consumption-investment problems with transaction costs: survey and open problems

Math Meth Oper Res (2000) 51 :43±68

9992000

Consumption-investment problems with transaction costs:Survey and open problems*

Abel Cadenillas

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1,Canada (e-mail: [email protected])

Abstract. We present a survey of problems and methods contained in variousworks on consumption-investment problems with transaction costs in contin-uous time. The methods are those of optimal stopping, stochastic singularcontrol, and stochastic impulse control. We also describe some open problemsin this active area of research.

Key words: Consumption-investment, optimal stopping, stochastic singularcontrol, stochastic impulse control, transaction costs

1 Introduction

The study of consumption-investment problems via stochastic processes incontinuous time was initiated by Merton (1969). He considered a model inwhich the prices of the risky securities were generated by a Brownian motion,and assumed that there were no transaction costs. The optimal strategy in hismodel consists of making an in®nite number of transactions in order to keepthe proportions invested in the risky securities equal to a constant vector.Referring to the Merton solution, Magill and Constantinides (1976) observedthat ``by combining the assumption that trading opportunities are availablecontinuously with the assumption that the trading opportunities are availablecostlessly the investor is led to a quite unrealistic type of portfolio behavior.''This observation motivated them to consider the ®rst model in continuoustime that incorporated transaction costs. This initial work was then extended

* While remaining responsible for any errors, I would like to thank Ioannis Karatzas and SureshP. Sethi for some suggestions. This work was supported by the Natural Sciences and EngineeringResearch Council of Canada grant OGP0194137.Manuscript received: April 1999/®nal version received: October 1999

in many diërent directions. The objective of this paper is to survey that lit-erature, and present some open problems.

In this survey we are also going to present the mathematical techniquesthat have been applied to model and solve consumption-investment problems.They include the methods of optimization, stochastic calculus, classical sto-chastic control, optimal stopping, stochastic singular control, and stochasticimpulse control.

We must indicate that we have not included in this survey models indiscrete time or about equilibrium. We have not covered either the manymodels of option pricing in continuous time that incorporate transactioncosts, but we have made a brief reference in section 4 about option pricingmodels related to consumption-investment problems. Good references forother topics in Mathematical Finance are the books of Du½e (1996), Elliottand Kopp (1999), Hull (1997), Karatzas (1996), Karatzas and Shreve (1998),Korn (1997), Merton (1990), Musiela and Rutkowski (1998), Pliska (1997),Sethi (1997), and Wilmott, Dewynne and Howison (1993, 1995).

In sections 2 and 3 we present two models of consumption-investmentwithout transaction costs. They will be helpful to understand the models withtransaction costs of sections 4, 5, 6, and 7. In those sections, we presentmodels with diërent transaction costs structures: proportional transactioncosts, ®xed and proportional transaction costs, transactions costs as a ®xedproportion of portfolio value, and a model that includes not only transactioncosts but also taxes. We present some open problems in section 8.

2 Maximizing the utility of consumption without transaction costs

Let us consider a ®nancial market in which m� 1 securities (®nancial assets)are traded continuously. One of them is a pure discount bond with price P0�t�at time t governed by the equation

dP0�t� � P0�t�r dt: �1�There are also m risky assets called stocks with prices-per-share Pi�t� at time tgoverned by the stochastic diërential equations

dPi�t� � Pi�t� mi dt�Xm

j�1

sij dW j�t�( )

; i A f1; 2; . . . ;mg: �2�

In this model f�W � � �W1; . . . ;Wm�;Ft�; t A �0;y�g is an m-dimensionalBrownian motion, with its natural ®ltration, and it represents the uncertaintyin the ®nancial market. Here, M � denotes de transpose of a matrix M. Wealso denote by

m� � �m1; . . . ; mm�the appreciation rate of the stocks, and by

S :� �sij�1Ui; jUm

the matrix of volatilities.

44 A. Cadenillas

Let us consider now an economic agent, who invests in the above secur-ities. We shall denote by

p� � �p1; . . . ; pm�

the proportions of his total wealth that he invests in each of the m stocks, andby c�t� the rate at which he withdraws funds for consumption. Since thenumber of shares of stock k held at time t is pk�t�X�t�=Pk�t�, the wealth X ofthe investor evolves according to

dX �t� � 1ÿXm

k�1

pk�t� !

rX �t� dt�Xm

k�1

pk�t�X�t� mk dt�Xm

j�1

skj dW j�t� !

ÿ c�t� dt;

� frX�t� ÿ c�t� � p��t�X�t��mÿ r~1�g dt� p��t�SX�t� dW �t� �3�

with initial condition

X �0� � x: �4�

Here, ~1 is the m-dimensional column vector whose entries are equal to 1.

De®nition 2.1. A pair �p; c� is an admissible strategy if p is an fFtg-adaptedstochastic process that satis®es

PfEt A �0;y� : jp�t�jUMg � 1;

where M <y is a constant that may vary from policy to policy, and c is a non-negative fFtg-adapted stochastic process that satis®es

P ET A �0;y� :

�T

0

ct dt <y

� �� 1:

We also want that the wealth X determined by the strategy �p; c� satis®es

PfEt A �0;y� : X�t�V 0g � 1:

We shall denote by A1�x� the class of admissible strategies.

Let us de®ne the time of bankruptcy by

t :� infftV 0 : X�t� � 0g:

We observe that �c; p� A A1�x� only if PfEt A �t;y� : ct � 0; pt � 0g � 1, andhence PfEt A �t;y� : X �t� � 0g � 1.

Problem 2.1. The investor's objective is to maximize the utility of consumptionas measured by the quantity

Consumption-investment problems with transaction costs 45

J1��p; c�; x� :� E x

�y0

eÿdtu�ct� dt

� �: �5�

Here, d > 0 is a discount factor and u is a utility function. That is, u is a func-tion that is strictly increasing and strictly concave. The value function is de®nedby

V�x� :� sup�p; c� AA1�x�

J��p; c�; x�: �6�

We shall denote the optimal policy by �p; c�.

We shall assume that the utility function u is de®ned by

u�c� � cg

g; g A �ÿy; 1� ÿ f0g; �7�

and the parameters of the model satisfy

d > g r� 1

2�mÿ r~1��SS��ÿ1�mÿ r~1� 1

�1ÿ g��

: �8�

The following result was obtained by Merton (1969, 1971, 1973).

Theorem 2.1. Suppose that the conditions (7) and (8) are satis®ed. Then, theoptimal consumption and the optimal portfolio are given by

c�t� � 1

1ÿ gdÿ grÿ g

2�1ÿ g� �mÿ r~1��SS��ÿ1�mÿ r~1��

X�t� �9�

p�t� � �SS��ÿ1�mÿ r~1� 1

�1ÿ g� : �10�

Furthermore, the value function is given by

V�x� � 1

g

1

1ÿ gdÿ grÿ g

2�1ÿ g� �mÿ r~1��SS��ÿ1�mÿ r~1�� gÿ1

xg: �11�

To compare this result with the results to be presented in the next sections, letus consider the special case in which there is only one stock (that is, m � 1).Let us denote by �B�t�;S�t�� the total amount of money that the investorinvests in the bond and the stock, respectively, so that X �t� � B�t� � S�t�represents the wealth of the investor at time t. We then observe (see (10)) thatis optimal to invest the same constant fraction of wealth in the stock at anyinstant in time. In addition, the investor should consume (see (9)) at a rate thatis proportional to the wealth of the investor.

The proportion p is called the Merton proportion. Let us denote by�B�t�; S�t�� the optimal amount of money that the investor invests in the bondand the stock, respectively. Since p does not depend on time, for every tV 0,

46 A. Cadenillas

S�t�B�t� �

p�t�1ÿ p�t� �

p

1ÿ p

is a constant. This means that the stochastic process �B�t�; S�t�� lies in thestraight line through the origin with slope p=�1ÿ p� for any instant in time.This is called the Merton line.

We observe that any attempt to apply the above strategy in the presence oftransaction costs would result in ruin, since incessant trading would be neces-sary to hold the portfolio on the Merton line. Hence, a diërent strategy mustbe considered in the presence of transaction costs.

The problem of maximizing utility from consumption, when the prices ofthe ®nancial assets are given by (1)±(2) and there are not transaction costs,was ®rst studied by Merton (1969, 1971, 1973). Previous or contemporarywork in discrete-time include the models of Arrow (1971), Lintner (1965),Markowitz (1952, 1959), Mossin (1968), Samuelson (1969), and Tobin (1965).

The work of Merton was extended and given a more rigorous mathemati-cal treatment by Karatzas, Lehoczky, Sethi, and Shreve (1986) and Sethi andTaksar (1988). This work inspired new research in this area, but it is impossi-ble to mention all that work in this limited space. The interested reader mayconsult Karatzas and Shreve (1998) or Sethi (1997) for diërent extensions ofthe Merton problem that do not include transaction costs.

In sections 4 and 5 we shall consider some versions of Problem 2.1 inwhich there are transaction costs.

3 Maximizing the growth rate without transaction costs

Let us consider the same ®nancial market as in the previous section. That is, a®nancial market in which there is one bond and m stocks with prices given byequations (1)±(2).

Let us also consider an economic agent who invests in the above m� 1assets. We shall denote by p� � �p1; . . . ; pm� the proportions of his totalwealth that he invests in each of the m stocks. In this model, we are not goingto consider consumption as in the previous section. Hence, the wealth X of theinvestor is the solution of the stochastic diërential equation (3) with c�t� � 0,or equivalently

Xt � X0 exp

� t

0

�1ÿ ~1�ps�r� p�s mÿ 1

2p�s SS�ps

� �ds�

� t

0

p�s S dWs

� ��12�

De®nition 3.1. An fFtg-adapted stochastic process p is an admissible strategy if

P Et A �0;y�; i A f1; 2; . . . ;mg : pi�t�>0 and ~1�p�Xm

i�1

pi�t� < 1

( )�1: �13�


This means that we are going to consider only those strategies in which thebond and all the stocks are active.


Problem 3.1. The investor's objective is to maximize the long run growth rate ofhis investment, as measured by the quantity

J2�p� :� lim infT!y

E�log X �T��T

: �14�

The optimal long run growth rate is de®ned by

R :� supp AA2

J2�p� �15�

This problem is presented in Morton and Pliska (1995). Applying Kuhn-Tucker's theorem, they obtained the following result.

Theorem 3.1. The optimal portfolio is given by

p :� �SS��ÿ1�mÿ r~1�; �16�

and the optimal growth rate by

R :� �1ÿ ~1�p�r� p�mÿ 12 p�SS�p: �17�

We observe that the optimal portfolio given by (16) is constant, so the optimalstrategy is qualitatively similar to the one described in the previous section.That is, the optimal strategy consists of incessant trading to keep the propor-tions invested in the m stocks equal to the constant vector on the right handside of (16). If there were transaction costs, then this strategy would lead tobankruptcy.

We shall return to Problem 3.1 in sections 6 and 7, but assuming that thereare transaction costs.

4 Proportional transaction costs

In this section, we shall incorporate proportional transaction costs in themodel described in section 2. Nevertheless, we are going to assume that thereis only one stock. Thus, let us consider a ®nancial market with one bond andone stock with prices modeled by equations (1)±(2) with m � 1. We shallsuppose that purchasing and selling the stock result in transaction coststhat are proportional to the actual amount traded. That is, the investorpays fractions l and a of the amount transacted, on purchase and sell of thestock respectively. We assume that the transaction costs and the consumptionare deducted from the bond holdings. Thus, the equations describing theevolution of the bond and stock holdings are

dB�t� � �rBt ÿ ct� dtÿ �1� l� dLt � �1ÿ a� dUt; B�0� � x �18�

dS�t� � mS�t� dt� sS�t� dWt � dLt ÿ dUt; S�0� � y; �19�

48 A. Cadenillas

where Lt and Ut represent the cumulative purchase and sale of the stock up totime t, respectively. We observe that purchase of dL units of stock requires apayment of �1� l� dL units of bond, while sale of dU units of stock realizesonly �1ÿ a� dU units of bond.

Let us de®ne now the solvency region

S :� f�x; y� A R2 : x� �1ÿ a�yV 0; and x� �1� l�yV 0g: �20�

The solvency region is the region where selling of all shares of the stock orclosing of the short position in the stock leads to non-negative bond holdingsafter transaction costs. According to DoleÂans-Dade (1976), equations (18)±(19) have a unique strong solution at least up to the bankruptcy time

t :� infftV 0 : �B�t�;S�t�� B Sg:

De®nition 4.1. A triple �c;L;U� is an admissible strategy if L and U are right-continuous and nondecreasing fFtg-adapted stochastic processes with L0 �U0 � 0. Furthermore the consumption c is a non-negative fFtg-adapted sto-chastic process that satis®es

P ET A �0;y� :

�T

0

ct dt <y

� �� 1:

We also want that

PfEt A �0;y� : �B�t�;S�t�� A Sg � 1:


For instance, an admissible strategy is one that does not jump outside Sand sell all the stock holdings and stop consumption the ®rst time that theboundary of S is reached.

Problem 4.1. The investor's objective is to maximize the utility of consumptionas measured by the quantity

J3��c;L;U�; x� :� E x

�y0

eÿdtu�ct� dt

� �; �21�

where u is a utility function. The value function is de®ned by

V�x� :� sup�c;L;U� AA3�x�

J3��c;L;U�; x�: �22�

We shall denote the optimal strategy by �c; L; U�.

This is a stochastic control problem in which c is a classical stochastic control,and L and U are stochastic singular controls (see chapter 8 of Fleming andSoner (1993) or Cadenillas and Haussmann (1994) for theory that covers themixed classical-singular stochastic control problem). As in section 2, we shall


suppose that the utility function u is de®ned by

u�c� � cg

g; g A �ÿy; 1� ÿ f0g; �23�

and the parameters of the model satisfy

d > g r� 1

2

mÿ r

s

� �2 1

�1ÿ g��

: �24�

Let us de®ne the constants

b1 :� ÿ 12 s2g�1ÿ g� � mgÿ d;

b2 :� s2�1ÿ g� � rÿ m;

b3 :� 12 s2:

Davis and Norman (1990) applied Dynamic Programming to obtain thefollowing result.

Theorem 4.1. Assume that conditions (23)±(24) are satis®ed. Suppose thatthere are constants A, B, xmin, xmax, and a function c : �ÿ�1ÿ a�;y� 7! R suchthat

0 < xmin < xmax <y; �25�

c A C 2 with c 0�x� > 0 for x A �ÿ�1ÿ a�;y�; �26�

c�x� � 1

gA�x� 1ÿ m�g for xU xmin �27�

b3x2c 00�x� � b2xc 0�x� � b1c�x� � 1ÿ g

g

� ��c 0�x��ÿg=�1ÿg� � 0

for xmin U xU xmax; �28�

and

c�x� � 1

gB�x� 1� l�g for xV xmax: �29�

Let us denote the regions

BR :� f�x; y� A S : yU xÿ1maxxg; �30�

SR :� f�x; y� A S : yV xÿ1minxg; �31�

NT :� f�x; y� A S : xÿ1maxxU yU xÿ1

minxg: �32�

50 A. Cadenillas

De®ne for every �x; y� A NTÿ f�0; 0�g

c��x; y� :� y�c 0�x=y��ÿ1=�1ÿg�: �33�

Let �c; L; U� be the strategy de®ned for t < t by

c�t� :� c��B�t�;S�t��; �34�

dB�t� � frBt ÿ c��B�t�;S�t��g dtÿ �1� l� dLt � �1ÿ a� dUt;

B�0� � x; �35�

dS�t� � mS�t� dt� sS�t� dWt � dLt ÿ dUt; S�0� � y; �36�

Lt �� t

0

If�B�r�; S�r�� A qBRg dLr �37�

Ut �� t

0

If�B�r�; S�r�� A qSRg dUr: �38�

Then, such �c; L; U� exists and is optimal. The value function is given by

v�x; y� � ygcx

y

� �: �39�

Theorem 4.1 says that the optimal strategy consists in dividing the solvencyregion S into three regions: the buying region BR, the selling region SR,and the no transaction region NT. Each region is a wedge. If at time t � 0,the pair �B�t�;S�t�� is in BR or SR, then the investor should make aninstantaneous ®nite transaction to the boundary of NT. In the interior ofthe no transaction region NT, the investor consumes but does not makeany transaction. However, when the pair �B�t�;S�t�� reaches the boundarybetween BR and NT, the investor buys stock to keep �B�t�;S�t�� containedin NT. Similarly, when the pair �B�t�;S�t�� reaches the boundary betweenSR and NT, the investor sells stock to keep �B�t�;S�t�� contained in NT.We can also say that the optimal strategy consists of doing minimal trading tokeep the fraction of wealth invested in the stock

p�t� � S�t�B�t� � S�t�

in the no transaction interval ��1� xmax�ÿ1; �1� xmin�ÿ1�.The initial work on the maximization of expected utility from consumption

when there are proportional transaction costs was done by Magill andConstantinides (1976). Even though they used only heuristic methods, theyobtained some qualitative properties of the optimal solution. Indeed, inthe Introduction of their paper, they write that in the optimal strategy ``theinvestor trades in securities when the variation in the underlying securityprices forces his portfolio proportions outside a certain region about the


optimal proportions in the absence of transaction costs.'' In a companionpaper, Magill (1976) studied mutual funds under transaction costs. He showedthat when proportional transaction costs are introduced in the Merton model,``a mutual fund can be formed such that individual investors prefer investmentthrough the mutual fund to individual investment on the capital market.''Davis and Norman (1990) then applied the theory of stochastic singularcontrol to the problem of maximizing expected utility from consumptionwhen there are proportional transaction costs. They made a rigorous analysisof that problem, providing an algorithm and numerical computations of theoptimal strategy. The model presented in this section was developed in theirpaper. They also gave su½cient conditions for the existence of a solution tothe free boundary problem (27)±(29), and showed a numerical example inwhich the no transaction interval ��1� xmax�ÿ1; �1� xmin�ÿ1� contains theMerton proportion p of section 2. In addition, they studied the case of loga-rithmic utility, but we have omitted the presentation of their results for thatcase, because the form of the solution is similar to the one corresponding to thepower utility given by (23). Shreve and Soner (1994) studied the same problemapplying the theory of viscosity solutions to Hamilton-Jacobi-Bellmannequations (see, for instance, Fleming and Soner (1993) for that theory). Theyremoved some of the assumptions of Davis and Norman (1990), studied thecases in which the Merton proportion p is outside the no transaction interval��1� xmax�ÿ1; �1� xmin�ÿ1�, and analyzed the liquidity premium. The liquid-ity premium under proportional transaction costs has also been studied byConstantinides (1986). Shreve, Soner and Xu (1991) have studied the problemwith two bonds under proportional transaction costs. Zariphopoulou (1992)has considered the problem of maximizing expected utility from consumptionwhen the price of the stock is given by a continuous time Markov chain.

Akian, Menaldi and Sulem (1996) have considered the more realisticproblem in which there are a ®nite number of stocks, but assuming that thenoise terms are uncorrelated (that is, the matrix S of section 2 is the identitymatrix). They characterized the value function as the unique viscosity solutionof a variational inequality. They solved numerically that variational inequal-ity, but without proving convergence. Collings and Haussmann (1998) studiedthe same problem, but allowing the noise terms to be correlated. They alsoshowed that the value function is the unique viscosity solution of a variationalinequality. In addition, they applied the method of the Markov chainapproximation (see, for instance, Kushner and Dupuis (1992)), to construct anumerical approximation, and proved the convergence of the approximation.

The consumption-investment problem with proportional transaction costsand random market coe½cients has been studied by CvitanicÂ and Karatzas(1996). They considered a ®nite horizon, together with the objective of max-imizing expected utility from terminal wealth. Their solution was given interms of the solution to a dual problem, whose existence has recently beenproved by CvitanicÂ and Wang (1999).

The model described in this section has also been applied to pricecontingent claims under proportional transaction costs. That idea wasinitiated by Hodges and Neuberger (1989), and extended by Davis, Panasand Zariphopoulou (1993), Davis and Zariphopoulou (1995), CvitanicÂand Karatzas (1996), Barles and Soner (1998), and Constantinides andZariphopoulou (1999). Other papers on option pricing under proportionaltransaction costs include Davis and Clark (1994), Dewynne, Whalley and

52 A. Cadenillas

Wilmott (1994), Soner, Shreve and CvitanicÂ (1995), CvitanicÂ, Pham and Touzi(1999), Levental and Skorohod (1997), Leland (1985), Hoggard, Whalley andWilmott (1994), Avellaneda and ParaÂs (1994), Kabanov and Safarian (1997),Whalley and Wilmott (1997), and Kabanov (1999).

The maximization of expected growth rate under proportional transactioncosts was ®rst studied by Taksar, Klass and Assaf (1988). They were the ®rstones in applying the theory of stochastic singular control to a consumption-investment problem with transaction costs. Roughly speaking, they assumedthat the dynamics of the bond and stock holdings were similar to (18)±(19),but their objective was to maximize the criterion of (14). The solutionobtained by Taksar, Klass and Assaf (1988) has the same qualitative propertyas the one obtained by Davis and Norman (1990): the optimal strategy con-sists of keeping the proportion invested in the stock within a certain intervalwith minimal e¨ort. Dumas and Luciano (1991) and Fleming, Grossman, Vilaand Zariphopoulou (1990) have also studied similar models. Duncan, Faul,Pasik-Duncan and Zane (1994a,1994b) and Duncan, Pasik-Duncan and Zane(1995) have studied the models of Taksar, Klass and Assaf (1988) and Davisand Norman (1991) when some of the parameters are unknown.

Grossman and Laroque (1990) and Cuoco and Liu (1998) have studiedmodels with proportional transaction costs for changes in level of consump-tion. Weerasinghe (1998) has considered a model with proportional transac-tion costs, and the objective of maximizing the probability of reaching a ®xedpositive total wealth before bankruptcy.

5 Fixed and proportional transaction costs

Although transaction costs are included in the model of the previous section,the results still do not re¯ect what happens in practice. Indeed, the optimalstrategy in the problem of the previous section involves doing in®nitesimalsmall transactions. In the presence of ®xed transaction costs, such strategywould lead to ruin.

Let us a consider a ®nancial market like the one described in section 2, sothe price dynamics of the bond and stock are given by equations (1)±(2).Then, the dynamics of the bond holdings and stock holdings between trans-action times are given by

dB�t� � B�t�r dt

dS�t� � S�t�fm dt� s dWtg:

Now, suppose that at each transaction time ti, the investor must pay thetransaction cost

K � kjDSij;

where K is a ®xed transaction cost, k is a proportional transaction cost, andDSi is the change in stock holdings at time ti. We assume that transactioncosts and consumption must be paid from the bond holdings. Thus,

Bi � Biÿ1 ÿ DSi ÿ K ÿ kjDSij ÿ ci;


where Bi and ci are the value of the bond holdings and the consumptionimmediately after the transaction time ti. We observe that the decisions of theinvestor are completely described by the sequence f�tn;DSn; cn�; n A Ng.

De®nition 5.1 (Admissible strategies). An admissible strategy is a sequence�T;DS;C� � f�tn;DSn; cn�; n A Ng, such that tn is a stopping time with respectto the ®ltration

s��B�sÿ�;S�sÿ��; 0U sU t; �ti;DSi; ci�; i < n�; tV 0;

and �DSn; cn� is measurable with respect to the s-®eld

s��B�tnÿ�;S�tnÿ��; �DSi; ci�; i < n�:

Furthermore,

PfEn A N : 0U tn U tn�1g � 1;

PfEn A N : cn V 0g � 1;

PfEt A �0;y� : B�t�V 0;S�t�V 0g � 1;

and

ET A �0;y� : P limn!y

tn UTn o

� 0:


Problem 5.1. The investor wants to select the admissible strategy �T;DS;C�that maximizes the functional J4 de®ned by

J4��T;DS;C�; x� :� E xXyn�1

eÿltn u�cn�Iftn<yg

" #; �40�

where u is a utility function. That is, u is a strictly increasing and strictly con-cave function. The value function is de®ned by

V�x� :� sup�T;DS;C� AA4�x�

J4��T;DS;C�; x�: �41�

We shall denote the optimal strategy by �T;dDS; C�.

In the stochastic control literature, the above problem is called a stochasticimpulse control problem (see, for instance, Bensoussan and Lions (1982)or Korn (1997a) for references on that theory, and Korn (1999) for a recentsurvey on its applications to Mathematical Finance). The maximum utilityoperator M of a function v : �0;y�2 7! R is de®ned by

54 A. Cadenillas

Mv�B;S� :� supfv�Bÿ DS ÿ K ÿ kjDSj ÿ c;S � DS�

�U�c� : �c;DS� A ~A�B;S�g;where

~A�B;S� :� fcV 0;Bÿ DS ÿ K ÿ kjDSj ÿ cV 0;S � DS V 0g:

MV�B;S� represents the value of the strategy that consists in choosing thebest consumption and change in stock holdings, when the bond and stockholdings are B and S, respectively.

Let us consider the operator L de®ned by

Lc�B;S� :� 12 s2S2cSS�B;S� � mScS�B;S� ÿ lc�B;S�:

Now we intend to ®nd the value function and an associated optimal strategy.Suppose there exists an optimal strategy for each initial point. Then, if the

process starts at �B;S� and follows the optimal strategy, the utility associatedwith this optimal strategy is V�B;S�. On the other hand, if the process startsat �B;S�, selects the best transaction (that is, selects the best consumption andchange in stock holdings), and then follows an optimal strategy, then theutility associated with this second strategy is MV�B;S�. Since the ®rst strategyis optimal, its utility is greater or equal than the utility associated with thesecond strategy. Furthermore, these two utilities are equal when it is optimalto consume and change the stock holdings. Hence, V�B;S�VMV�B;S�, withequality when it is optimal to make a transaction. In the continuation region,that is, when the investor does not consume or change the stock holdings, wemust have LV�B;S� � 0 (this is an heuristic application of the dynamicprogramming principle to the problem we are considering). These intuitiveobservations can be applied to give a characterization of the value function.We formalize this intuition in the next two de®nitions and theorem.

De®nition 5.2 (QVI). We say that a function v : �0;y�2 7! �0;y� satis®es thequasi-variational inequalities for Problem 5.1 if for every �B;S� A �0;y�2,

Lv�B;S�U 0;

v�B;S�VMv�B;S�;

�v�B;S� ÿMv�B;S��Lv�B;S� � 0:

We observe that a solution v of the QVI separates the region �0;y�2 into twodisjoint regions: a continuation region

C :� f�B;S� A �0;y�2 : v�B;S� > Mv�B;S� and Lv�B;S� � 0g

and an intervention region

S :� f�B;S� A �0;y�2 : v�B;S� �Mv�B;S� and Lv�B;S� < 0g:


From a solution to the QVI it is possible to construct the following impulsestrategy.

De®nition 5.3. Let v be a solution of the QVI. Then the following impulse con-trol is called the QVI-strategy determined by v:

�t0;DS0; c0� � �0; 0; 0�;

and, for every n A N:

tn :� infftV tnÿ1 : v�B�t�;S�t�� Mv�B�t�;S�t��g

�DSn; cn� :� arg supfv�B�tn� ÿ DS ÿ K ÿ kjDSj ÿ c;S�tn� � DS�

�U�c� : �c;DS� A ~A�B�tn�;S�tn��g

This means that the investor makes a transaction whenever v and Mv coincideand the size of the consumption and change in stock holdings is the solution tothe optimization problem corresponding to Mv�B;S�.

The following result was obtained by Korn (1998).

Theorem 5.1. Let v A C1��0;y�2; �0;y�� be a solution of the QVI that satis®esItoÃ's formula in a ``certain sense'' (see Korn (1998)). Suppose that v satis®esthe growth conditions

E

�y0

feÿltS�t�vS�B�t�;S�t��g2 dt

� �<y:

and

limT!y

E�eÿlT v�B�T�;S�T�� 0:

for all stochastic processes �B�t�;S�t�� determined by admissible impulse strat-

egies. Then, for every �B;S� A �0;y�2,

V�B;S�U v�B;S�:

Furthermore, if the QVI-strategy corresponding to v is admissible then it is anoptimal impulse strategy, and for every �B;S� A �0;y�2

V�B;S� � v�B;S�:

Theorem 5.1 says that the solution to the problem of maximizing expectedutility from consumption, when there are ®xed and proportional transactioncosts, can be characterized by a system of quasi-variational inequalities.

In section 1.6.2 of Bensoussan and Lions (1982), there is a reference to apotential application of the theory of stochastic impulse controls to a portfoliomanagement problem. They characterize the value function as a solution to asystem of quasi-variational inequalities, but there is no indication about thesolution of that system of inequalities.

56 A. Cadenillas

The ®rst application of the theory of stochastic impulse controls to a con-sumption-investment problem was done by Eastham and Hastings (1988),who studied a ®nite horizon version of the problem described in this section.They developed a general theory, and obtained an approximate solution for asimple problem in which the appreciation rate of the stock and the interestrate of the bond were in the relation m < r, the discount factor was l � 0, theutility function u was linear, and the number of available shares of the stockwas bounded. That initial work was extended by Hastings (1992), whoallowed jumps (generated by a Poisson process) in the price of the stock,m > r, l > 0, and no bounds on available shares of the stock. He obtainedmany properties of the optimal strategy, but did not give numerical examples.Korn (1998) studied the in®nite horizon version of the Eastham and Hastingsmodel. He presented not only an approach based on quasi-variationalinequalities, but also a formal optimal stopping approach. In addition, hepresented some ®nite horizon models, and obtained an asymptotic solution tothe problem of maximizing exponential utility of terminal wealth. The modeldescribed in this section is the one developed by Korn (1998) (see also section5.2 of Korn (1997b)). Cadenillas (1999) has solved a problem of maximizingexpected utility from consumption in an in®nite horizon model with one stockand no bond.

6 Transaction cost as a ®xed proportion of wealth

Let us consider the same ®nancial market of section 3, but assume thatwhenever a trade is made, the investor has to pay a transaction cost equal to a®xed fraction a of his entire portfolio value.

Suppose that at the initial time t0 � 0, the investor allocates hisinitial wealth X�0� in the proportions p0 :� p�t0�. That is, he invests

�1ÿPmi�1 p0

i �X�0� dollars in the bond and p0i X �0� dollars in the i-th stock

�1U iUm�. He then holds this portfolio until he decides to make a transac-tion at time t1. At time t1 he pays the transaction fee, so that X �t1� ��1ÿ a�X�t1ÿ�, and allocates his wealth X �t1� in the proportions p1 :� p�t1�.He then holds this portfolio until he decides to trade at time t2 choosingp2 :� p�t2�, etcetera.

With X�tn� denoting the portfolio value after the n-th transaction fee hasbeen paid, we have

X�tn� � �1ÿ a�X �tnÿ1�

� �1ÿ ~1�pnÿ1� expfr�tn ÿ tnÿ1�g �Xm

i�1

pnÿ1i

Si�tn�Si�tnÿ1�

!

� �1ÿ a�nX�0�Yn

j�1

� �1ÿ ~1�p jÿ1� expfr�t j ÿ tjÿ1�g �Xm

i�1

pjÿ1i

Si�t j�Si�tjÿ1�

!:


De®nition 6.1. An admissible strategy is a sequence �T;P� � f�tk; pk�; k A Ng,

where 0 � t0 U t1 U � � � U tk U tk�1 U � � � are stopping times with respect tothe ®ltration fFt; t A �0;y�g and each pk is Ftk

-measurable. Furthermore,

P Et A �0;y�; k A N : pki �t� > 0 and ~1�pk �

Xm

i�1

pki �t� < 1

( )� 1:


Problem 6.1. The investor's objective is to maximize the long run growth rate ofhis investment, as measured by the quantity

J5�T;P� :� lim infT!y

E�log X �T��T

: �42�

The optimal long run growth rate is de®ned by

R :� sup�T;P� AA5

J5�T;P� �43�

Let us denote the m-dimensional region

�0; 1�m1 :� p A �0; 1�m :Xm

i�1

pi < 1

( )

and de®ne the function g : W� �0;y� � �0; 1�m1 7! R by

g�t; p��o� :� log�1ÿ a� � log

(�1ÿ ~1� p� expfrtg

�Xm

i�1

pi exp mi ÿ1

2

Xm

j�1

s2ij

!t�Xm

j�1

sijW j�t��o�( ))

Let us denote by S the class of stopping times with respect to the ®ltrationfFt; t A �0;y�g and

~S :� ft A S : E�t� <yg:

The following result was obtained by Morton and Pliska (1995). It says thatthe original problem (43) is equivalent to problem (44), which consists ofselecting only one constant for the proportion and only one stopping time.

Proposition 6.1. Let the pair �t; p� be a solution of the problem

supt A ~S;p A �0;1�m

1

E�g�t; p��E�t� : �44�

58 A. Cadenillas

Then

R � supt A ~S; p A �0;1�m

1

E�g�t; p��E�t� ; �45�

or equivalently

0 � supt A ~S;p A �0;1�m

1

fE�g�t; p�� ÿ RE�t�g: �46�

Furthermore, the optimal strategy is to choose, for every n A N, pn � p andtn ÿ tnÿ1 as the solution of problem (46).

Let us denote by f�~pt;Ft�; t A �0;y�g the m-dimensional adapted stochasticprocess that represents the fraction of wealth held by the investor in each ofthe m assets at time t. Applying ItoÃ 's formula, we observe that the risky frac-tion process ~p satis®es the following stochastic di¨erential equation

d ~p�t� � Diag�~p�t��I ÿ ~1~p��t��mÿ r~1ÿ SS�~p�t�� dt� S dW �t��;

where I is the m by m identity matrix, and Diag�~p�t�� is an m by m matrix withthe elements of ~p�t� on the diagonal. Then, it is possible to transform problem(46) into the equivalent problem

0 � supp A �0;1�m

1

flog�1ÿ a� � log�1ÿ ~1� p� � fR�p�g; �47�

where

fR�p� :� supt A ~S

fÿE p�log�1ÿ ~1�~pt�� ÿ �Rÿ r�E p�t�g:

Let us consider now the operator L de®ned by

Lf�x� :� 1

2

Xm

i�1

Xm

j�1

q

qxiqx jf�x�xix j��e�i ÿ x��SS��e j ÿ x��

�Xm

i�1

q

qxif�x�xi��e�i ÿ x��mÿ r~1ÿ SS�x��:

The following result was obtained by Morton and Pliska (1995).

Theorem 6.1. The value function fR is the smallest function f in the domain ofthe generator L that satis®es for every p

f�p�Vÿlog�1ÿ ~1� p� and Lf�p�URÿ r: �48�

Furthermore, the optimal stopping time is


t � infftV 0 : fR�~p�t�� ÿlog�1ÿ ~1�~p�t��g:

� infftV 0 : ~p�t� B Cg �49�

where the continuation region is

C :� fp A �0; 1�m1 : fR�p� > ÿlog�1ÿ ~1�p�g: �50�

Proposition 6.1 and Theorem 6.1 say that the optimal strategy is characterizedby the constant vector of portfolio proportions p together with the stoppingtime t of Theorem 6.1. Indeed, once the function fR is known, the investorshould make a transaction the ®rst time that the vector ~p of proportions in-vested in the risky assets leaves the continuation region C. At that time, theinvestor should rebalance his portfolio, so that the new vector of proportionsis the solution p of problem (47). Morton and Pliska (1995) proved that thecontinuation region C is nonempty and, as expected, contains the optimalvector of proportions p of section 3 when there are not transaction costs.However, they obtained fR explicitly only in the case of one stock �m � 1�. Inparticular, they presented the following example.

Example 6.1. Let us consider a numerical example in which r � 0:07, m �0:182, s � 0:4, and a � 0:001. Then, the optimal proportion p is slightlyhigher than p � 0:7, and the continuation region is C � �0:540; 0:837�.

In addition, Pliska and Selby (1994) and Morton and Pliska (1995) presenteda numerical approximation for the case of two stocks, while Atkinson andWilmott (1995) presented an asymptotic analysis for the case of a ®nite num-ber of stocks when the transaction cost parameter a was small.

Bielecki and Pliska (2000) have generalized the above model in threedi¨erent directions. First, they have considered a transaction cost structurethat includes not only the ®xed proportion of portfolio value model of thissection, but also the transaction costs models of sections 4 and 5. Second, theyhave allowed the prices of the securities to be a¨ected by economic factorssuch as dividend yields, interest rates, unemployment rates, etc. Third, theyhave considered a risk sensitive criterion which is more general than the longrun growth rate criterion of Problem 6.1 (see, for instance, Whittle (1990) forthe theory of risk-sensitive control). Under these assumptions, they haveproved that the optimal strategy can be obtained by solving a system of quasi-variational inequalities.

Du½e and Sun (1990) have also considered a model in which, at eachtransaction, the investor is charged a ®xed proportion of his current wealth.Their objective is to maximize expected utility from consumption. However,they assume that the investor observes his current wealth only when making atransaction.

7 Taxes and transaction costs

Let us consider a ®nancial market in which there is only one stock, with pricemodeled by a geometric Brownian motion

60 A. Cadenillas

dPt � Ptfm dt� s dWtg: �51�

We shall suppose that the long-run growth rate l of the stock satis®es

l :� mÿ 1

2s2 > 0: �52�

We de®ne the stochastic process Y by

Yt :� Pt

P0� exp mt� sWt ÿ 1

2s2t

� �:

We shall assume that, whenever a stock is sold, the proportion a times thevalue of the stock is paid as the transaction cost, and that taxes must be paidat a rate b. To model this, suppose an investor purchases one share of stock attime 0. If t denotes the amount of time that the investor owns this stock, thenthe after-tax pro®t per share is f�1ÿ a�Pt ÿ P0g�1ÿ b�, the after-tax returnover t is f�1ÿ a��Pt=P0� ÿ 1g�1ÿ b�, and the factor by which his wealth isincreased during �0; t� is

1� f�1ÿ a��Pt=P0� ÿ 1g�1ÿ b� � b � �1ÿ b��1ÿ a�Pt

P0

� b � �1ÿ b��1ÿ a�Yt:

Now suppose that the investor wants to select a sequence T� ftn; n A N0gof transactions times 0 � t0 < t1 < t2 < � � � : The initial funds, denoted by X0,are all invested in the ®rst stock at time t0. At time t1, this position is closedout and all the after-tax proceeds are invested in the second stock. Thistransaction cycle is repeated at times t2; t3; . . . :

Let the random variable

Mi :� b � �1ÿ b��1ÿ a�Yti=Ytiÿ1

represent the multiplicative factor by which the value of the investment in-creases over the i-th transaction cycle, so that X0M1M2 . . . Mi is the value ofthe investment just after the i-th transaction. We shall denote by X�t� thewealth of the investor at time t.

Problem 7.1. The investor wants to select the sequence of transaction times Tthat maximize the long-run growth rate of his investment, as de®ned by

J6�T� :� lim infT!y

E�log X�T��T

: �53�

It is convenient to de®ne the function g : �0;y� 7! R by

g�y� :� log fb � �1ÿ b��1ÿ a�yg:


As in the previous section, we shall denote by ~S the class of stopping timeswith positive but ®nite expectation. Applying renewal theory, it is possible toprove the following result.

Lemma 7.1. The optimal time between transactions has the same probabilitydistribution that t A ~S that maximizes

y � supt A ~S

E�logfb � �1ÿ b��1ÿ a�Ytg�E�t� � sup

t A ~S

E�g�Yt��E�t� : �54�

This means that the investor needs to solve only one problem of optimalstopping.

Since there is only one stock available for investment, it is clear that if theinvestor does not make any transactions, then the long-run growth rate of hisinvestment is equal to l, the long-run growth rate of the stock. Such strategy iscalled a buy-and-hold strategy. Since we are considering an in®nite horizon, itis possible to prove that it gives the same growth rate as any cut-losses-short-and-let-pro®ts-run strategy, i.e. any strategy of the form x�a� :� infftV 0 :X �t� B �a;y�g, where 0 < a < 1. Cadenillas and Pliska (1999) have provedthat any of those equivalent strategies are optimal under the followingconditions.

Theorem 7.1. Suppose that b � 0, or that mV s2 and b > 0. Then, for everyt A ~S,

E�g�Yt��E�t� U l;

with strict inequality when a A �0; 1�.

Now that we know the solution for the restrictive assumptions of Theorem7.1, let us consider from now on the interesting case in which

0 < a < 1; 0 < b < 1; and 12 s2 < m < s2: �55�

To solve problem (54) under this more realistic assumption, it is convenient tonote that (54) is equivalent to

Problem 7.2. First, for each yV l, solve the optimal stopping problem withvalue

H�y� :� supt A ~S

E�g�Xt� ÿ yt�: �56�

Second, select yV l such that

H�y� � 0: �57�

To apply the theory of Dynamic Programming to Problem 7.2, we de®ne,for each yV l, the function s : �0;y� 7! R by

62 A. Cadenillas

s�x� � sy�x� :� supt A ~S

E x�g�Xt� ÿ yt�:

We observe that selecting y satisfying (57) is the same as choosing y satisfying

sy�1� � 0:

Let us denote the continuation and stopping region, respectively, by

C :� fx A �0;y� : s�x� > g�x�g

S :� fx A �0;y� : s�x� � g�x�g:

According to the theory of optimal stopping, (see, for instance, Shiryaev(1978)) the function s is a solution of the following free boundary problem:

mxds

dx� 1

2s2x2 d 2s

dx2� y if x A C �58�

and

s�x� � g�x� if x A S: �59�

The general solution of the ordinary di¨erential equation (58) is

s�x� � c� dx1ÿ�2m=s2� � y

�mÿ 12 s2� log x;

where c and d are real numbers.If we conjecture that the smooth-®tting condition holds and that C �

�a; b�, where 0 < a < 1 < b <y, then we may ®nd y, c, d, a, and b from thefollowing equations:

s�1� � 0; �60�

s�a� � g�a�; �61�

s�b� � g�b�; �62�

s 0�a� � g 0�a�; �63�

s 0�b� � g 0�b�: �64�

Equations (63)±(64) are the smooth-®tting condition.The following result was proved by Cadenillas and Pliska (1999).

Theorem 7.2. Let us suppose that the parameters of the model satisfy condition(55). Let a, b, c, d, and y, where 0 < a < 1 < b <y, yV l, and c; d A R, be asolution of equations (60)±(64). We de®ne the function V by


V�y� :� f �y� if y A �a; b�,g�y� if y B �a; b�,

��65�

where

f �y� � c� dy1ÿ�2m=s2� � y

�mÿ 12 s2� log y: �66�

If the conditions

f �y�V g�y�; Ey A �a; b�; �67�

aUbfmÿ 2yÿ �m2 ÿ 2ys2�1=2g2�1ÿ a��1ÿ b��yÿ m� 1

2 s2� ; �68�

and

bVbfmÿ 2y� �m2 ÿ 2ys2�1=2g2�1ÿ a��1ÿ b��yÿ m� 1

2 s2� �69�

are satis®ed, then V is the value function. That is,

V�y� � supt A ~S

E y�g�Yt� ÿ yt�: �70�

Furthermore, the solution of Problem 7.2 (or equivalently, of Problem 7.1) is

t � t�a; b� :� infft A �0;y� : Yt B �a; b�g: �71�

Thus, Theorem 7.2 says that in the interesting case (55) it is optimal to cut-short-both-the-losses-and-the-pro®ts. This result is rather surprising, becauseit is di¨erent from the ones obtained in Constantinides (1983, 1984) andDammon and Spatt (1996), in which the optimal strategy is to cut-losses-short-and-let-pro®ts-run.

Cadenillas and Pliska (1999) have conjectured that the conditions (67)±(69) are irrelevant, because they considered many di¨erent numerical exam-ples, and in all of them those conditions were satis®ed. For instance,

Example 7.1. Let us consider a numerical example in which m � 0:065, s �0:3, a � 0:02, and b � 0:3. Then, the solution of the system (60)±(64) isa � 0:3032934964, b � 7:186001883, y � 0:02231148251, d � 0:9348432672,and c � ÿ0:9348432672. It is easy to verify that conditions (67)±(69) are sat-is®ed. Therefore, the solution of Problem 7.2, or equivalently Problem 7.1, isgiven by the stopping time of (71), with a, b, and y given above. We note thatl � mÿ 1

2 s2 � 0:02, so this trading strategy beats buy-and-hold as well as anycut-losses-short-and-let-pro®ts-run strategy.

The second surprising result obtained by Cadenillas and Pliska (1999) is that itmight be preferable to invest in a taxable world rather than in a non-taxableworld! They also present some heuristic interpretations of their results.

64 A. Cadenillas

8 Conclusions and open problems

From this survey, it is clear that a realistic consumption-investment modelmust include transaction costs. Nevertheless, although a lot of progress hasbeen made recently in this area, there are still many open questions.

We start by noticing that most of the results presented in this survey arelimited to the case of one bond and only one stock. It is then important to seeif these results can be extended to cover a realistic number of stocks (at leastthirty). Since there is little hope to solve explicitly a multi-stock version of theproblems described in the previous sections, it is then important to develope½cient numerical methods to handle a consumption-investment problemwith many stocks. In particular, it would be interesting to see if the existingfew numerical methods can be improved.

It is also important to see if the above models can be generalized to coverthe case of incomplete markets, or more generally of constrained consumptionand investment (see, for instance, chapter 6 of Karatzas and Shreve (1998) fora description of this problem without transaction costs).

Another interesting direction of research is the solution of consumption-investment problems in which the prices of the securities depend on economicfactors, as proposed by Bielecki and Pliska (2000). Their model include casesin which the parameters of the system are unknown, and also cases in which,even if the parameters are known, the prices of the stocks do not necessarilyfollow a geometric Brownian motion. They characterize the optimal strategyin terms of a system of quasi-variational inequalities, but there is not any in-dication about the solution of this system. It would be interesting to solve thatgeneral problem, or at least some particular cases. In this context, it wouldalso be interesting to solve problems with the risk-sensitive criterion.

It is still an open problem to incorporate bankruptcy reward/penalty in aproblem with transaction costs (see Sethi (1997) for some open problems onconsumption-investment with bankruptcy). In addition, it would be interest-ing to incorporate jumps in the prices of the securities (see, for instance,Eastham and Hastings (1988)).

Finally, the consumption-investment problems with transaction costsgenerate open problems in the areas of option pricing and equilibrium (see, forinstance, sections 2 and 4, respectively, of Karatzas and Shreve (1998)). Inparticular, it would be interesting to see if the utility maximization approachto price options with proportional transaction costs, could be extended toprice options with other structures of transaction costs. Finally, it is still anopen problem to determine the impact of transaction costs on capital marketequilibrium.

References

[1] Akian M, Menaldi JL, Sulem A (1996) On an investment-consumption model with transac-tion costs. SIAM J. Control and Optimization 34:329±364

[2] Arrow KJ (1971) Essays in the theory of risk-bearing. Markham Publishing Company,Chicago

[3] Atkinson C, Wilmott P (1995) Portfolio management with transaction costs: an asymptoticanalysis of the Morton and Pliska model. Mathematical Finance 5:357±367

[4] Avellaneda M, ParaÂs A (1994) Dynamic hedging portfolios for derivative securities in thepresence of large transaction costs. Applied Mathematical Finance 1:165±194


[5] Barles G, Soner HM (1998) Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance and Stochastics 2:369±397

[6] Bensoussan A, Lions JL (1982) ControÃ le impulsionnel et ineÂquations quasi variationneles.Dunod, Paris

[7] Bielecki T, Pliska S (2000) Risk sensitive asset management with transaction costs. Financeand Stochastics 4

[8] Cadenillas A (1999) Optimal consumption when there are ®xed and proportional transactioncosts. Preprint, Department of Mathematical Sciences, University of Alberta

[9] Cadenillas A, Haussmann UG (1994) The stochastic maximum principle for a singular con-trol problem. Stochastics and Stochastics Reports 49:211±237

[10] Cadenillas A, Pliska S (1999) Optimal trading of a security when there are taxes and trans-action costs. Finance and Stochastics 3:137±165

[11] Collings P, Haussmann UG (1998) Optimal portfolio selection with transaction costs. In:Proceedings of the Conference on Control of Distributed and Stochastic Systems. Kluwer

[12] Constantinides GM (1983) Capital market equilibrium with personal tax. Econometrica51:611±636

[13] Constantinides GM (1984) Optimal stock trading with personal taxes: implications for pricesand the abnormal January returns. Journal of Financial Economics 13:65±89

[14] Constantinides GM (1986) Capital market equilibrium with transaction costs. Journal ofPolitical Economy 94:842±862

[15] Constantinides GM, Zariphopoulou T (1999) Bounds on prices of contingent claims in anintertemporal economy with proportional transaction costs and general preferences. Financeand Stochastics 3:345±369

[16] Cuoco D, Liu H (1998) Optimal consumption of a divisible durable good. Journal of Eco-nomic Dynamics and Control, to appear

[17] CvitanicÂ J, Karatzas I (1996) Hedging and portfolio optimization under transaction costs: amartingale approach. Mathematical Finance 6:133±165

[18] CvitanicÂ J, Pham H, Touzi N (1999) A closed form solution to the problem of super-replication under transaction costs. Finance and Stochastics 3:35±54

[19] CvitanicÂ J, Wang H (1999) On optimal terminal wealth under transaction costs. Preprint,Department of Statistics, Columbia University

[20] Dammon R, Spatt C (1996) The optimal trading and pricing of securities with asymmetriccapital gains taxes and transaction costs. The Review of Financial Studies 9:921±952

[21] Davis MHA, Clark JMC (1994) A note on super-replicating strategies. Philosophical Trans-actions of the Royal Society of London 347:485±494

[22] Davis MHA, Norman A (1990) Portfolio selection with transaction costs. Mathematics ofOperations Research 15:676±713

[23] Davis MHA, Panas VG, Zariphopoulou T(1993) European option pricing with transactioncosts. SIAM J. Control and Optimization 31:470±493

[24] Davis MHA, Zariphopoulou T (1995) American options and transaction fees. In: Davis MHA,Du½e D, Fleming W, Shreve S (ed.) Mathematical Finance. Springer-Verlag, New York

[25] Dewynne JN, Whalley AE, Wilmott P (1994) Path dependent options and transaction costs.Philosophical Transactions of the Royal Society of London A347:517±529

[26] DoleÂans-Dade C (1976) On the existence and unicity of solutions of stochastic integralequations. Zeitschrift fuÈr Wahrscheinlichkeitstheorie und verwandte Gebiete 36:93±101

[27] Du½e D (1996) Dynamic asset pricing theory. Second Edition. Princeton University Press,Princeton (New Jersey)

[28] Du½e D, Sun TS (1990) Transaction costs and portfolio choice in a discrete-continuous-timesetting. Journal of Economic Dynamics and Control 14:35±51

[29] Dumas B, Luciano E (1991) An exact solution to a dynamic portfolio choice problem undertransaction costs. The Journal of Finance 46:577±595

[30] Duncan TE, Faul M, Pasik-Duncan B, Zane O (1994a) On the stochastic adaptive control ofan investment model with transaction fees. Ulam Quarterly 4:1±15

[31] Duncan TE, Faul M, Pasik-Duncan B, Zane O (1994b) Computational methods for the sto-chastic adaptive control for an investment model with transaction fees. Proceedings of the33rd IEEE Conference on Decision and Control: 2813±2816

[32] Duncan TE, Pasik-Duncan B, Zane O (1995) Numerical methods for the stochastic adaptivecontrol of an investment model and consumption model with transaction fees. Proceedings ofthe 34rd IEEE Conference on Decision and Control: 3360±3365

66 A. Cadenillas

[33] Eastham J, Hastings K (1988) Optimal impulse control of portfolios. Mathematics of Oper-ations Research 13:588±605

[34] Elliott R, Kopp PE (1999) Mathematics of Financial Markets. Springer-Verlag, New York[35] Fleming W, Grossman S, Vila JL, Zariphopoulou T (1990) Optimal portfolio rebalancing

with transaction costs. Preprint, Division of Applied Mathematics, Brown University[36] Fleming W, Soner HM (1993) Controlled Markov processes and viscosity solutions.

Springer-Verlag, New York[37] Grossman S, Laroque G (1990) Asset pricing and optimal portfolio choice in the presence of

illiquid durable consumption goods. Econometrica 58:25±51[38] Hastings K (1992) Impulse control of portfolios with jumps and transaction costs. Commu-

nications in Statistics ± Stochastic Models 8:59±72[39] Hodges SD, Neuberger A (1989) Optimal replication of contingent claims under transaction

costs. Rev. Futures Markets 8:222±239[40] Hoggard T, Whalley A, Wilmott P (1994) Hedging option portfolios in the presence of

transaction costs. Adv. Futures and Options Research 7:21±35[41] Hull JC (1997) Options, futures, and other derivatives. Third Edition. Prentice-Hall, Engle-

wood Cli¨s (New Jersey)[42] Kabanov YM (1999) Hedging and liquidation under transaction costs in currency markets.

Finance and Stochastics 3:237±248[43] Kabanov YM, Safarian MM (1997) On Leland's strategy of option pricing with transaction

costs. Finance and Stochastics 1:239±250[44] Karatzas I (1996) Lectures on the mathematics of ®nance. CRM Monographs 8, American

Mathematical Society[45] Karatzas I, Lehoczky JP, Sethi SP, Shreve SP (1986) Explicit solution of a general con-

sumption-investment problem. Mathematics of Operations Research 11:261±294[46] Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. Second Edition.

Springer-Verlag, New York[47] Karatzas I, Shreve SE (1998) Methods of mathematical ®nance. Springer-Verlag, New York[48] Korn R (1997a) Optimal impulse control when control actions have random consequences.

Mathematics of Operations Research 22:639±667[49] Korn R (1997b) Optimal portfolios. World Scienti®c, Singapore[50] Korn R (1998) Portfolio optimization with strictly positive transaction costs and impulse

controls. Finance and Stochastics 2:85±114[51] Korn R (1999) Some applications of impulse control in mathematical ®nance. Mathematical

Methods of Operations Research 50[52] Kushner HJ, Dupuis PG (1992) Numerical methods for stochastic control problems in con-

tinuous time. Springer-Verlag, New York[53] Leland H (1985) Option pricing and replication with transaction costs. The Journal of

Finance 40:1283±1301[54] Levental S, Skorohod AV (1997) On the possibility of hedging options in the presence of

transaction costs. Annals of Applied Probability 7:410±443[55] Lintner J (1965) The valuation of risk assets and the selection of risky investments in stock

portfolios and capital budgets. The Review of Economics and Statistics 47:13±37[56] Magill MJP (1976) The preferability of investment through a mutual fund. Journal of

Economic Theory 13:264±271[57] Magill MJP, Constantinides GM (1976) Portfolio selection with transaction costs. Journal of

Economic Theory 13:245±263[58] Markowitz HM (1952) Portfolio selection. The Journal of Finance 7:77±91[59] Markowitz HM (1959) Portfolio selection: E½cient diversi®cation of investments. Wiley,

New York[60] Merton RC (1969) Lifetime portfolio selection under uncertainty: the continuous-time case.

The Review of Economics and Statistics 51:247±257[61] Merton RC (1971)(1973) Optimum consumption and portfolio rules in a continuous-time

model. Journal of Economic Theory 3:373±413. Erratum. Journal of Economic Theory6:213±214

[62] Merton RC (1990) Continuous-time ®nance. Basil Blackwell, Oxford and Cambridge[63] Morton A, Pliska S (1995) Optimal portfolio management with ®xed transaction costs.

Mathematical Finance 5:337±356[64] Mossin J (1968) Optimal multiperiod portfolio policies. Journal of Business 41:215±229


[65] Musiela M, Rutkowski M (1998) Martingale methods in ®nancial modelling. Springer-Verlag[66] Pliska SR (1997) Introduction to mathematical ®nance: Discrete time models. Blackwell

Publishers, Oxford[67] Pliska S, Selby MJP (1994) On a free boundary problem that arises in portfolio management.

Philosophical Transactions of the Royal Society of London 347:555±561[68] Samuelson PA (1969) Lifetime portfolio selection by dynamic stochastic programming. The

Review of Economics and Statistics 51:239±246[69] Sethi SP (1997) Optimal Consumption and Investment with Bankruptcy. Kluwer, Boston[70] Sethi SP, Taksar M (1988) A note on Merton's ``Optimal consumption and portfolio rules in

a continuous-time model''. Journal of Economic Theory 46:395±401[71] Shiryaev A (1978) Optimal stopping rules. Springer-Verlag, New York[72] Shreve S, Soner HM (1994) Optimal investment and consumption with transaction costs.

Annals of Applied Probability 4:609±692[73] Shreve SE, Soner HM, Xu GL (1991) Optimal investment and consumption with two bonds

and transaction costs. Mathematical Finance 1:53±84[74] Soner HM, Shreve S, CvitanicÂ J (1995) There is no nontrivial hedging portfolio for option

pricing with transaction costs. Annals of Applied Probability 5:327±355[75] Taksar M, Klass MJ, Assaf D (1988) A di¨usion model for optimal portfolio selection in the

presence of brokerage fees. Mathematics of Operations Research 13:277±294[76] Tobin J (1965) The theory of portfolio selection. In: Hahn FH, Brechling FPR (eds.) The

theory of interest rates. McMillan, London[77] Weerasinghe A (1998) Singular optimal strategies for investment with transaction costs. The

Annals of Applied Probability 8:1312±1330[78] Whalley AE, Wilmott P (1997) An asymptotic analysis of an optimal hedging model for

option pricing with transaction costs. Mathematical Finance 7:307±324[79] Whittle P (1990) Risk-sensitive optimal control. J. Wiley, Chichester, England[80] Wilmott P, Dewynne JN, Howison S (1993) Option pricing: Mathematical models and

computation. Oxford University Press, Oxford[81] Wilmott P, Dewynne JN, Howison S (1995) The mathematics of ®nancial derivatives: A

student edition. Cambridge University Press[82] Zariphopoulou T (1992) Investment/consumption models with transaction costs and

Markov-chain parameters. SIAM J. Control Optimization 30:616±636

68 A. Cadenillas

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