parameter estimation for fractional diffusion...

Research ArticleParameter Estimation for Fractional Diffusion Process withDiscrete Observations

Yuxia Su 1 and YutianWang2

1School of Statistics Qufu Normal University Jining Shandong 273165 China2School of Software Engineering Qufu Normal University Jining Shandong 273165 China

Correspondence should be addressed to Yuxia Su qfsyx163com

Received 2 August 2018 Revised 27 November 2018 Accepted 24 December 2018 Published 8 January 2019

Academic Editor Yong H Wu

Copyright copy 2019 Yuxia Su and Yutian Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper deals with the problem of estimating the parameters for fractional diffusion process from discrete observations when theHurst parameter119867 is unknownWith combination of several methods such as the Donsker type approximate formula of fractionalBrownian motion quadratic variation method and the maximum likelihood approach we give the parameter estimations of theHurst index diffusion coefficients and volatility and then prove their strong consistency Finally an extension for generalizedfractional diffusion process and further work are briefly discussed

1 Introduction

In recent years many scholars have found that some financialtime series data tend to be shown as biased randomwalk longmemory and self-similarity etc which made the stochasticdifferential equation model driven by Brownian motion nolonger applicable to describe financial data Perhaps themost popular approach for modeling long memory is theuse of fractional Brownian motion (hereafter fBm) that hasbeen verified as a good model to describe the long memoryproperty of some time series

Compared with the traditional efficient market hypoth-esis theory fractional market theory can accurately depictthe actual law of financial market such as the Ornstein-Uhlenbeck process driven by fractional Brownian motionwhich ismore consistent with the characteristics of long-termmemory in place of Vasicek model that is suitable to simulatethe short-term interest rate model

Although the study of fractional Brownian motion hasbeen going on for decades statistical inference problemsrelated are just in its infancy Such questions have beenrecently treated in several papers [1ndash3] in general thetechniques used to construct maximum likelihood estima-tors (MLE) for the drift parameter are based on Girsanov

transforms for fBm and depend on the properties of thedeterministic fractional operators related to the fBm Gen-erally speaking these papers focused on the problems ofestimating the unknown parameters in the continuous-timecase Prakasa Rao [4] gave an extensive review on most ofthe recent developments related to the parametric and otherinference procedures for stochastic models driven by fBmThe latest study can be found in Xiao and Yu [5 6] whodeveloped the asymptotic theory for least square estimatorsfor two parameters in the drift function in the fractionalVasicek model with a continuous record of observationsAnother possibility is to use Euler-type approximations forthe solution of the above equation and to construct an MLEestimator based on the density of the observations givenrdquothe pastrdquo for the case of stochastic equations driven byBrownian motion ldquoReal-worldrdquo data is however typicallydiscretely sampled (eg stock prices collected once a dayor at best at every tick) Therefore statistical inference fordiscretely observed diffusions is of great interest for practicalpurposes and at the same time it poses a challenging problemSome papers are devoted to the parameter estimation for themodels with fBm and discrete observations see eg Hu andNualart[1] Hu and Song [7] Mishura and Ralchenko [8]Zhang Xiao Zhang and Niu [9] and Sun and Shi [10]

HindawiJournal of Function SpacesVolume 2019 Article ID 9036285 6 pageshttpsdoiorg10115520199036285

2 Journal of Function Spaces

In this paper we shall consider the parameter estima-tion problem for fractional linear diffusion process (FLDP)Assume that we have the model

119889119883119905 = (120572 minus 120573119883119905) 119889119905 + 120590119889119861119867119905 (1)

which can describe the intrinsic characteristics of interest ratemore accurately in practical problem The drift parameter 120572120573 can characterize respectively the long-term equilibriuminterest rate level and the rate of the short-term interestrates deviate from long-term interest rates In general theparameters of long-term equilibrium level of short-terminterest rate are unknown We assume 120573 gt 0 throughoutthe paper so that the process is ergodic (when 120573 lt 0 thesolution to (1) will diverge) 120590 describes the volatility ofinterest rates and (119861119867119905 )119905ge0 is a fBm with Hurst parameter119867 isin (0 1) In this paper we suppose the Hurst index 119867 thediffusion coefficients 120572 120573 and the volatility 120590 are unknownparameters to be estimated We will furthermore show thestrong consistence of these estimators

In the case of diffusion process driven by Brownianmotion the most important methods are either maximumlikelihood estimation or least square estimation Since fBmis not a Markov process the Kalman filter method cannotbe applied to estimate the parameters of stochastic processdriven by fBm Consequently it is a convenient way to handlethe estimation problem by replacing fBm with its associateddisturbed random walk In this paper we follow Zhang etal [9] to use discrete expressions of fractional Bronwnianmotion with Donsker type approximate formula whichcan to some extent simplify calculation and simulationAlthough we do not have martingales in the model thisconstruction involving random walks allows using martin-gales arguments to obtain the asymptotic behaviour of theestimators

Our paper is organized as follows In Section 2 we pro-pose MLE estimators for FLDP from discrete observationsThe almost sure convergence of the estimators is provided inthe latter part of this section In Section 4 an extension forgeneralized fractional diffusion process is briefly discussedFinally Section 5 includes conclusions and directions offurther work

2 Estimation Procedure

It is worth emphasizing that the solution of (1) is given by

119883119905 = 1198830 + int1199050(120572 minus 120573119883119904) 119889119904 + 120590119861119867119905 (2)

where the unknown parameters included 120572 120573 120590 and 119867 Wenowproceed to estimate these parameters based on quadraticvariation method and maximum likelihood approach

Let 119883119905 119905 isin 119877 be the FLDP with 119867 gt 12 and supposethat 120587119899 = 120591119899119896 119896 = 0 1 119894119899 119899 ge 1 119894119899 uarr infin be a sequenceof partitions of the interval [0 119879] If partition 120587119899 is uniformthen 120591119899119896 = 119896119879119894119899 for all 119896 isin 0 1 119894119899 If 119894119899 equiv 119899 we write119905119899119896 instead of 120591119899119896 Assume that process 119883119905 is observed at timepoints (119894119898119899)119879 119894 = 1 2 119898119899 where119898119899 = 119899119896119899 and 119896119899 grows

faster than 119899119897119899119899 but the growth does not exceed polynomialeg 119896119899 = 119899119897119899120579119899 120579 gt 1 or 119896119899 = 1198992

In applications the estimation of 119867 isin (0 1) (called theHurst index) is a fundamental problem Its solution dependson the theoretical structure of a model under considerationTherefore particularmodels usually deserve separate analysis

According to the notation of Kubilius Skorniakov [11]suppose there are two hypotheses

(1198621) Δ119883120591119899119896= 119883120591119899

119896minus 119883120591119899

119896minus1= 119874120596 (119889119867minus120576119899 )

(1198622) Δ(2)119883120591119899119896= Δ119883120591119899

119896minus Δ119883120591119899

119896minus1

= 120590Δ(2)119861119867120591119899119896minus1

+ 119874120596 (1198892(119867minus120576)119899 ) (3)

for all 120576 isin (0119867 minus 12) where 119889119899 = 1198981198861199091le119896le119898119899(120591119899119896minus120591119899119896minus1)119884119899 =119874120596(119886119899) means for a sequence of rv 119884119899 and 119886119899 sub (0infin) andthere exists as non-negative rv 120589 such that |119884119899 | le 120589sdot119886119899Thesetwo conditions are used to prove the strongly consistent andasymptotically normality of the estimator 119867 from discreteobservatios

Denote

119882119899119896 = 119896119899sum119894=minus119896119899+2

(Δ(2)119883119904119899119894+119905119899119896)2

= 119896119899sum119894=minus119896119899+2

(119883119904119899119894+119905119899119896minus 2119883119904119899

119894minus1+119905119899119896+ 119883119904119899

119894minus2+119905119899119896)2

(4)

where 1 le 119896 le 119899 and 119904119899119894 = (119894119898119899)119879Then the estimator of Hurst parameter 119867 can be written

as

= 12 + 12 ln 119896119899 ln( 2119898119899119898119899sum119896=2

(Δ(2)119883119905119899119896)2119882119899119896minus1 ) (5)

Next we turn to the estimation problem of the diffusionparameter 1205902 When 119867 is known Xiao et al [12] obtainedthe estimators based on approximating integrals via Riemannsums with Hurst index 119867 isin (12 34) In contrast wesuppose in this paper the Hurst index is unknown Thereforein the next estimation the estimator of 119867 will be embeddedin the equation For simplicity denote 119883120591119899

119894≜ 119883119894ℎ 119861119867120591119899

119896minus1=119861119867119898119899119894 119894 = 1 119898119899 ℎ = 119879119898119899 Thus the full sequence of119898119899 observations can be written as 119883ℎ 1198832ℎ 119883119898119899ℎ

For the diffusion parameter we easily obtain an estimatorfor the diffusion parameter by using quadratic variationssuch

1205902 = sum119898119899minus1119894=1 (119883(119894+1)ℎ minus 119883119894ℎ)2(119898119899 minus 1) ℎ2 (6)

which converges (in 1198712 and almost surely) to 1205902Finally we are in a position to estimate the drift param-

eter Note that 119861119867119905 minus 119861119867119905minus1 is not independent and the process119861119867119905 is not a semimartingale therefore the martingale typetechniques cannot be used to study this estimator This

Journal of Function Spaces 3

problem will be avoided by the use of the random walksthat approximate 119861119867119905 Based on the results on Sottinen [13]the fractional Brownian motion can be approximated by ardquodisturbedrdquo random walk which was called Donsker typeapproximation for fBm

Lemma 1 The fBm with Hurst parameter 119867 gt 12 can berepresented by its associated disturbed random walk

119861119867119898119899119905 = lfloor119898119899119905rfloorsum119894=1

radic119898119899 (int119894119898119899(119894minus1)119898119899

119870119867(lfloor119898119899119905rfloor119898119899 119904) 119889119904) 120576119894 (7)

with 119870119867(119905 119904) = 119888119867(119867 minus 12)11990412minus119867 int119905119904(119906 minus 119904)119867minus12119906119867minus12119889119906

which is the kernel function that transforms the standardBrownian motion into a fractional one 119888119867 is the normalizingconstant 119888119867 = radic2119867Γ(23 minus 119867)Γ(119867 + 12)Γ(2 minus 2119867) and 120576119894are iid random variables with 119864120576119894 = 0 and var 120576119894 = 1 andlfloor119909rfloor denotes the greatest integer not exceeding 119909

Sottinen (2011) proved that 119861119867119898119899119905 converges weakly inthe skorohod topology to the fractional Brownian motionWith the estimators 120590 plug-in the replacing model stillkept the main properties of the original process such as longrange dependence and asymptotic self-similar Therefore themartingales can be used to treat this replacing model

In general numerical approximation of model (1) can bepresented by Euler scheme

119883(119894+1)ℎ = 119883119894ℎ + (120572 minus 120573119883119894ℎ) ℎ + 120590 (119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ ) 119894 = 1 119898119899 minus 1 (8)

Set

119891119894 (1205761 1205762 120576119894)= radic119898119899 119894sum

119895=1

[int119895ℎ(119895minus1)ℎ

(119870119867 ((119894 + 1) ℎ 119904) minus 119870119867 (119894ℎ 119904)) 119889119904] 120576119895 (9)

to denote the contribution of the 119899 minus 1 first jumps of therandom walk and

119865119894 = radic119898119899 int(119894+1)ℎ119894ℎ

119870119867 ((119894 + 1) ℎ 119904) 119889119904 (10)

to denote the contribution of the last jump

With the approximation of fBm (Lemma 1) we can write

119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ = 119891119894 (1205761 1205762 120576119894) + 119865119894120576119894+1119894 = 1 119898119899 minus 1 (11)

with which (8) can be written as

119883(119894+1)ℎ = 119883119894ℎ + (120572 minus 120573119883119894ℎ) ℎ+ 120590 [119891119894 (1205761 1205762 120576119894) + 119865119894120576119894+1] (12)

Hence we have

119864 [119883(119894+1)ℎ minus 119883119894ℎ | 119883119894ℎ]= (120572 minus 120573119883119894ℎ) ℎ + 120590 (119891119894 (1205761 1205762 120576119894))

var [119883(119894+1)ℎ minus 119883119894ℎ | 119883119894ℎ] = 21198652119894 (13)

We assume that random variables 120576119894 follow a standardnormal law 119873(0 1) Then the random variable 119883(119894+1)ℎ isconditionally Gaussian and the conditional density of119883(119894+1)ℎgiven 119883ℎ 1198832ℎ 119883119894ℎ can be written as

119891119883(119894+1)ℎ |119883ℎ1198832ℎ119883119894ℎ (119909(119894+1)ℎ | 119909ℎ 1199092ℎ 119909119894ℎ) = 1radic212058712059021198652119894

sdot expminus12sdot (119909(119894+1)ℎ minus 119909119894ℎ minus (120572 minus 120573119909119894ℎ) ℎ minus 120590119891119894 (1205761 1205762 120576119894))21198652119894

(14)

The likelihood function can be expressed as

119871 (120572 120573) = 119891119883ℎ (119909ℎ) 1198911198832ℎ|119883ℎ (1199092ℎ | 119909ℎ)sdot sdot sdot 119891119883119898119899ℎ|119883ℎ1198832ℎ119883(119898119899minus1)ℎ (119909119898119899ℎ | 119909ℎ 1199092ℎ 119909(119898119899minus1)ℎ)= 119898119899prod119894=1

1radic212058712059021198652119894 expminus12

sdot (119909(119894+1)ℎ minus 119909119894ℎ minus (120572 minus 120573119909119894ℎ) ℎ minus 120590119891119894 (1205761 1205762 120576119894))21198652119894

(15)

This leads to the MLE of 120572 and 120573

= sum119898119899minus1119894=0 ((119910119894ℎ minus 120573ℎ119909119894ℎ) 1198652119894 )sum119873minus1119894=0 (ℎ1198652119894 ) (16)

120573 = (1ℎ) sum119898119899minus1119894=0 (119909119894ℎ1198652119894 )sum119898119899minus1119894=0 (119910119894ℎ1198652119894 ) minus sum119898119899minus1119894=0 (11198652119894 )sum119898119899minus1119894=0 (119910119894ℎ119909119894ℎ1198652119894 )sum119898119899minus1119894=0 (1199092119894ℎ1198652119894 )sum119898119899minus1119894=0 (11198652119894 ) minus (sum119898119899minus1119894=0 (119909119894ℎ1198652119894 ))2 (17)

where119910119894ℎ = 119909(119894+1)ℎminus119909119894ℎminus120590119891119894(1205761 1205762 120576119894) = (120572minus120573119883119894ℎ)ℎ+V119894 V119894 = 120590119865119894120576119894+1 119894 = 1 2 119898119899 minus 1 Remark 2 Note that the parameter estimators of drift coeffi-

cients are related to the volatility 120590 while in fact 1205902 can be


(at least theoretically) computed on any finite time intervalFurthermore fBm is self-similar to stationary increments andit satisfies 119864|119861119867119905 minus119861119867119904 | = |119905minus119904|2119867 for every 119904 119905 isin [0 119879] For thisreason we may assume that the diffusion coefficient is equalto 1

3 The Asymptotic Properties

In this section we turn to study the strong consistency ofthese estimators by (5) (6) (16) and (17)

Theorem 3 Assume that solution of (1) satisfies hypotheses(C1) and (C2) then estimator converges to 119867 almost surelyas119898119899 goes to infinity

Detailed proof can be found inKubilus and Skorniakov [11]

Theorem 4 The estimator 2 converges to 1205902 almost surely as119898119899 goes to infinityProof With the strong consistency of to 119867 and that 2 ≜(sum119898119899minus1119894=1 (119883(119894+1)ℎ minus 119883119894ℎ)2)(119898119899 minus 1)ℎ2119867 997888rarr 1205902 with probability1 as 119898119899 goes to infinity it can be easily shown that estimator2 converges to 1205902 almost surely as 119898119899 997888rarr infin

Theorem 5 With probability one 997888rarr 120572 120573 997888rarr 120573 as119898119899 997888rarr infin

Proof Clearly the consistency of can be inferred combinedwith (16) and consistency of 120573 We just prove that 120573 is strongconsistent

A simple calculation shows that

120573 minus 120573 = 1ℎ sum119898119899minus1119894=0 (V1198941198652119894 )sum119898119899minus1119894=0 (119909119894ℎ1198652119894 ) minus sum119898119899minus1119894=0 (11198652119894 )sum119873minus1119894=0 (V119894119909119894ℎ1198652119894 )sum119898119899minus1119894=0 (1199092119894ℎ1198652119894 )sum119898119899minus1119894=0 (11198652119894 ) minus (sum119898119899minus1119894=0 (119909119894ℎ1198652119894 ))2

= (119879119898119899)sum119898119899minus1119894=0 (11198652119894 )sum119898119899minus1119894=0 (V119894119909119894ℎ1198652119894 ) minus sum119898119899minus1119894=0 (V1198941198652119894 )sum119898119899minus1119894=0 (119909119894ℎ1198652119894 )(119879119898119899)2sum119898119899minus1119894=0 (1199092119894ℎ1198652119894 )sum119898119899minus1119894=0 (11198652119894 ) minus (sum119898119899minus1119894=0 (119909119894ℎ1198652119894 ))2 = 119872119899lt 119872gt119899

(18)

where 119872119899 = (119879119898119899) sum119898119899minus1119894=0 (11198652119894 )sum119898119899minus1119894=0 (V119894ℎ119909119894ℎ1198652119894 ) minussum119898119899minus1119894=0 (119910119894ℎ1198652119894 ) sum119898119899minus1119894=0 (119909119894ℎ1198652119894 ) is a square-integrable martin-gle and lt 119872gt119899 = (119879119898119899)2sum119898119899minus1119894=0 (1199092119894ℎ1198652119894 ) sum119898119899minus1119894=0 (11198652119894 ) minus(sum119898119899minus1119894=0 (119909119894ℎ1198652119894 ))2 is quadratic characteristic of119872119899

Using the assumption of 119867 isin (12 1) and fractionalintegral we have the explicit solution of (1) that can beexpressed as

119883119905 = (1 minus 119890minus120573119905) 120572120573 + 120590int1199050119890minus120573(119905minus119904)119889119861119867119904 119905 ge 0 (19)

where the integral can be understood in the Skorohod senseAs a consequence for any 119894 we have

119864 [1198832119894ℎ] = 119864[(1 minus 119890minus120573119894ℎ) 120572120573]2

+ (120590int119894ℎ0

119890minus120573(119905minus119904)119889119861119867119904 )2

+ 2 (1 minus 119890minus120573119894ℎ) 120572120573120590int119894ℎ0

119890minus120573(119905minus119904)119889119861119867119904 ]le 2[(1 minus 119890minus120573119894ℎ) 120572120573]2 + 1205902119890minus2120573119894ℎ119867Γ (2119867)1205732119867

(20)

Hence for any 119894 we obtain that 119864[1198832119894ℎ] is boundedMoreover by using Cauchy-Schwartz inequality weshow that (see also in [14] with a slight modificationbelow)

119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ

100381610038161003816100381610038162 = 119864[[lfloor119898119899(119894+1)ℎrfloorsum119895=lfloor119898119899119894ℎrfloor

radic119898119899 int119895119898119899(119895minus1)119898119899

(119870119867(lfloor119898119899 (119894 + 1) ℎrfloor119898119899 119904) minus 119870119867(lfloor119898119899119894ℎrfloor119898119899 119904))119889119904120576119895]]2

= [[lfloor119898119899(119894+1)ℎrfloorsum119895=lfloor119898119899119894ℎrfloor

radic119898119899 int119895119898119899(119895minus1)119898119899


le lfloor119898119899(119894+1)ℎrfloorsum119895=lfloor119898119899119894ℎrfloor

119898119899 int119895119898119899(119895minus1)119898119899

((119870119867(lfloor119898119899 (119894 + 1) ℎrfloor119898119899 119904) minus 119870119867(lfloor119898119899119894ℎrfloor119898119899 119904)))2 119889119904le 119898119899 int(119894+1)ℎ

119894ℎ((119870119867(lfloor119898119899 (119894 + 1) ℎrfloor119898119899 119904) minus 119870119867(lfloor119898119899119894ℎrfloor119898119899 119904)))2 119889119904


le int1198790((119870119867(lfloor119898119899 (119894 + 1) ℎrfloor119898119899 119904) minus 119870119867(lfloor119898119899119894ℎrfloor119898119899 119904)))2 119889119904 = 100381610038161003816100381610038161003816100381610038161003816

lfloor119898119899 (119894 + 1) ℎrfloor119898119899 minus lfloor119898119899119894ℎrfloor1198981198991003816100381610038161003816100381610038161003816100381610038162119867 le 11198982119867119899

(21)

By standard calculations we will have

1198652119894 + 119864 [1198912119894 ] = 119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ 100381610038161003816100381610038162 le 11198982119867119899 (22)

and it holds that

1198652119895 le 11198982119867119899 (23)

Now (20) combined with (23) shows that 119872119899 lt 119872gt119899 997888rarr0 119886119904 as119898119899 997888rarr infinRemark 6 The asymptotic normality of estimators is notinvolved in the results of this paper In fact Kubilius andSkorniakov [11] proposed the asymptotic normality of theestimators in view of Remark 2 the asymptotic of 120590 istrivial For the parameter estimation of fractional diffusionprocess (1) there are usually two key challenges the likeli-hood is intractable and the data is not Markovian With theDonsker type approximation formula the statistical inferenceof fractional diffusion process (FDP) can be simplified toa certain extent It has proved that the estimator of driftparameter is 119871119901(119901 ge 1)- consistent and the asymptoticnormality may be obtained with more complex operations bythe future studies of this area

4 Extension

Fractional stochastic differential equations have been widelyused in the fields of finance hydrology information andstochastic networks Although model (1) is concerned ofsimpler linear function our method can be expected to beapplicable for general fractional diffusion processes such as

119889119883119905 = 120583 (119883119905 120579) 119889119905 + 120590119889119861119867119905 (24)

where 120583( 120579) is drift functions representing the conditionalmean of the infinitesimal change of 119883119905 at time 119905 120590119889119861119867119905 is therandom perturbation Here we suppose the diffusion func-tion is constant for simplicity As far as we know for a generalsmooth and elliptic coefficient 120590() only the uniqueness ofthe invariant measure is shown in Haier and Ohashi [15]with an interesting extension to the hypoelliptic case in Haierand Pillai [16] Nothing is known about the convergenceof estimate equation not to mention rates Suppose 119883119905 isobserved at a discrete set of instants 120591119899119894 = (119894119898119899)119879 119894 =1 2 119898119899With the Donsker type approximate formula andthe above estimation procedure we use the following globalestimation equations to estimate parameter 120579

119902119899 (120579) = 119898119899minus1sum119894=1

119892 (120579 119883119894ℎ)sdot 119883(119894+1)ℎ minus 119883119894ℎ minus 119892 (120579 119883119894ℎ) minus 120590119891119894 (1205761 120576119894) = 0

(25)

where 119892(120579 119883119894ℎ) is the derivative of 119892(120579 119883119894ℎ) on 120579 Theasymptotic property of the estimators is expected to bestudied in the future and how to obtain the asymptotic theoryis still an open question

On the other hand our method can extend to anotherself-similar process still with long memory (but not Gaus-sian) which is called Rosenblatt process 119885119867119905 In contrastto the fBm model the density of Rosenblatt process is notexplicitly known any more However it can be written as adouble integral of a two-variable deterministic with respectto the Wiener process The method based on random walksapproximation offers a solution to the problem of estimatingthe parameters in fractional diffusion process driven byRosenblatt process

5 Concluding Remarks

In this paper we proposed the estimators of FLDP such astheHurst index drift coefficients and volatility and providedthe strong consistency for these estimators With the Donskerrepresentation of fractional Brownian motion the statisticalinference of FLDPmay be simplifiedHowever it is importantto note that this approximation is satisfied in the sense ofweak convergence This means only when with large numberof samples can the simulation be much better On the otherhand the approximate representation of FLDP is based onthe Euler scheme which is the main source of the errorin the computations There is always a trade-off betweenthe number of Euler steps and the number of simulationsbut what is usually computationally costly is the number ofEuler steps The rate of convergence depended on 119867 and thecloser the value of 119867 to 12 This study also suggests severalimportant directions for further research How to estimateparameters in FDP from discrete time observations and howto obtain the asymptotic theory are open questions

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The paper is supported by National Science FoundationProject (11701318) of China


References

[1] Y Hu and D Nualart ldquoParameter estimation for fractionalOrnstein-Uhlenbeck processesrdquo Statistics amp Probability Lettersvol 80 no 11-12 pp 1030ndash1038 2010

[2] M L Kleptsyna and D Melichov ldquoQuadratic variations andestimation of the Hurst index of the solution of SDE driven by afractional Brownianmotionrdquo LithuanianMathematical Journalvol 50 no 4 pp 401ndash417 2010

[3] P W Srivastava and N Mittal ldquoOptimum step-stress partiallyaccelerated life tests for the truncated logistic distribution withcensoringrdquo Applied Mathematical Modelling vol 34 no 10 pp3166ndash3178 2010

[4] B L Prakasa Rao Statistical Inference for Fractional DiffusionProcess Wiley Series in Probability and Statistics A JohnWileyand Sons 2010

[5] W Xiao and J Yu ldquoAsymptotic theory for estimating driftparameters in the fractional Vasicek modelsrdquo EconometricTheory pp 1ndash34 2018

[6] W Xiao and J Yu ldquoAsymptotic theory for rough fractionalVasicek modelsrdquo Economics and Statistics working papers 7-2018 Singapore Management University School of Economics2018 httplinklibrarysmuedusgsoe-research2158

[7] Y Z Hu and J Song ldquoParameter Estimation for FractionalOrnstein-Uhlenbeck Processes with Discrete ObservationsrdquoApplied Mathematical Modeling vol 35 no 9 pp 4196ndash42072011

[8] Y Mishura and K Ralchenko ldquoOn Drift Parameter EstimationinModels with Fractional BrownianMotion byDiscreteObser-vationsrdquoAustrian Journal of Statistics vol 43 no (3-4) pp 217ndash228 2014

[9] P Zhang W-L Xiao X-L Zhang and P-Q Niu ldquoParame-ter identification for fractional Ornstein-Uhlenbeck processesbased on discrete observationrdquo Economic Modelling vol 36 pp198ndash203 2014

[10] X X Sun and Y H Shi ldquoEstimations for parameters offractional diffusion models and their applicationsrdquo AppliedMathematics A Journal of Chinese University vol 32 no 3 pp295ndash305 2017

[11] K Kubilius andV Skorniakov ldquoOn some estimators of theHurstindex of the solution of SDE driven by a fractional Brownianmotionrdquo Statistics amp Probability Letters vol 109 pp 159ndash1672016

[12] W L Xiao W G Zhang and W D Xu ldquoParameter estima-tion for fractional Ornstein-Uhlenbeckrdquo Applied MathematicalModelling vol 35 no 9 pp 4196ndash4207 2011

[13] T Sottinen ldquoFractional Brownian motion random walks andbinary marketmodelsrdquo Finance and Stochastics vol 5 no 3 pp343ndash355 2001

[14] K Bertin S Torres and C A Tudor ldquoMaximum-likelihoodestimators and random walks in long memory modelsrdquo Statis-tics A Journal ofTheoretical and Applied Statistics vol 45 no 4pp 361ndash374 2011

[15] M Haier and A Ohashi ldquoErgodicity theory of SDEs withextrinsic memoryrdquoAnnals of Probability vol 35 no 5 pp 1950ndash1977 2007

[16] M Haier and N S Pillai ldquoErgodicity theory of hypoellipticSDEs driven by fractional BrownianmotionrdquoAnnales de lrsquoIHPProbabilites et statistiques vol 47 no 2 pp 601ndash628 2011

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


In this paper we shall consider the parameter estima-tion problem for fractional linear diffusion process (FLDP)Assume that we have the model

119889119883119905 = (120572 minus 120573119883119905) 119889119905 + 120590119889119861119867119905 (1)

which can describe the intrinsic characteristics of interest ratemore accurately in practical problem The drift parameter 120572120573 can characterize respectively the long-term equilibriuminterest rate level and the rate of the short-term interestrates deviate from long-term interest rates In general theparameters of long-term equilibrium level of short-terminterest rate are unknown We assume 120573 gt 0 throughoutthe paper so that the process is ergodic (when 120573 lt 0 thesolution to (1) will diverge) 120590 describes the volatility ofinterest rates and (119861119867119905 )119905ge0 is a fBm with Hurst parameter119867 isin (0 1) In this paper we suppose the Hurst index 119867 thediffusion coefficients 120572 120573 and the volatility 120590 are unknownparameters to be estimated We will furthermore show thestrong consistence of these estimators

In the case of diffusion process driven by Brownianmotion the most important methods are either maximumlikelihood estimation or least square estimation Since fBmis not a Markov process the Kalman filter method cannotbe applied to estimate the parameters of stochastic processdriven by fBm Consequently it is a convenient way to handlethe estimation problem by replacing fBm with its associateddisturbed random walk In this paper we follow Zhang etal [9] to use discrete expressions of fractional Bronwnianmotion with Donsker type approximate formula whichcan to some extent simplify calculation and simulationAlthough we do not have martingales in the model thisconstruction involving random walks allows using martin-gales arguments to obtain the asymptotic behaviour of theestimators

Our paper is organized as follows In Section 2 we pro-pose MLE estimators for FLDP from discrete observationsThe almost sure convergence of the estimators is provided inthe latter part of this section In Section 4 an extension forgeneralized fractional diffusion process is briefly discussedFinally Section 5 includes conclusions and directions offurther work

2 Estimation Procedure

It is worth emphasizing that the solution of (1) is given by

119883119905 = 1198830 + int1199050(120572 minus 120573119883119904) 119889119904 + 120590119861119867119905 (2)

where the unknown parameters included 120572 120573 120590 and 119867 Wenowproceed to estimate these parameters based on quadraticvariation method and maximum likelihood approach

Let 119883119905 119905 isin 119877 be the FLDP with 119867 gt 12 and supposethat 120587119899 = 120591119899119896 119896 = 0 1 119894119899 119899 ge 1 119894119899 uarr infin be a sequenceof partitions of the interval [0 119879] If partition 120587119899 is uniformthen 120591119899119896 = 119896119879119894119899 for all 119896 isin 0 1 119894119899 If 119894119899 equiv 119899 we write119905119899119896 instead of 120591119899119896 Assume that process 119883119905 is observed at timepoints (119894119898119899)119879 119894 = 1 2 119898119899 where119898119899 = 119899119896119899 and 119896119899 grows

faster than 119899119897119899119899 but the growth does not exceed polynomialeg 119896119899 = 119899119897119899120579119899 120579 gt 1 or 119896119899 = 1198992

In applications the estimation of 119867 isin (0 1) (called theHurst index) is a fundamental problem Its solution dependson the theoretical structure of a model under considerationTherefore particularmodels usually deserve separate analysis

According to the notation of Kubilius Skorniakov [11]suppose there are two hypotheses

(1198621) Δ119883120591119899119896= 119883120591119899

119896minus 119883120591119899

119896minus1= 119874120596 (119889119867minus120576119899 )

(1198622) Δ(2)119883120591119899119896= Δ119883120591119899

119896minus Δ119883120591119899

119896minus1

= 120590Δ(2)119861119867120591119899119896minus1

+ 119874120596 (1198892(119867minus120576)119899 ) (3)

for all 120576 isin (0119867 minus 12) where 119889119899 = 1198981198861199091le119896le119898119899(120591119899119896minus120591119899119896minus1)119884119899 =119874120596(119886119899) means for a sequence of rv 119884119899 and 119886119899 sub (0infin) andthere exists as non-negative rv 120589 such that |119884119899 | le 120589sdot119886119899Thesetwo conditions are used to prove the strongly consistent andasymptotically normality of the estimator 119867 from discreteobservatios

Denote

119882119899119896 = 119896119899sum119894=minus119896119899+2

(Δ(2)119883119904119899119894+119905119899119896)2

= 119896119899sum119894=minus119896119899+2

(119883119904119899119894+119905119899119896minus 2119883119904119899

119894minus1+119905119899119896+ 119883119904119899

119894minus2+119905119899119896)2

(4)

where 1 le 119896 le 119899 and 119904119899119894 = (119894119898119899)119879Then the estimator of Hurst parameter 119867 can be written

as

= 12 + 12 ln 119896119899 ln( 2119898119899119898119899sum119896=2

(Δ(2)119883119905119899119896)2119882119899119896minus1 ) (5)

Next we turn to the estimation problem of the diffusionparameter 1205902 When 119867 is known Xiao et al [12] obtainedthe estimators based on approximating integrals via Riemannsums with Hurst index 119867 isin (12 34) In contrast wesuppose in this paper the Hurst index is unknown Thereforein the next estimation the estimator of 119867 will be embeddedin the equation For simplicity denote 119883120591119899

119894≜ 119883119894ℎ 119861119867120591119899

119896minus1=119861119867119898119899119894 119894 = 1 119898119899 ℎ = 119879119898119899 Thus the full sequence of119898119899 observations can be written as 119883ℎ 1198832ℎ 119883119898119899ℎ

For the diffusion parameter we easily obtain an estimatorfor the diffusion parameter by using quadratic variationssuch

1205902 = sum119898119899minus1119894=1 (119883(119894+1)ℎ minus 119883119894ℎ)2(119898119899 minus 1) ℎ2 (6)

which converges (in 1198712 and almost surely) to 1205902Finally we are in a position to estimate the drift param-

eter Note that 119861119867119905 minus 119861119867119905minus1 is not independent and the process119861119867119905 is not a semimartingale therefore the martingale typetechniques cannot be used to study this estimator This





radic119898119899 (int119894119898119899(119894minus1)119898119899

119870119867(lfloor119898119899119905rfloor119898119899 119904) 119889119904) 120576119894 (7)






Set

119891119894 (1205761 1205762 120576119894)= radic119898119899 119894sum

119895=1

[int119895ℎ(119895minus1)ℎ

(119870119867 ((119894 + 1) ℎ 119904) minus 119870119867 (119894ℎ 119904)) 119889119904] 120576119895 (9)


119865119894 = radic119898119899 int(119894+1)ℎ119894ℎ

119870119867 ((119894 + 1) ℎ 119904) 119889119904 (10)



119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ = 119891119894 (1205761 1205762 120576119894) + 119865119894120576119894+1119894 = 1 119898119899 minus 1 (11)


119883(119894+1)ℎ = 119883119894ℎ + (120572 minus 120573119883119894ℎ) ℎ+ 120590 [119891119894 (1205761 1205762 120576119894) + 119865119894120576119894+1] (12)

Hence we have


var [119883(119894+1)ℎ minus 119883119894ℎ | 119883119894ℎ] = 21198652119894 (13)


119891119883(119894+1)ℎ |119883ℎ1198832ℎ119883119894ℎ (119909(119894+1)ℎ | 119909ℎ 1199092ℎ 119909119894ℎ) = 1radic212058712059021198652119894


(14)



1radic212058712059021198652119894 expminus12


(15)


















(18)





119864 [1198832119894ℎ] = 119864[(1 minus 119890minus120573119894ℎ) 120572120573]2

+ (120590int119894ℎ0

119890minus120573(119905minus119904)119889119861119867119904 )2



(20)


119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ


radic119898119899 int119895119898119899(119895minus1)119898119899



radic119898119899 int119895119898119899(119895minus1)119898119899



119898119899 int119895119898119899(119895minus1)119898119899






(21)


1198652119894 + 119864 [1198912119894 ] = 119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ 100381610038161003816100381610038162 le 11198982119867119899 (22)

and it holds that

1198652119895 le 11198982119867119899 (23)


4 Extension


119889119883119905 = 120583 (119883119905 120579) 119889119905 + 120590119889119861119867119905 (24)


119902119899 (120579) = 119898119899minus1sum119894=1


(25)





Data Availability




Acknowledgments



References
























Journal of












Journal of







Volume 2018






Volume 2018












radic119898119899 (int119894119898119899(119894minus1)119898119899

119870119867(lfloor119898119899119905rfloor119898119899 119904) 119889119904) 120576119894 (7)






Set

119891119894 (1205761 1205762 120576119894)= radic119898119899 119894sum

119895=1

[int119895ℎ(119895minus1)ℎ

(119870119867 ((119894 + 1) ℎ 119904) minus 119870119867 (119894ℎ 119904)) 119889119904] 120576119895 (9)


119865119894 = radic119898119899 int(119894+1)ℎ119894ℎ

119870119867 ((119894 + 1) ℎ 119904) 119889119904 (10)



119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ = 119891119894 (1205761 1205762 120576119894) + 119865119894120576119894+1119894 = 1 119898119899 minus 1 (11)


119883(119894+1)ℎ = 119883119894ℎ + (120572 minus 120573119883119894ℎ) ℎ+ 120590 [119891119894 (1205761 1205762 120576119894) + 119865119894120576119894+1] (12)

Hence we have


var [119883(119894+1)ℎ minus 119883119894ℎ | 119883119894ℎ] = 21198652119894 (13)


119891119883(119894+1)ℎ |119883ℎ1198832ℎ119883119894ℎ (119909(119894+1)ℎ | 119909ℎ 1199092ℎ 119909119894ℎ) = 1radic212058712059021198652119894


(14)



1radic212058712059021198652119894 expminus12


(15)


















(18)





119864 [1198832119894ℎ] = 119864[(1 minus 119890minus120573119894ℎ) 120572120573]2

+ (120590int119894ℎ0

119890minus120573(119905minus119904)119889119861119867119904 )2



(20)


119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ


radic119898119899 int119895119898119899(119895minus1)119898119899



radic119898119899 int119895119898119899(119895minus1)119898119899



119898119899 int119895119898119899(119895minus1)119898119899






(21)


1198652119894 + 119864 [1198912119894 ] = 119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ 100381610038161003816100381610038162 le 11198982119867119899 (22)

and it holds that

1198652119895 le 11198982119867119899 (23)


4 Extension


119889119883119905 = 120583 (119883119905 120579) 119889119905 + 120590119889119861119867119905 (24)


119902119899 (120579) = 119898119899minus1sum119894=1


(25)





Data Availability




Acknowledgments



References
























Journal of












Journal of







Volume 2018






Volume 2018




















(18)





119864 [1198832119894ℎ] = 119864[(1 minus 119890minus120573119894ℎ) 120572120573]2

+ (120590int119894ℎ0

119890minus120573(119905minus119904)119889119861119867119904 )2



(20)


119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ


radic119898119899 int119895119898119899(119895minus1)119898119899



radic119898119899 int119895119898119899(119895minus1)119898119899



119898119899 int119895119898119899(119895minus1)119898119899






(21)


1198652119894 + 119864 [1198912119894 ] = 119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ 100381610038161003816100381610038162 le 11198982119867119899 (22)

and it holds that

1198652119895 le 11198982119867119899 (23)


4 Extension


119889119883119905 = 120583 (119883119905 120579) 119889119905 + 120590119889119861119867119905 (24)


119902119899 (120579) = 119898119899minus1sum119894=1


(25)





Data Availability




Acknowledgments



References
























Journal of












Journal of







Volume 2018






Volume 2018











(21)


1198652119894 + 119864 [1198912119894 ] = 119864 10038161003816100381610038161003816119861119867119898119899(119894+1)ℎ minus 119861119867119898119899119894ℎ 100381610038161003816100381610038162 le 11198982119867119899 (22)

and it holds that

1198652119895 le 11198982119867119899 (23)


4 Extension


119889119883119905 = 120583 (119883119905 120579) 119889119905 + 120590119889119861119867119905 (24)


119902119899 (120579) = 119898119899minus1sum119894=1


(25)





Data Availability




Acknowledgments



References
























Journal of












Journal of







Volume 2018






Volume 2018









References
























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








parameter estimation for fractional diffusion...

Documents