclusion holding Only if the Sharpe

ratio is greater than the product of inflation volatility rate andcorrelation coefficient 120588(119905) does the traditional conclusionhold (c) the expected inflation rate and the utility coefficienthave no impact on the optimal investment strategy

For the optimal consumption strategy (a) when the riskaversion is close to zero the consumption proportion isalmost zero When the risk aversion is relatively small (big)the consumption proportion decreases (increases) with time


0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)


(b) when correlation coefficient 120588(119905) is a constant in [0 119879] andthe risk aversion is greater than 1 the consumption propor-tion is increasing according to the correlation coefficient ifthe Sharpe ratio of the stock is high enough (c) when the riskaversion is greater than 1 the consumption proportiondecreases according to an increasing expected inflation rate(d) the higher the volatility rate of the inflation is the higher

the consumption proportion is (e) a larger coefficient ofutility 120577(119894) results in a higher consumption proportion at state119894 but a lower consumption proportion at state 119895 = 119894

Although our model is rather general it still deservesfurther extension as future research For example in mostexisting literature including our paper only the coefficient ofthe utility depends on the market states but the risk aversion


0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)


is independent of themarket state So the future researchmayfocus on the optimal investment-consumption problem witha state-dependent risk aversion

Competing Interests

The author declares that they have no competing interests

Acknowledgments

This research is supported by grants of the National NaturalScience Foundation of China (no 11301562) the Programfor Innovation Research in Central University of Financeand Economics and Beijing Social Science Foundation (no15JGB049)

References

[1] P A Samuelson ldquoLifetime portfolio selection by dynamic sto-chastic programmingrdquo The Review of Economics and Statisticsvol 51 no 3 pp 239ndash246 1969

[2] N H Hakansson ldquoOptimal investment and consumptionstrategies under risk for a class of utility functionsrdquo Economet-rica vol 38 no 5 pp 587ndash607 1970

[3] E F Fama ldquoMultiperiod consumption-investment decisionsrdquoTheAmerican Economic Review vol 60 no 1 pp 163ndash174 1970

[4] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[5] T Zariphopoulou ldquoInvestment-consumption models withtransaction fees and Markov-chain parametersrdquo SIAM Journalon Control and Optimization vol 30 no 3 pp 613ndash636 1992


[6] M Akian J L Menaldi and A Sulem ldquoOn an investment-consumption model with transaction costsrdquo SIAM Journal onControl and Optimization vol 34 no 1 pp 329ndash364 1996

[7] H Liu ldquoOptimal consumption and investment with transactioncosts and multiple risky assetsrdquo The Journal of Finance vol 59no 1 pp 289ndash338 2004

[8] X-Y Zhao and Z-K Nie ldquoMulti-asset investment-consump-tion model with transaction costsrdquo Journal of MathematicalAnalysis and Applications vol 309 no 1 pp 198ndash210 2005

[9] M Dai L Jiang P Li and F Yi ldquoFinite horizon optimalinvestment and consumption with transaction costsrdquo SIAMJournal on Control and Optimization vol 48 no 2 pp 1134ndash1154 2009

[10] M Taksar and S Sethi ldquoInfinite-horizon investment consum-ption model with a nonterminal bankruptcyrdquo Journal of Opti-mization Theory and Applications vol 74 no 2 pp 333ndash3461992

[11] T Zariphopoulou ldquoConsumption-investment models withconstraintsrdquo SIAM Journal on Control andOptimization vol 32no 1 pp 59ndash85 1994

[12] C Munk and C Soslashrensen ldquoOptimal consumption and invest-ment strategies with stochastic interest ratesrdquo Journal of Bankingamp Finance vol 28 no 8 pp 1987ndash2013 2004

[13] X KWang and Y Q Yi ldquoAn optimal investment and consump-tion model with stochastic returnsrdquo Applied Stochastic Modelsin Business and Industry vol 25 no 1 pp 45ndash55 2009

[14] C Munk ldquoOptimal consumptioninvestment policies withundiversifiable income risk and liquidity constraintsrdquo Journalof Economic Dynamics and Control vol 24 no 9 pp 1315ndash13432000

[15] P H Dybvig and H Liu ldquoLifetime consumption and invest-ment retirement and constrained borrowingrdquo Journal of Eco-nomic Theory vol 145 no 3 pp 885ndash907 2010

[16] S R Pliska and J Ye ldquoOptimal life insurance purchase andconsumptioninvestment under uncertain lifetimerdquo Journal ofBanking amp Finance vol 31 no 5 pp 1307ndash1319 2007

[17] M Kwak Y H Shin and U J Choi ldquoOptimal investmentand consumption decision of a family with life insurancerdquoInsurance Mathematics amp Economics vol 48 no 2 pp 176ndash1882011

[18] M R Hardy ldquoA regime-switching model of long-term stockreturnsrdquoNorth American Actuarial Journal vol 5 no 2 pp 41ndash53 2001

[19] X Y Zhou and G Yin ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching a continuous-time modelrdquoSIAM Journal on Control and Optimization vol 42 no 4 pp1466ndash1482 2003

[20] U Cakmak and S Ozekici ldquoPortfolio optimization in stochasticmarketsrdquoMathematicalMethods of Operations Research vol 63no 1 pp 151ndash168 2006

[21] U Celikyurt and S Ozekici ldquoMultiperiod portfolio optimiza-tion models in stochastic markets using the mean-varianceapproachrdquo European Journal of Operational Research vol 179no 1 pp 186ndash202 2007

[22] S-Z Wei and Z-X Ye ldquoMulti-period optimization portfoliowith bankruptcy control in stochastic marketrdquo Applied Math-ematics and Computation vol 186 no 1 pp 414ndash425 2007

[23] H L Wu and Z F Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science and Complexity vol 24 no1 pp 140ndash155 2011

[24] H L Wu and Z F Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsurance Mathematics and Economics vol 50 no 3 pp 371ndash384 2012

[25] H Wu and Y Zeng ldquoMulti-period mean-variance portfolioselection in a regime-switchingmarket with a bankruptcy staterdquoOptimal Control Applications ampMethods vol 34 no 4 pp 415ndash432 2013

[26] P Chen H L Yang and G Yin ldquoMarkowitzrsquos mean-vari-ance asset-liability management with regime switching a con-tinuous-time modelrdquo Insurance Mathematics and Economicsvol 43 no 3 pp 456ndash465 2008

[27] K C Cheung and H L Yang ldquoAsset allocation with regime-switching discrete-time caserdquo ASTIN Bulletin vol 34 pp 247ndash257 2004

[28] E Canakoglu and S Ozekici ldquoPortfolio selection in stochasticmarkets with HARA utility functionsrdquo European Journal ofOperational Research vol 201 no 2 pp 520ndash536 2010

[29] E Canakoglu and S Ozekici ldquoHARA frontiers of optimal port-folios in stochastic marketsrdquo European Journal of OperationalResearch vol 221 no 1 pp 129ndash137 2012

[30] K C Cheung and H Yang ldquoOptimal investment-consumptionstrategy in a discrete-time model with regime switchingrdquoDiscrete and Continuous Dynamical Systems Series B vol 8 no2 pp 315ndash332 2007

[31] Z Li K S Tan and H Yang ldquoMultiperiod optimal investment-consumption strategies with mortality risk and environmentuncertaintyrdquo North American Actuarial Journal vol 12 no 1pp 47ndash64 2008

[32] Y Zeng H Wu and Y Lai ldquoOptimal investment and con-sumption strategies with state-dependent utility functions anduncertain time-horizonrdquo Economic Modelling vol 33 pp 462ndash470 2013

[33] P Gassiat F Gozzi and H Pham ldquoInvestmentconsumptionproblems in illiquid markets with regime-switchingrdquo SIAMJournal on Control and Optimization vol 52 no 3 pp 1761ndash1786 2014

[34] T A Pirvu andH Y Zhang ldquoInvestment and consumptionwithregime-switching discount ratesrdquo Working Paper httparxivorgabs13031248

[35] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo The Journal of Finance vol 57 no 3 pp 1201ndash12382002

[36] C Munk C Soslashrensen and T Nygaard Vinther ldquoDynamic assetallocation under mean-reverting returns stochastic interestrates and inflation uncertainty are popular recommendationsconsistent with rational behaviorrdquo International Review ofEconomics and Finance vol 13 no 2 pp 141ndash166 2004

[37] C Chiarella C Y Hsiao and W Semmler IntertemporalInvestment Strategies under Inflation Risk vol 192 of ResearchPaper Series Quantitative Finance Research Centre Universityof Technology Sydney Australia 2007

[38] F Menoncin ldquoOptimal real investment with stochastic incomea quasi-explicit solution for HARA investorsrdquo Working PaperUniversite Catholique de Louvain Louvain-la-Neuve Belgium2003

[39] A Mamun and N Visaltanachoti ldquoInflation expectation andasset allocation in the presence of an indexed bondrdquo WorkingPaper 2006

[40] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004


[41] A H Zhang and C-O Ewald ldquoOptimal investment for apension fund under inflation riskrdquo Mathematical Methods ofOperations Research vol 71 no 2 pp 353ndash369 2010

[42] N-W Han and M-W Hung ldquoOptimal asset allocation for DCpension plans under inflationrdquo Insurance Mathematics andEconomics vol 51 no 1 pp 172ndash181 2012

[43] P Battocchio and F Menoncin ldquoOptimal portfolio strategieswith stochastic wage income and inflation the case of a definedcontribution pension planrdquo Working Paper 2002

[44] A Zhang R Korn and C-O Ewald ldquoOptimal managementand inflation protection for defined contribution pensionplansrdquo Blatter der DGVFM vol 28 no 2 pp 239ndash258 2007

[45] F de Jong ldquoPension fund investments and the valuation of lia-bilities under conditional indexationrdquo Insurance Mathematicsand Economics vol 42 no 1 pp 1ndash13 2008

[46] F Menoncin ldquoOptimal real consumption and asset allocationfor aHARA investor with labour incomerdquoWorking Paper 2003httpideasrepecorgpctllouvir2003015html

[47] Y-Y Chou N-W Han and M-W Hung ldquoOptimal portfolio-consumption choice under stochastic inflation with nominaland indexed bondsrdquo Applied Stochastic Models in Business andIndustry vol 27 no 6 pp 691ndash706 2011

[48] A Paradiso P Casadio and B B Rao ldquoUS inflation and con-sumption a long-term perspective with a level shiftrdquo EconomicModelling vol 29 no 5 pp 1837ndash1849 2012

[49] R Korn T K Siu and A H Zhang ldquoAsset allocation for a DCpension fund under regime switching environmentrdquo EuropeanActuarial Journal vol 1 supplement 2 pp S361ndashS377 2011

[50] H K Koo ldquoConsumption and portfolio selection with laborincome a continuous time approachrdquo Mathematical Financevol 8 no 1 pp 49ndash65 1998

[51] N V Krylov Controlled Diffusion Processes vol 14 of StochasticModelling and Applied Probability Springer Berlin Germany1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


that is the utility is being of the form 119880(119894 119909) = 119870(119894) +

119862(119894)(119909 minus 120573)120574120574 As for the optimal investment-consumption

problem Cheung and Yang [30] considered a multiperiodmodel where the return of the risky assets depends onthe economic environments with an absorbing state whichrepresents the bankruptcy state Li et al [31] and Zeng etal [32] investigated a discrete-time investment-consumptionproblem with regime switching and uncertain time horizonGassiat et al [33] studied this problem in an illiquid financialmarket where the asset trading has time restriction Pirvuand Zhang [34] considered a continuous-time investment-consumption problem with regime-switching discount rateand asset returns In this paper enlightened by the existingliterature we assume that the utility function is of this form119880(119909) = 120577(119894)119909

120574120574 where 120577(119894) is dependent on the current

market state 119894 Under this assumption the utility functionis changed according to the market states over time In thissense our paper has adopted the similar assumption aboutthe movement of the financial market and the utility param-eters as some existing literature For example Canakogluand Ozekici [28 29] also assumed that the parametersof the power utility are state-dependent However thereare some differences between our paper and the existingliterature Firstly the optimization problem considered isdifferent Canakoglu and Ozekici [28 29] investigated anoptimal portfolio selection problem while our paper studiesan investment-consumption strategy Secondly the above-mentioned research papers do not analyze the effects of theinflation In contrast in addition to the financial risk theinflation risk is also considered in our paper

In recent years the phenomenon of inflation has beencausing grave concern in developed and developing coun-tries When the continuous increase in price level exceedsa tolerable limit the inflation can cause many distortionsin investment behavior and effect greatly on daily life ofthe people The persistence of inflation can diminish theinvestment enthusiasm on the normal financial productssince the investors are not really earning money They preferacquisition of land and other assets which yield quick capitalgains When inflation continues over a period of time italso erodes the motivation for saving due to the fact thatthe money is worth more presently than in the future Forexample if the return of the bank account was 4 and theinflation was 5 then the real return on investment wouldbe minus 1 In addition when commodity price is raisedthe consumers cannot buy as much as they could previouslyand hence they have opted for major cuts in their dailybudget Nowadays the problem of inflation is quite commonfor the people all over the world Therefore we think thatit is theoretically and practically important to consider theinvestment-consumption problem under inflation

However none of the above papers allow for stochasticinflationHerewe introduce some existing literature allowingfor inflation For the optimal portfolio selection problemunder inflation Brennan and Xia [35] Munk et al [36] andChiarella et al [37] aimed to maximize the expected powerutility from terminal real wealth and obtained the closed-form investment strategies for the investors Menoncin [38]studied an optimal portfolio selection problem for a HARA

utility investor under stochastic inflation and wage incomeHe obtained a quasiexplicit solution for this problemMamunand Visaltanachoti [39] analyzed numerically how the antic-ipated rate of inflation affected the investment strategy ofUS investors under the assumption that the assets availableincluded treasury inflation protected securities equity realestate treasury bonds and corporate bonds Their studyindicated that when the anticipated rate of inflation is higherthe investor should allocate more wealth to the treasuryinflation protected securities For the defined contribu-tion management problem under inflation Battocchio andMenoncin [40] considered an optimal pension managementunder stochastic interest rate wage income and inflationThey wanted to maximize the expected exponential utilityfrom terminal wealth and found a closed-form solutionfor this problem Zhang and Ewald [41] assumed that thefinancial market consists of a money account a stock and aninflation linked bondThey wanted tomaximize the expectedpower utility from the terminal wealth and obtained theoptimal investment strategy using the martingale methodHan and Hung [42] assumed that the retired individualreceived a guarantee as a downside protection The closed-form solution is obtained under the power utility functionFor more information refer to Battocchio and Menoncin[43] Zhang et al [44] and de Jong [45] For the optimalconsumption problem under inflation Brennan and Xia [35]investigated a problem for the interim consumption underthe power utility and obtained the explicit expression ofthe consumption Menoncin [46] generalizedMenoncin [38]to the case with intertemporal consumption He aimed tomaximize the expected HARA utility of the intertemporalconsumption plus the terminal wealth under the stochasticincome and inflation and computed a quasiexplicit solutionfor both optimal consumption and investment Chou et al[47] considered an optimal portfolio-consumption problemunder stochastic inflation with nominal and indexed bondsThey studied respectively an optimization problem that aimsto maximize the expected terminal wealth at a fixed terminaltime 119879 and an optimization problem that maximizes theintertemporal consumption utility with infinite time horizonParadiso et al [48] studied the existence and stability ofthe consumption function in the United States of Americasince the 1950s They introduced inflation as an additionalexplanatory variable to analyze the life-cycle consumptionfunction

We can see that literature on optimal investment-con-sumption under inflation is so limitedMoreover the existingliterature has not studied how the commodity price levelaffects the optimal investment-consumption decision of theinvestors in a Markovian regime-switching market as men-tioned above This paper aims to bridge the gap Referring toKorn et al [49] we assume that the instantaneous expectedrate and volatility rate of inflation are also dependent on themarket states

The rest of our paper is organized as followsThe problemformulation and the verification theory are presented inSection 2 The explicit expressions of the investment strategyand consumption are obtained in Section 3The properties ofinvestment strategy are analyzedmathematically in Section 4






1198891198600 (119905) = 1198600 (119905) 119903 (119905 120585 (119905)) 119889119905 119860 (0) = 1198860 (1)


1198891198601 (119905) = 1198601 (119905) [120583 (119905 120585 (119905)) 119889119905 + 120590 (119905 120585 (119905)) 119889119882 (119905)]

1198601 (0) = 1198861

(2)



119889119868 (119905) = 119868 (119905) [120583119868 (119905 120585 (119905)) 119889119905 + 120590119868 (119905 120585 (119905)) 119889119882119868 (119905)]

119868 (0) = 1198862 gt 0

(3)


119889119868 (119905) = 119868 (119905) [120583119868 (119905 120585 (119905)) 119889119905 + 120590119868 (119905 120585 (119905)) 120588 (119905) 119889119882 (119905)

+ 120590119868 (119905 120585 (119905))radic1 minus 120588

2(119905)1198891198820 (119905)]

(4)







119889119883120579119888

(119905) = 119883120579119888

(119905) (1 minus 120579 (119905)) 119903 (119905 119894) 119889119905

+ 119883120579119888

(119905) 120579 (119905) [120583 (119905 119894) 119889119905 + 120590 (119905 119894) 119889119882 (119905)]

minus 119888 (119905) 119868 (119905) 119889119905

= 119883120579119888

(119905) 119903 (119905 119894) 119889119905

+ 119883120579119888

(119905) 120579 (119905) [ (119905 119894) 119889119905 + 120590 (119905 119894) 119889119882 (119905)]

minus 119888 (119905) 119868 (119905) 119889119905

(5)

where (119905 119894) = 120583(119905 119894) minus 119903(119905 119894)

Denote by 119883120579119888



119889119883120579119888

(119905) = 119889(119883120579119888

(119905)

119868 (119905)) = 119883

120579119888

(119905) [119903 (119905 119894) minus 120583119868 (119905 119894)

+ (120590119868 (119905 119894))2

+ 120579 (119905) ( (119905 119894) minus 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894))] 119889119905

minus 119888 (119905) 119889119905 + 119883120579119888

(119905) 120579 (119905) 120590 (119905 119894) 119889119882 (119905) minus 119883120579119888

(119905)

sdot 120590119868 (119905 119894) (120588 (119905) 119889119882 (119905) + radic1 minus 1205882(119905)1198891198820 (119905))

(6)


by the following

maxA(0119879)

E011989401199090 [int119879

0

119890minus120575119905

119880 (120585 (119905) 119888 (119905)) 119889119905 + 119890minus120575119879

119880(120585 (119879) 119883120579119888

(119879))] (7)



120574120574 where 120574 lt 1 120574 = 0

and 120577(119894) gt 0



120579119888




119905isin[0119879]|119883120579119888



0(119888(119905))120574119889119905) lt +infin for all

120574 le 1

(iv) 119883120579119888(119879) gt 0 as



119881 (119905 119909 119894) = maxA(119905119879)

E119905119894119909 [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(8)



minus 120575119881 (119905 119909 119894) + 119881119905 (119905 119909 119894) +

119871

sum

119895=1

119902119894119895119881 (119905 119909 119895) +1

2

sdot 1199092119881119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2+ sup120579(119905)119888(119905)ge0

119880 (119894 119888 (119905))

+ 119881119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092119881119909119909 (119905 119909 119894)

sdot [1205792(119905) 1205902(119905 119894) minus 2120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)] = 0

(9)






E119905119894119909 ( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

) lt +infin (10)

then we have

(a) V(119905 119909 119894) ge 119881(119905 119909 119894)



119880(120585 (119879) 119883120579119888

(119879)) = V (119879119883120579119888

(119879) 120585 (119879))

= 119890120575(119879minus119905)V (119905 119909 119894)

+ int

119879

119905

minus120575119890120575(119879minus119904)V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890120575(119879minus119904)V119905 (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))

sdot [119883120579119888

(119904) (120578 (119904 120585 (119904)) + 120579 (119904) 120581 (119904 120585 (119904))) minus 119888 (119904)] 119889119904

+1

2int

119879

119905

119890120575(119879minus119904)

(119883120579119888

(119904))

2

V119909119909 (119904 119883120579119888

(119904) 120585 (119904))

times [(120579 (119904) 120590 (119904 120585 (119904)) minus 120590119868 (119904 120585 (119904)) 120588 (119904))2

+ (120590119868 (119904 120585 (119904)))2(1 minus 120588

2(119904))] 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot

119871

sum

119895=1

119902120585(119904)119895V (119904 119883120579119888

(119904) 119895) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904) minus int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(11)


Denote

A120579119888V (119905 119909 119894) = 119880 (119894 119888 (119905)) minus 120575V (119905 119909 119894) + V119905 (119905 119909 119894)

+

119871

sum

119895=1

119902119894119895V (119905 119909 119895) +1

21199092V119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2

+ V119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092V119909119909 (119905 119909 119894)

sdot [(120579 (119905))2(120590 (119905 119894))

2minus 2120579 (119905) 120588120590 (119905 119894) 120590119868 (119905 119894)]

(12)

Thus we have

int

119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904 + 119890minus120575(119879minus119905)

119880(120585 (119879)

119883120579119888

(119879)) = V (119905 119909 119894)

+ int

119879

119905

119890minus120575(119904minus119905)

119860120579119888V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904)

sdot [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

minus int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(13)



int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(14)


119860120579119888V(119904 119883

120579119888

(119904) 120585(119904))119889119904] lt


E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))] le V (119905 119909 119894)

(15)



O119901 = O



(16)



120579119888

(119904))119904ge119905 from 119876119901


E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))] le V (119905 119909 119894)

(17)



E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))]

= E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(18)






119880 (119894 119909) = 120577 (119894)119909120574

120574 (19)


V (119905 119909 119894) = 120577 (119905 119894)119909120574

120574

V (119879 119909 119894) = 120577 (119894)119909120574

120574

(20)



minus 120575120577 (119905 119894)119909120574

120574+ 120577119905 (119905 119894)

119909120574

120574minus1

2120577 (119905 119894) (1 minus 120574)

sdot 119909120574(120590119868 (119905 119894))

2+ 120577 (119905 119894) 120578 (119905 119894) 119909

120574+119909120574

120574

119871

sum

119895=1

119902119894119895120577 (119905 119895)

+ sup120579(119905)119888(119905)ge0

120577 (119894)119888 (119905)120574

120574minus 120577 (119905 119894) 119909

120574minus1119888 (119905)

+ 120577 (119905 119894) 119909120574120579 (119905) 120581 (119905 119894)

minus1

2120577 (119905 119894) (1 minus 120574) 119909

1205741205792(119905) 1205902(119905 119894)

+ 120577 (119905 119894) (1 minus 120574) 119909120574120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894) = 0

(21)



120579lowast(119905 119894) =

120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

(22)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (23)


0 = 120577119905 (119905 119894) + (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

+

119871

sum

119895=1

119902119894119895120577 (119905 119895) + 120577 (119905 119894) (120574120578 (119905 119894) minus 120575

+1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2)

120577 (119879 119894) = 120577 (119894) gt 0

(24)






Proof (a) Denote

120601 (119905 119894) = 120574120578 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2

= 120574119903 (119905 119894) minus 120574120583119868 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

2(119905 119894)

1205902(119905 119894)

+1

2120574 (1 + 120574) (120590119868 (119905 119894))

2

minus1205742

1 minus 120574

120588 (119905) (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

+1

2

1205743

1 minus 1205741205882(119905) (120590119868 (119905 119894))

2

(25)

119870 (119905 119904) = exp [int119904

119905

120601 (119906 120585 (119906)) 119889119906] (26)

119872(119905 119904) = sum


minus int

119904

119905

119871

sum

119895=1

119902120585(Vminus)119895120577 (V 119895) 119889V(27)


119889 [119870 (119905 119904) 120577 (119904 120585 (119904))] = 120577 (119904 120585 (119904)) 119870119904 (119905 119904)

+ 119870 (119905 119904) 119889120577 (119904 120585 (119904)) = 119870 (119905 119904)

sdot [120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) 119889119904 + 119889120577 (119904 120585 (119904))] = 119870 (119905 119904)

sdot [

[

120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) + 120577119904 (119904 120585 (119904))

+

119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895)]

]

119889119904 + 119870 (119905 119904) [120577 (119904 120585 (119904))


119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895) 119889119904 = minus (1

minus 120574)119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119904 120585 (119904))

)

120574(1minus120574)

119889119904

+ 119870 (119905 119904) 119889119872 (119905 119904)

(28)



119870 (119905 119879) 120577 (120585 (119879)) = 120577 (119905 119894) minus (1 minus 120574)

sdot int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119905 120585 (119904))

)

120574(1minus120574)

119889119904

+ int

119879

119905

119870 (119905 119904) 119889119872 (119905 119904)

(29)


120577 (119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [

[

int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (119904 120585 (119904))

120577 (120585 (119904)))

120574(120574minus1)

119889119904]

]

(30)


(119896)

(119905 119894) 119896 = 0 1 2 as follows

120577(0)

(119905 119894) = 120577 (119894)

120577(119896+1)

(119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [int119879

119905

119870 (119905 119904) [120577 (120585 (119904))]1(1minus120574)

sdot (120577(119896)

(119904 120585 (119904)))

120574(120574minus1)

119889119904]

(31)


120577(119896)

(119905 119894) ge E119905119894 [120577 (120585 (119879))119870 (119905 119879)] gt 0 119896 = 1 2 (32)


(119896)


(119896)


(b) For 119894 = 1 2 119871 denote

119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

= minus (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

minus

119871

sum

119895=1

119902119894119895120577 (119905 119895)

minus 120577 (119905 119894) 120601 (119905 119894)

(33)

We have

120577119905 (119905 119894) = 119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

119894 = 1 2 119871

(34)


100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816


sdot [(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)

] minus

119871

sum

119895=1

119902119894119895



(119905 119894)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198601

10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

+ 1198602

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(35)


120597 (120577 (119905 119894))120574(120574minus1)

120597120577 (119905 119894)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

120577 (119905 119894)

)

1(1minus120574)

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574)

(36)

Then10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574) 100381610038161003816100381610038161003816120577 (119905 119894) minus 120577

lowast

(119905 119894)

100381610038161003816100381610038161003816

(37)

Therefore100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198603

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(38)


sum

119894=1

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198604

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(39)





lowast(119905 119909 119894) in (23) has a







E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572



119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)


2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)



(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)




(119905)|120572) lt +infin


1015840119889(119904)


119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)


119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)



E(sup119905isin[0119879]


E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)


(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)


(119883120579lowast119888lowast

(119905))120572) lt +infin





lowast(119905 119894)




2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198891198600 (119905) = 1198600 (119905) 119903 (119905 120585 (119905)) 119889119905 119860 (0) = 1198860 (1)


1198891198601 (119905) = 1198601 (119905) [120583 (119905 120585 (119905)) 119889119905 + 120590 (119905 120585 (119905)) 119889119882 (119905)]

1198601 (0) = 1198861

(2)



119889119868 (119905) = 119868 (119905) [120583119868 (119905 120585 (119905)) 119889119905 + 120590119868 (119905 120585 (119905)) 119889119882119868 (119905)]

119868 (0) = 1198862 gt 0

(3)


119889119868 (119905) = 119868 (119905) [120583119868 (119905 120585 (119905)) 119889119905 + 120590119868 (119905 120585 (119905)) 120588 (119905) 119889119882 (119905)

+ 120590119868 (119905 120585 (119905))radic1 minus 120588

2(119905)1198891198820 (119905)]

(4)







119889119883120579119888

(119905) = 119883120579119888

(119905) (1 minus 120579 (119905)) 119903 (119905 119894) 119889119905

+ 119883120579119888

(119905) 120579 (119905) [120583 (119905 119894) 119889119905 + 120590 (119905 119894) 119889119882 (119905)]

minus 119888 (119905) 119868 (119905) 119889119905

= 119883120579119888

(119905) 119903 (119905 119894) 119889119905

+ 119883120579119888

(119905) 120579 (119905) [ (119905 119894) 119889119905 + 120590 (119905 119894) 119889119882 (119905)]

minus 119888 (119905) 119868 (119905) 119889119905

(5)

where (119905 119894) = 120583(119905 119894) minus 119903(119905 119894)

Denote by 119883120579119888



119889119883120579119888

(119905) = 119889(119883120579119888

(119905)

119868 (119905)) = 119883

120579119888

(119905) [119903 (119905 119894) minus 120583119868 (119905 119894)

+ (120590119868 (119905 119894))2

+ 120579 (119905) ( (119905 119894) minus 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894))] 119889119905

minus 119888 (119905) 119889119905 + 119883120579119888

(119905) 120579 (119905) 120590 (119905 119894) 119889119882 (119905) minus 119883120579119888

(119905)

sdot 120590119868 (119905 119894) (120588 (119905) 119889119882 (119905) + radic1 minus 1205882(119905)1198891198820 (119905))

(6)


by the following

maxA(0119879)

E011989401199090 [int119879

0

119890minus120575119905

119880 (120585 (119905) 119888 (119905)) 119889119905 + 119890minus120575119879

119880(120585 (119879) 119883120579119888

(119879))] (7)



120574120574 where 120574 lt 1 120574 = 0

and 120577(119894) gt 0



120579119888




119905isin[0119879]|119883120579119888



0(119888(119905))120574119889119905) lt +infin for all

120574 le 1

(iv) 119883120579119888(119879) gt 0 as



119881 (119905 119909 119894) = maxA(119905119879)

E119905119894119909 [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(8)



minus 120575119881 (119905 119909 119894) + 119881119905 (119905 119909 119894) +

119871

sum

119895=1

119902119894119895119881 (119905 119909 119895) +1

2

sdot 1199092119881119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2+ sup120579(119905)119888(119905)ge0

119880 (119894 119888 (119905))

+ 119881119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092119881119909119909 (119905 119909 119894)

sdot [1205792(119905) 1205902(119905 119894) minus 2120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)] = 0

(9)






E119905119894119909 ( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

) lt +infin (10)

then we have

(a) V(119905 119909 119894) ge 119881(119905 119909 119894)



119880(120585 (119879) 119883120579119888

(119879)) = V (119879119883120579119888

(119879) 120585 (119879))

= 119890120575(119879minus119905)V (119905 119909 119894)

+ int

119879

119905

minus120575119890120575(119879minus119904)V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890120575(119879minus119904)V119905 (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))

sdot [119883120579119888

(119904) (120578 (119904 120585 (119904)) + 120579 (119904) 120581 (119904 120585 (119904))) minus 119888 (119904)] 119889119904

+1

2int

119879

119905

119890120575(119879minus119904)

(119883120579119888

(119904))

2

V119909119909 (119904 119883120579119888

(119904) 120585 (119904))

times [(120579 (119904) 120590 (119904 120585 (119904)) minus 120590119868 (119904 120585 (119904)) 120588 (119904))2

+ (120590119868 (119904 120585 (119904)))2(1 minus 120588

2(119904))] 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot

119871

sum

119895=1

119902120585(119904)119895V (119904 119883120579119888

(119904) 119895) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904) minus int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(11)


Denote

A120579119888V (119905 119909 119894) = 119880 (119894 119888 (119905)) minus 120575V (119905 119909 119894) + V119905 (119905 119909 119894)

+

119871

sum

119895=1

119902119894119895V (119905 119909 119895) +1

21199092V119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2

+ V119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092V119909119909 (119905 119909 119894)

sdot [(120579 (119905))2(120590 (119905 119894))

2minus 2120579 (119905) 120588120590 (119905 119894) 120590119868 (119905 119894)]

(12)

Thus we have

int

119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904 + 119890minus120575(119879minus119905)

119880(120585 (119879)

119883120579119888

(119879)) = V (119905 119909 119894)

+ int

119879

119905

119890minus120575(119904minus119905)

119860120579119888V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904)

sdot [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

minus int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(13)



int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(14)


119860120579119888V(119904 119883

120579119888

(119904) 120585(119904))119889119904] lt


E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))] le V (119905 119909 119894)

(15)



O119901 = O



(16)



120579119888

(119904))119904ge119905 from 119876119901


E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))] le V (119905 119909 119894)

(17)



E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))]

= E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(18)






119880 (119894 119909) = 120577 (119894)119909120574

120574 (19)


V (119905 119909 119894) = 120577 (119905 119894)119909120574

120574

V (119879 119909 119894) = 120577 (119894)119909120574

120574

(20)



minus 120575120577 (119905 119894)119909120574

120574+ 120577119905 (119905 119894)

119909120574

120574minus1

2120577 (119905 119894) (1 minus 120574)

sdot 119909120574(120590119868 (119905 119894))

2+ 120577 (119905 119894) 120578 (119905 119894) 119909

120574+119909120574

120574

119871

sum

119895=1

119902119894119895120577 (119905 119895)

+ sup120579(119905)119888(119905)ge0

120577 (119894)119888 (119905)120574

120574minus 120577 (119905 119894) 119909

120574minus1119888 (119905)

+ 120577 (119905 119894) 119909120574120579 (119905) 120581 (119905 119894)

minus1

2120577 (119905 119894) (1 minus 120574) 119909

1205741205792(119905) 1205902(119905 119894)

+ 120577 (119905 119894) (1 minus 120574) 119909120574120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894) = 0

(21)



120579lowast(119905 119894) =

120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

(22)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (23)


0 = 120577119905 (119905 119894) + (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

+

119871

sum

119895=1

119902119894119895120577 (119905 119895) + 120577 (119905 119894) (120574120578 (119905 119894) minus 120575

+1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2)

120577 (119879 119894) = 120577 (119894) gt 0

(24)






Proof (a) Denote

120601 (119905 119894) = 120574120578 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2

= 120574119903 (119905 119894) minus 120574120583119868 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

2(119905 119894)

1205902(119905 119894)

+1

2120574 (1 + 120574) (120590119868 (119905 119894))

2

minus1205742

1 minus 120574

120588 (119905) (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

+1

2

1205743

1 minus 1205741205882(119905) (120590119868 (119905 119894))

2

(25)

119870 (119905 119904) = exp [int119904

119905

120601 (119906 120585 (119906)) 119889119906] (26)

119872(119905 119904) = sum


minus int

119904

119905

119871

sum

119895=1

119902120585(Vminus)119895120577 (V 119895) 119889V(27)


119889 [119870 (119905 119904) 120577 (119904 120585 (119904))] = 120577 (119904 120585 (119904)) 119870119904 (119905 119904)

+ 119870 (119905 119904) 119889120577 (119904 120585 (119904)) = 119870 (119905 119904)

sdot [120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) 119889119904 + 119889120577 (119904 120585 (119904))] = 119870 (119905 119904)

sdot [

[

120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) + 120577119904 (119904 120585 (119904))

+

119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895)]

]

119889119904 + 119870 (119905 119904) [120577 (119904 120585 (119904))


119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895) 119889119904 = minus (1

minus 120574)119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119904 120585 (119904))

)

120574(1minus120574)

119889119904

+ 119870 (119905 119904) 119889119872 (119905 119904)

(28)



119870 (119905 119879) 120577 (120585 (119879)) = 120577 (119905 119894) minus (1 minus 120574)

sdot int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119905 120585 (119904))

)

120574(1minus120574)

119889119904

+ int

119879

119905

119870 (119905 119904) 119889119872 (119905 119904)

(29)


120577 (119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [

[

int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (119904 120585 (119904))

120577 (120585 (119904)))

120574(120574minus1)

119889119904]

]

(30)


(119896)

(119905 119894) 119896 = 0 1 2 as follows

120577(0)

(119905 119894) = 120577 (119894)

120577(119896+1)

(119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [int119879

119905

119870 (119905 119904) [120577 (120585 (119904))]1(1minus120574)

sdot (120577(119896)

(119904 120585 (119904)))

120574(120574minus1)

119889119904]

(31)


120577(119896)

(119905 119894) ge E119905119894 [120577 (120585 (119879))119870 (119905 119879)] gt 0 119896 = 1 2 (32)


(119896)


(119896)


(b) For 119894 = 1 2 119871 denote

119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

= minus (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

minus

119871

sum

119895=1

119902119894119895120577 (119905 119895)

minus 120577 (119905 119894) 120601 (119905 119894)

(33)

We have

120577119905 (119905 119894) = 119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

119894 = 1 2 119871

(34)


100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816


sdot [(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)

] minus

119871

sum

119895=1

119902119894119895



(119905 119894)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198601

10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

+ 1198602

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(35)


120597 (120577 (119905 119894))120574(120574minus1)

120597120577 (119905 119894)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

120577 (119905 119894)

)

1(1minus120574)

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574)

(36)

Then10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574) 100381610038161003816100381610038161003816120577 (119905 119894) minus 120577

lowast

(119905 119894)

100381610038161003816100381610038161003816

(37)

Therefore100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198603

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(38)


sum

119894=1

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198604

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(39)





lowast(119905 119909 119894) in (23) has a







E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572



119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)


2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)



(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)




(119905)|120572) lt +infin


1015840119889(119904)


119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)


119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)



E(sup119905isin[0119879]


E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)


(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)


(119883120579lowast119888lowast

(119905))120572) lt +infin





lowast(119905 119894)




2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574120574 where 120574 lt 1 120574 = 0

and 120577(119894) gt 0



120579119888




119905isin[0119879]|119883120579119888



0(119888(119905))120574119889119905) lt +infin for all

120574 le 1

(iv) 119883120579119888(119879) gt 0 as



119881 (119905 119909 119894) = maxA(119905119879)

E119905119894119909 [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(8)



minus 120575119881 (119905 119909 119894) + 119881119905 (119905 119909 119894) +

119871

sum

119895=1

119902119894119895119881 (119905 119909 119895) +1

2

sdot 1199092119881119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2+ sup120579(119905)119888(119905)ge0

119880 (119894 119888 (119905))

+ 119881119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092119881119909119909 (119905 119909 119894)

sdot [1205792(119905) 1205902(119905 119894) minus 2120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)] = 0

(9)






E119905119894119909 ( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

) lt +infin (10)

then we have

(a) V(119905 119909 119894) ge 119881(119905 119909 119894)



119880(120585 (119879) 119883120579119888

(119879)) = V (119879119883120579119888

(119879) 120585 (119879))

= 119890120575(119879minus119905)V (119905 119909 119894)

+ int

119879

119905

minus120575119890120575(119879minus119904)V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890120575(119879minus119904)V119905 (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))

sdot [119883120579119888

(119904) (120578 (119904 120585 (119904)) + 120579 (119904) 120581 (119904 120585 (119904))) minus 119888 (119904)] 119889119904

+1

2int

119879

119905

119890120575(119879minus119904)

(119883120579119888

(119904))

2

V119909119909 (119904 119883120579119888

(119904) 120585 (119904))

times [(120579 (119904) 120590 (119904 120585 (119904)) minus 120590119868 (119904 120585 (119904)) 120588 (119904))2

+ (120590119868 (119904 120585 (119904)))2(1 minus 120588

2(119904))] 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot

119871

sum

119895=1

119902120585(119904)119895V (119904 119883120579119888

(119904) 119895) 119889119904 + int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904) minus int

119879

119905

119890120575(119879minus119904)

sdot V119909 (119904 119883120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(11)


Denote

A120579119888V (119905 119909 119894) = 119880 (119894 119888 (119905)) minus 120575V (119905 119909 119894) + V119905 (119905 119909 119894)

+

119871

sum

119895=1

119902119894119895V (119905 119909 119895) +1

21199092V119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2

+ V119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092V119909119909 (119905 119909 119894)

sdot [(120579 (119905))2(120590 (119905 119894))

2minus 2120579 (119905) 120588120590 (119905 119894) 120590119868 (119905 119894)]

(12)

Thus we have

int

119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904 + 119890minus120575(119879minus119905)

119880(120585 (119879)

119883120579119888

(119879)) = V (119905 119909 119894)

+ int

119879

119905

119890minus120575(119904minus119905)

119860120579119888V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904)

sdot [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

minus int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(13)



int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(14)


119860120579119888V(119904 119883

120579119888

(119904) 120585(119904))119889119904] lt


E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))] le V (119905 119909 119894)

(15)



O119901 = O



(16)



120579119888

(119904))119904ge119905 from 119876119901


E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))] le V (119905 119909 119894)

(17)



E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))]

= E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(18)






119880 (119894 119909) = 120577 (119894)119909120574

120574 (19)


V (119905 119909 119894) = 120577 (119905 119894)119909120574

120574

V (119879 119909 119894) = 120577 (119894)119909120574

120574

(20)



minus 120575120577 (119905 119894)119909120574

120574+ 120577119905 (119905 119894)

119909120574

120574minus1

2120577 (119905 119894) (1 minus 120574)

sdot 119909120574(120590119868 (119905 119894))

2+ 120577 (119905 119894) 120578 (119905 119894) 119909

120574+119909120574

120574

119871

sum

119895=1

119902119894119895120577 (119905 119895)

+ sup120579(119905)119888(119905)ge0

120577 (119894)119888 (119905)120574

120574minus 120577 (119905 119894) 119909

120574minus1119888 (119905)

+ 120577 (119905 119894) 119909120574120579 (119905) 120581 (119905 119894)

minus1

2120577 (119905 119894) (1 minus 120574) 119909

1205741205792(119905) 1205902(119905 119894)

+ 120577 (119905 119894) (1 minus 120574) 119909120574120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894) = 0

(21)



120579lowast(119905 119894) =

120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

(22)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (23)


0 = 120577119905 (119905 119894) + (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

+

119871

sum

119895=1

119902119894119895120577 (119905 119895) + 120577 (119905 119894) (120574120578 (119905 119894) minus 120575

+1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2)

120577 (119879 119894) = 120577 (119894) gt 0

(24)






Proof (a) Denote

120601 (119905 119894) = 120574120578 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2

= 120574119903 (119905 119894) minus 120574120583119868 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

2(119905 119894)

1205902(119905 119894)

+1

2120574 (1 + 120574) (120590119868 (119905 119894))

2

minus1205742

1 minus 120574

120588 (119905) (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

+1

2

1205743

1 minus 1205741205882(119905) (120590119868 (119905 119894))

2

(25)

119870 (119905 119904) = exp [int119904

119905

120601 (119906 120585 (119906)) 119889119906] (26)

119872(119905 119904) = sum


minus int

119904

119905

119871

sum

119895=1

119902120585(Vminus)119895120577 (V 119895) 119889V(27)


119889 [119870 (119905 119904) 120577 (119904 120585 (119904))] = 120577 (119904 120585 (119904)) 119870119904 (119905 119904)

+ 119870 (119905 119904) 119889120577 (119904 120585 (119904)) = 119870 (119905 119904)

sdot [120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) 119889119904 + 119889120577 (119904 120585 (119904))] = 119870 (119905 119904)

sdot [

[

120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) + 120577119904 (119904 120585 (119904))

+

119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895)]

]

119889119904 + 119870 (119905 119904) [120577 (119904 120585 (119904))


119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895) 119889119904 = minus (1

minus 120574)119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119904 120585 (119904))

)

120574(1minus120574)

119889119904

+ 119870 (119905 119904) 119889119872 (119905 119904)

(28)



119870 (119905 119879) 120577 (120585 (119879)) = 120577 (119905 119894) minus (1 minus 120574)

sdot int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119905 120585 (119904))

)

120574(1minus120574)

119889119904

+ int

119879

119905

119870 (119905 119904) 119889119872 (119905 119904)

(29)


120577 (119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [

[

int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (119904 120585 (119904))

120577 (120585 (119904)))

120574(120574minus1)

119889119904]

]

(30)


(119896)

(119905 119894) 119896 = 0 1 2 as follows

120577(0)

(119905 119894) = 120577 (119894)

120577(119896+1)

(119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [int119879

119905

119870 (119905 119904) [120577 (120585 (119904))]1(1minus120574)

sdot (120577(119896)

(119904 120585 (119904)))

120574(120574minus1)

119889119904]

(31)


120577(119896)

(119905 119894) ge E119905119894 [120577 (120585 (119879))119870 (119905 119879)] gt 0 119896 = 1 2 (32)


(119896)


(119896)


(b) For 119894 = 1 2 119871 denote

119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

= minus (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

minus

119871

sum

119895=1

119902119894119895120577 (119905 119895)

minus 120577 (119905 119894) 120601 (119905 119894)

(33)

We have

120577119905 (119905 119894) = 119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

119894 = 1 2 119871

(34)


100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816


sdot [(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)

] minus

119871

sum

119895=1

119902119894119895



(119905 119894)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198601

10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

+ 1198602

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(35)


120597 (120577 (119905 119894))120574(120574minus1)

120597120577 (119905 119894)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

120577 (119905 119894)

)

1(1minus120574)

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574)

(36)

Then10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574) 100381610038161003816100381610038161003816120577 (119905 119894) minus 120577

lowast

(119905 119894)

100381610038161003816100381610038161003816

(37)

Therefore100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198603

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(38)


sum

119894=1

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198604

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(39)





lowast(119905 119909 119894) in (23) has a







E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572



119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)


2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)



(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)




(119905)|120572) lt +infin


1015840119889(119904)


119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)


119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)



E(sup119905isin[0119879]


E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)


(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)


(119883120579lowast119888lowast

(119905))120572) lt +infin





lowast(119905 119894)




2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Denote

A120579119888V (119905 119909 119894) = 119880 (119894 119888 (119905)) minus 120575V (119905 119909 119894) + V119905 (119905 119909 119894)

+

119871

sum

119895=1

119902119894119895V (119905 119909 119895) +1

21199092V119909119909 (119905 119909 119894) (120590119868 (119905 119894))

2

+ V119909 (119905 119909 119894) [119909 (120578 (119905 119894) + 120579 (119905) 120581 (119905 119894)) minus 119888 (119905)] +1

2

sdot 1199092V119909119909 (119905 119909 119894)

sdot [(120579 (119905))2(120590 (119905 119894))

2minus 2120579 (119905) 120588120590 (119905 119894) 120590119868 (119905 119894)]

(12)

Thus we have

int

119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904 + 119890minus120575(119879minus119905)

119880(120585 (119879)

119883120579119888

(119879)) = V (119905 119909 119894)

+ int

119879

119905

119890minus120575(119904minus119905)

119860120579119888V (119904 119883

120579119888

(119904) 120585 (119904)) 119889119904

+ int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904)

sdot [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

minus int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(13)



int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) [120579 (119904) 120590 (119904 120585 (119904))

minus 120590119868 (119904 120585 (119904)) 120588 (119904)] 119889119882 (119904)

int

119879

119905

119890minus120575(119904minus119905)V119909 (119904 119883

120579119888

(119904) 120585 (119904))119883120579119888

(119904) 120590119868 (119904 120585 (119904))

sdot radic1 minus 1205882(119904) 1198891198820 (119904)

(14)


119860120579119888V(119904 119883

120579119888

(119904) 120585(119904))119889119904] lt


E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))] le V (119905 119909 119894)

(15)



O119901 = O



(16)



120579119888

(119904))119904ge119905 from 119876119901


E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))] le V (119905 119909 119894)

(17)



E [int120603119901

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(120603119901minus119905)V (120603119901 119883

120579119888

(120603119901) 120585 (120603119901))]

= E [int119879

119905

119890minus120575(119904minus119905)

119880 (120585 (119904) 119888 (119904)) 119889119904

+ 119890minus120575(119879minus119905)

119880(120585 (119879) 119883120579119888

(119879))]

(18)






119880 (119894 119909) = 120577 (119894)119909120574

120574 (19)


V (119905 119909 119894) = 120577 (119905 119894)119909120574

120574

V (119879 119909 119894) = 120577 (119894)119909120574

120574

(20)



minus 120575120577 (119905 119894)119909120574

120574+ 120577119905 (119905 119894)

119909120574

120574minus1

2120577 (119905 119894) (1 minus 120574)

sdot 119909120574(120590119868 (119905 119894))

2+ 120577 (119905 119894) 120578 (119905 119894) 119909

120574+119909120574

120574

119871

sum

119895=1

119902119894119895120577 (119905 119895)

+ sup120579(119905)119888(119905)ge0

120577 (119894)119888 (119905)120574

120574minus 120577 (119905 119894) 119909

120574minus1119888 (119905)

+ 120577 (119905 119894) 119909120574120579 (119905) 120581 (119905 119894)

minus1

2120577 (119905 119894) (1 minus 120574) 119909

1205741205792(119905) 1205902(119905 119894)

+ 120577 (119905 119894) (1 minus 120574) 119909120574120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894) = 0

(21)



120579lowast(119905 119894) =

120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

(22)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (23)


0 = 120577119905 (119905 119894) + (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

+

119871

sum

119895=1

119902119894119895120577 (119905 119895) + 120577 (119905 119894) (120574120578 (119905 119894) minus 120575

+1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2)

120577 (119879 119894) = 120577 (119894) gt 0

(24)






Proof (a) Denote

120601 (119905 119894) = 120574120578 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2

= 120574119903 (119905 119894) minus 120574120583119868 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

2(119905 119894)

1205902(119905 119894)

+1

2120574 (1 + 120574) (120590119868 (119905 119894))

2

minus1205742

1 minus 120574

120588 (119905) (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

+1

2

1205743

1 minus 1205741205882(119905) (120590119868 (119905 119894))

2

(25)

119870 (119905 119904) = exp [int119904

119905

120601 (119906 120585 (119906)) 119889119906] (26)

119872(119905 119904) = sum


minus int

119904

119905

119871

sum

119895=1

119902120585(Vminus)119895120577 (V 119895) 119889V(27)


119889 [119870 (119905 119904) 120577 (119904 120585 (119904))] = 120577 (119904 120585 (119904)) 119870119904 (119905 119904)

+ 119870 (119905 119904) 119889120577 (119904 120585 (119904)) = 119870 (119905 119904)

sdot [120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) 119889119904 + 119889120577 (119904 120585 (119904))] = 119870 (119905 119904)

sdot [

[

120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) + 120577119904 (119904 120585 (119904))

+

119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895)]

]

119889119904 + 119870 (119905 119904) [120577 (119904 120585 (119904))


119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895) 119889119904 = minus (1

minus 120574)119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119904 120585 (119904))

)

120574(1minus120574)

119889119904

+ 119870 (119905 119904) 119889119872 (119905 119904)

(28)



119870 (119905 119879) 120577 (120585 (119879)) = 120577 (119905 119894) minus (1 minus 120574)

sdot int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119905 120585 (119904))

)

120574(1minus120574)

119889119904

+ int

119879

119905

119870 (119905 119904) 119889119872 (119905 119904)

(29)


120577 (119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [

[

int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (119904 120585 (119904))

120577 (120585 (119904)))

120574(120574minus1)

119889119904]

]

(30)


(119896)

(119905 119894) 119896 = 0 1 2 as follows

120577(0)

(119905 119894) = 120577 (119894)

120577(119896+1)

(119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [int119879

119905

119870 (119905 119904) [120577 (120585 (119904))]1(1minus120574)

sdot (120577(119896)

(119904 120585 (119904)))

120574(120574minus1)

119889119904]

(31)


120577(119896)

(119905 119894) ge E119905119894 [120577 (120585 (119879))119870 (119905 119879)] gt 0 119896 = 1 2 (32)


(119896)


(119896)


(b) For 119894 = 1 2 119871 denote

119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

= minus (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

minus

119871

sum

119895=1

119902119894119895120577 (119905 119895)

minus 120577 (119905 119894) 120601 (119905 119894)

(33)

We have

120577119905 (119905 119894) = 119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

119894 = 1 2 119871

(34)


100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816


sdot [(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)

] minus

119871

sum

119895=1

119902119894119895



(119905 119894)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198601

10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

+ 1198602

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(35)


120597 (120577 (119905 119894))120574(120574minus1)

120597120577 (119905 119894)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

120577 (119905 119894)

)

1(1minus120574)

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574)

(36)

Then10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574) 100381610038161003816100381610038161003816120577 (119905 119894) minus 120577

lowast

(119905 119894)

100381610038161003816100381610038161003816

(37)

Therefore100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198603

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(38)


sum

119894=1

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198604

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(39)





lowast(119905 119909 119894) in (23) has a







E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572



119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)


2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)



(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)




(119905)|120572) lt +infin


1015840119889(119904)


119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)


119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)



E(sup119905isin[0119879]


E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)


(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)


(119883120579lowast119888lowast

(119905))120572) lt +infin





lowast(119905 119894)




2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus 120575120577 (119905 119894)119909120574

120574+ 120577119905 (119905 119894)

119909120574

120574minus1

2120577 (119905 119894) (1 minus 120574)

sdot 119909120574(120590119868 (119905 119894))

2+ 120577 (119905 119894) 120578 (119905 119894) 119909

120574+119909120574

120574

119871

sum

119895=1

119902119894119895120577 (119905 119895)

+ sup120579(119905)119888(119905)ge0

120577 (119894)119888 (119905)120574

120574minus 120577 (119905 119894) 119909

120574minus1119888 (119905)

+ 120577 (119905 119894) 119909120574120579 (119905) 120581 (119905 119894)

minus1

2120577 (119905 119894) (1 minus 120574) 119909

1205741205792(119905) 1205902(119905 119894)

+ 120577 (119905 119894) (1 minus 120574) 119909120574120579 (119905) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894) = 0

(21)



120579lowast(119905 119894) =

120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

(22)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (23)


0 = 120577119905 (119905 119894) + (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

+

119871

sum

119895=1

119902119894119895120577 (119905 119895) + 120577 (119905 119894) (120574120578 (119905 119894) minus 120575

+1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2)

120577 (119879 119894) = 120577 (119894) gt 0

(24)






Proof (a) Denote

120601 (119905 119894) = 120574120578 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

1205812(119905 119894)

1205902(119905 119894)

+ 120574120588 (119905) 120581 (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

minus1

2120574 (1 minus 120574) (1 minus 120588

2(119905)) (120590119868 (119905 119894))

2

= 120574119903 (119905 119894) minus 120574120583119868 (119905 119894) minus 120575 +1

2

120574

1 minus 120574

2(119905 119894)

1205902(119905 119894)

+1

2120574 (1 + 120574) (120590119868 (119905 119894))

2

minus1205742

1 minus 120574

120588 (119905) (119905 119894) 120590119868 (119905 119894)

120590 (119905 119894)

+1

2

1205743

1 minus 1205741205882(119905) (120590119868 (119905 119894))

2

(25)

119870 (119905 119904) = exp [int119904

119905

120601 (119906 120585 (119906)) 119889119906] (26)

119872(119905 119904) = sum


minus int

119904

119905

119871

sum

119895=1

119902120585(Vminus)119895120577 (V 119895) 119889V(27)


119889 [119870 (119905 119904) 120577 (119904 120585 (119904))] = 120577 (119904 120585 (119904)) 119870119904 (119905 119904)

+ 119870 (119905 119904) 119889120577 (119904 120585 (119904)) = 119870 (119905 119904)

sdot [120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) 119889119904 + 119889120577 (119904 120585 (119904))] = 119870 (119905 119904)

sdot [

[

120601 (119904 120585 (119904)) 120577 (119904 120585 (119904)) + 120577119904 (119904 120585 (119904))

+

119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895)]

]

119889119904 + 119870 (119905 119904) [120577 (119904 120585 (119904))


119871

sum

119895=1

119902120585(119904minus)119895120577 (119904 119895) 119889119904 = minus (1

minus 120574)119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119904 120585 (119904))

)

120574(1minus120574)

119889119904

+ 119870 (119905 119904) 119889119872 (119905 119904)

(28)



119870 (119905 119879) 120577 (120585 (119879)) = 120577 (119905 119894) minus (1 minus 120574)

sdot int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119905 120585 (119904))

)

120574(1minus120574)

119889119904

+ int

119879

119905

119870 (119905 119904) 119889119872 (119905 119904)

(29)


120577 (119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [

[

int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (119904 120585 (119904))

120577 (120585 (119904)))

120574(120574minus1)

119889119904]

]

(30)


(119896)

(119905 119894) 119896 = 0 1 2 as follows

120577(0)

(119905 119894) = 120577 (119894)

120577(119896+1)

(119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [int119879

119905

119870 (119905 119904) [120577 (120585 (119904))]1(1minus120574)

sdot (120577(119896)

(119904 120585 (119904)))

120574(120574minus1)

119889119904]

(31)


120577(119896)

(119905 119894) ge E119905119894 [120577 (120585 (119879))119870 (119905 119879)] gt 0 119896 = 1 2 (32)


(119896)


(119896)


(b) For 119894 = 1 2 119871 denote

119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

= minus (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

minus

119871

sum

119895=1

119902119894119895120577 (119905 119895)

minus 120577 (119905 119894) 120601 (119905 119894)

(33)

We have

120577119905 (119905 119894) = 119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

119894 = 1 2 119871

(34)


100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816


sdot [(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)

] minus

119871

sum

119895=1

119902119894119895



(119905 119894)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198601

10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

+ 1198602

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(35)


120597 (120577 (119905 119894))120574(120574minus1)

120597120577 (119905 119894)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

120577 (119905 119894)

)

1(1minus120574)

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574)

(36)

Then10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574) 100381610038161003816100381610038161003816120577 (119905 119894) minus 120577

lowast

(119905 119894)

100381610038161003816100381610038161003816

(37)

Therefore100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198603

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(38)


sum

119894=1

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198604

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(39)





lowast(119905 119909 119894) in (23) has a







E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572



119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)


2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)



(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)




(119905)|120572) lt +infin


1015840119889(119904)


119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)


119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)



E(sup119905isin[0119879]


E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)


(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)


(119883120579lowast119888lowast

(119905))120572) lt +infin





lowast(119905 119894)




2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870 (119905 119879) 120577 (120585 (119879)) = 120577 (119905 119894) minus (1 minus 120574)

sdot int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (120585 (119904))

120577 (119905 120585 (119904))

)

120574(1minus120574)

119889119904

+ int

119879

119905

119870 (119905 119904) 119889119872 (119905 119904)

(29)


120577 (119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [

[

int

119879

119905

119870 (119905 119904) 120577 (120585 (119904)) (120577 (119904 120585 (119904))

120577 (120585 (119904)))

120574(120574minus1)

119889119904]

]

(30)


(119896)

(119905 119894) 119896 = 0 1 2 as follows

120577(0)

(119905 119894) = 120577 (119894)

120577(119896+1)

(119905 119894) = E119905119894 (120577 (120585 (119879))119870 (119905 119879)) + (1 minus 120574)

sdot E119905119894 [int119879

119905

119870 (119905 119904) [120577 (120585 (119904))]1(1minus120574)

sdot (120577(119896)

(119904 120585 (119904)))

120574(120574minus1)

119889119904]

(31)


120577(119896)

(119905 119894) ge E119905119894 [120577 (120585 (119879))119870 (119905 119879)] gt 0 119896 = 1 2 (32)


(119896)


(119896)


(b) For 119894 = 1 2 119871 denote

119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

= minus (1 minus 120574) 120577 (119894) (120577 (119894)

120577 (119905 119894)

)

120574(1minus120574)

minus

119871

sum

119895=1

119902119894119895120577 (119905 119895)

minus 120577 (119905 119894) 120601 (119905 119894)

(33)

We have

120577119905 (119905 119894) = 119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

119894 = 1 2 119871

(34)


100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816


sdot [(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)

] minus

119871

sum

119895=1

119902119894119895



(119905 119894)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198601

10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

+ 1198602

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(35)


120597 (120577 (119905 119894))120574(120574minus1)

120597120577 (119905 119894)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

120577 (119905 119894)

)

1(1minus120574)

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574)

(36)

Then10038161003816100381610038161003816100381610038161003816

(120577 (119905 119894))120574(120574minus1)

minus (120577lowast

(119905 119894))

120574(120574minus1)10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

(1

)

1(1minus120574) 100381610038161003816100381610038161003816120577 (119905 119894) minus 120577

lowast

(119905 119894)

100381610038161003816100381610038161003816

(37)

Therefore100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198603

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(38)


sum

119894=1

100381610038161003816100381610038161003816119891119894 (119905 120577 (119905 1) 120577 (119905 2) 120577 (119905 119871))

minus 119891119894 (119905 120577lowast

(119905 1) 120577lowast

(119905 2) 120577lowast

(119905 119871))

100381610038161003816100381610038161003816

le 1198604

119871

sum

119895=1

100381610038161003816100381610038161003816120577 (119905 119895) minus 120577

lowast

(119905 119895)

100381610038161003816100381610038161003816

(39)





lowast(119905 119909 119894) in (23) has a







E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572



119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)


2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)



(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)




(119905)|120572) lt +infin


1015840119889(119904)


119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)


119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)



E(sup119905isin[0119879]


E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)


(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)


(119883120579lowast119888lowast

(119905))120572) lt +infin





lowast(119905 119894)




2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












lowast(119905 119909 119894) in (23) has a







E( suptisin[0T]

1003816100381610038161003816100381610038161003816X120579lowastclowast

(t)1003816100381610038161003816100381610038161003816

120572



119889(119883120579lowast119888lowast

(119905)) = 119883120579lowast119888lowast

(119905) 120603 (119905 119894) 119889119905

+120581 (119905 119894)

(1 minus 120574) 120590 (119905 119894)119889119882 (119905)


2(119905)1198891198820 (119905)

(41)

where

120603 (119905 119894) = 120578 (119905 119894)

+120581 (119905 119894) + (1 minus 120574) 120588 (119905) 120590 (119905 119894) 120590119868 (119905 119894)

(1 minus 120574) 1205902(119905 119894)

120581 (119905 119894)

minus (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(42)



(119905) = 1199090

sdot expint119905

0

[120603 (119904 120585 (119904)) minus1

2(

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904)))

2

]119889119904

minus int

119905

0

1

2(120590119868 (119904 120585 (119904)))

2(1 minus 120588

2(119904)) 119889119904

+ int

119905

0

120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904)

minus int

119905

0

120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(43)




(119905)|120572) lt +infin


1015840119889(119904)


119885 (119905) = expint119905

0

ℎ (119904 120585 (119904))1015840119889 (119904)

= expint119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

times expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) le 1198671

sdot expminusint

119905

0

1

2

1003817100381710038171003817ℎ (119904 120585 (119904))

1003817100381710038171003817

2119889119904

+ int

119905

0

ℎ (119904 120585 (119904))1015840119889 (119904) fl 1198671 (119905)

(44)


119889 (119905) = (119905) ℎ (119905 120585 (119905))1015840119889 (119905) (45)



E(sup119905isin[0119879]


E( sup119905isin[0119879]

exp(int119905

0

ℎ (119904 120585 (119904))1015840119889 (119904))) lt +infin (46)


(119883120579lowast119888lowast

(119905))

120572

le 1198673 expint119905

0

120572120581 (119904 120585 (119904))

(1 minus 120574) 120590 (119904 120585 (119904))119889119882 (119904) minus int

119905

0

120572120590119868 (119904 120585 (119904))radic1 minus 120588

2(119904) 1198891198820 (119904)

(47)


(119883120579lowast119888lowast

(119905))120572) lt +infin





lowast(119905 119894)




2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2) lt


E(int119879

0


(119905) 120585 (119905))

1003816100381610038161003816100381610038161003816

120574

119889119905)

= E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

(120577 (120585 (119905))

120577 (119905 120585 (119905))

)

120574(1minus120574)

119889119905)

le 1198721E(int119879

0

(119883120579lowast119888lowast

(119905))

120574

119889119905)

le 1198721E(int119879

0

sup119905isin[0119879]

(119883120579lowast119888lowast

(119905))

120574

119889119905) lt +infin

(48)




E( sup119904isin[119905119879]

100381610038161003816100381610038161003816V (119904 119883

120579119888

(119904) 120585 (119904))

100381610038161003816100381610038161003816

120573

)

= E( sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577 (119904 120585 (119904))

(119883120579119888

(119904))

120574

120574

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198722E( sup119904isin[119905119879]

(119883120579119888

(119904))

2120574

) lt +infin

(49)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

minus120574

1 minus 120574

120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894) (50)

119888lowast(119905 119909 119894) = (

120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

119909 (51)


119881 (119905 119909 119894) =120577 (119905 119894) 119909

120574

120574 (52)



120579lowast(119905 119894) =

(119905 119894)

(1 minus 120574) 1205902(119905 119894)

(53)






120579lowast(119905 119894) =

1

1 minus 120574

(119905 119894) minus 120588 (119905) 120590119868 (119905 119894) 120590 (119905 119894)

1205902(119905 119894)

+120588 (119905) 120590119868 (119905 119894)

120590 (119905 119894)

(54)










Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Denote by

cp (119905 119894) fl (120577 (119894)

120577 (119905 119894)

)

1(1minus120574)

(55)



119902 = (

minus25 25

4 minus4

) (56)













120597120601

120597120588= minus

1205742

1 minus 120574

120590119868 (119905 119894)

120590 (119905 119894)[ (119905 119894) minus 120574120588120590 (119905 119894) 120590119868 (119905 119894)] (57)



(119896)

119896 = 0 1 2 decreases as 119870(119905 119894)










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

07

08

09

1

11

12

13

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06120574 = 05

120574 = minus04120574 = minus03

120574 = minus02120574 = minus01

120574 = 04

120574 = 03

120574 = 02

120574 = 01

(a)

0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120574 = 07

120574 = 09

120574 = 095

120574 = 08

(b)

0 1 2 3 4 5

07

08

09

1

11

12

13

14

15

16

17

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06120574 = 07

120574 = 05


120574 = minus01

120574 = 04

120574 = 03120574 = 02

120574 = 01

(c)

0 1 2 3 4 50

02

04

06

08

1

12

14

16

18

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 07

120574 = 08

120574 = 09

120574 = 095

(d)







6 Conclusion



0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 505

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(a)

0 1 2 3 4 5

05

055

06

065

07

075

08

085

09

095

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus08

120588 = 10120588 = 05120588 = 00

120588 = minus05120588 = minus10

(b)


0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120588 = minus1120588 = minus05

120574 = minus4

(a)

0 1 2 3 4 502

03

04

05

06

07

08

09

1

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120588 = minus1120588 = minus05

120574 = minus4

(b)





0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 05 1 15 2 25 3 35 4 45055

06

065

07

075

08

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = minus05

005007009

011013015

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)


0 1 2 3 4 51

105

11

115

12

125

13

135

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

009007005

120583I(1) =120583I(1) =120583I(1) =

120583I(1) =120583I(1) =120583I(1) =

(b)






0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 51

105

11

115

12

125

13

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 05

015013011

120583I(2) =120583I(2) =120583I(2) =

009007005

120583I(2) =120583I(2) =120583I(2) =

(b)


0 1 2 3 4 507

075

08

085

09

095

1

105

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(a)

0 1 2 3 4 51

105

11

115

12

125

13

135

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 08

021023025

015017019

120590I(1) =120590I(1) =120590I(1) = 120590I(1) =

120590I(1) =

120590I(1) =

(b)






0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 50

02

04

06

08

1

12

14

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(a)

0 1 2 3 4 51

15

2

25

3

35

4

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)

120574 = 06

120577(1) = 02

120577(1) = 04

120577(1) = 06

120577(1) = 08

120577(1) = 1

(b)


0 1 2 3 4 51

12

14

16

18

2

22

24

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 1

)

120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(a)

0 1 2 3 4 50

05

1

15

Time

Opt

imal

cons

umpt

ion

prop

ortio

n cp

(t 2

)120574 = 06

120577(2) = 02

120577(2) = 04

120577(2) = 06

120577(2) = 08

120577(2) = 1

(b)



Competing Interests


Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article optimal investment-consumption strategy under...

Documents