Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional
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ar X iv :m at h. PR /0 60 10 38 v 3 2 8 N ov 2 00 6 Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion Andreas Neuenkirch Johann Wolfgang Goethe Universitat, Fachbereich Mathematik, Robert-Mayer-Straße 10, 60325 Frankfurt am Main, Germany Ivan Nourdin LPMA, Universite Pierre et Marie Curie Paris 6, Boıte courrier 188, 4 Place Jussieu, 75252 Paris Cedex 5, France Abstract In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H . We consider two cases. If H > 1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in [17] and [18]. When 1/6 < H < 1/2, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.
Voir
- euler scheme
- brownian motion
- russo-vallois
- motion - russo-vallois integrals - doss-sussmann
- crank-nicholson scheme
- differential equations driven
- standard brownian
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Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion
Andreas Neuenkirch JohannWolfgangGoetheUniversit¨at,FachbereichMathematik, Robert-Mayer-Straße 10, 60325 Frankfurt am Main, Germany neuenkirch@math.uni-frankfurt.de
Ivan Nourdin LPMA,Universite´PierreetMarieCurieParis6, Boıˆtecourrier188,4PlaceJussieu,75252ParisCedex5,France nourdin@ccr.jussieu.fr
Abstract
In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst indexH If consider two cases.. WeH >12, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in [17] and [18]. When 16< H <12, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.
Key words:Fractional Brownian motion - Russo-Vallois integrals - Doss-Sussmann type transformation - Stochastic differential equations - Euler scheme - Crank-Nicholson scheme - Mixing law.
2000 Mathematics Subject Classification:60G18, 60H05, 60H20.
1
1
Introduction
LetB= (Bt t∈[0 fBm) with Hurst1]) be a fractional Brownian motion (in short: parameterH∈(01), i.e.,Bis a continuous centered Gaussian process with covariance function RH(s t)=12(s2H+t2H− |t−s|2H) s t∈[01] ForH= 12,Bis a standard Brownian motion, while forH6= 12, it is neither a semi-martingale nor a Markov process. Moreover, it holds (E|Bt−Bs|2)12=|t−s|H t s∈[01] and almost all sample paths ofBedrnyorsofanuouonticredlo¨Heraα∈(0 H). In this paper, we are interested in the pathwise approximation of the equation t t Xt=x0+Zσ(Xs)dBs+Zb(Xs)ds t∈[01](1) 0 0 with a deterministic initial valuex0∈R. Here,σandbsatisfy some standard smoothness assumptions and the integral equation (1) is understood in the sense of Russo-Vallois. Let us recall briefly the significant points of this theory.
ochastic Definition 1(following [27]) LetZ= (Zt)t∈[01] process with continuousbe a st paths.
•A family of processes(Ht(ε))t∈[01]is said to converge to the process(Ht)t∈[01]in theucp sense, ifsupt∈[01]|Ht(ε)−Ht|goes to 0 in probability, asε→0. •The(Russo-Vallois) forward integralR0tZsd−Bsis defined by t lεi→m0−ucZ0t)dt(2) pε−1Zt(Bt+ε−B
provided the limit exists. •The(Russo-Vallois) symmetric integralR0tZsd◦Bsis defined by li→m0−ucp (2ε)−1Z0t(Zt+ε+Zt)(Bt+ε−Bt)dt ε provided the limit exists.
(3)
Now we state the exact meaning of equation (1) and give conditions for the existence and uniqueness of its solution. We consider two cases, according to the value ofH:
2
•
CaseH >12.
Here the integral with respect toBis defined by the forward integral (2). Proposition 1Ifσ∈ Cb2and ifbsatisfies a global Lipschitz condition, then the equation Z0tσ(Xs)d−BsZtb(Xs)ds t∈[01] ( Xt=x0 4)+ + 0 admits a unique solutionXsontinuouswsesehoprofesoclo¨Hcredhtaperasseteniht of orderα >1−H we have a Doss-Sussmann type [7, 29] representation:. Moreover,
Xt=φ(At Bt) t∈[01] whereφandAare given respectively by
and
∂φ ∂x2(x1 x2) =σ(φ(x1 x2)) φ(x10) =x1 x1 x2∈R A′t= exp−Z0Btds(φ(At Bt)) A0=x0 t∈[01] σ′(φ(At s))b
(5)
(6)
(7)
Proof. IfXandYinuousldercontecorplaesohwsessarhsatep¨o.H.seatworare of indexα >0 andβ >0 withα+β >1, thenR0tYsd−Xscoincides with the Young integralR0tYsdXs(see [28], Proposition 2.12). Consequently, Proposition 1 is a consequence of, e.g., [11] or [24]. ✷
•Case16< H <12. WhenH <12, in particular the forward integralR0tBsd−Bsdoes not exist. Thus, in this case, the use of the symmetric integral (3) is more adequate. Here we consider only the caseb the general case see [19], [21] and Remark 1. for= 0:
Proposition 2IfH >16and ifσ∈ C5(R)satisfies a global Lipschitz condition, then the equation Xt=x0+Z0tσ(Xs)d◦Bs t∈[01] (8) admits a unique solutionXin the set of processes of the formXt=f(Bt)with f∈ C5(R). The solution is given byXt=φ(x0 Bt) t∈[01]whereφis defined by (6).
Proof. See [19], Theorem 2.10.
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