Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion

被引：39

作者：

Garrido-Atienza, Maria J. ^{[1
]}

Kloeden, Peter E. ^{[2
]}

Neuenkirch, Andreas ^{[2
]}

机构：

[1] Univ Seville, Dpto Ecuac Diferenciales & Anal Numer, E-41080 Seville, Spain

[2] Univ Frankfurt, Inst Math, D-60325 Frankfurt, Germany

来源：

APPLIED MATHEMATICS AND OPTIMIZATION | 2009年 / 60卷 / 02期

关键词：

Fractional Brownian motion; Random dynamical system; Random attractor; One-sided dissipative Lipschitz condition; Implicit Euler scheme; DIFFERENTIAL-EQUATIONS DRIVEN; APPROXIMATION; CONVERGENCE;

D O I：

10.1007/s00245-008-9062-9

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this article we study the behavior of dissipative systems with additive fractional noise of any Hurst parameter. Under a one-sided dissipative Lipschitz condition on the drift the continuous stochastic system is shown to have a unique stationary solution, which pathwise attracts all other solutions. The same holds for the discretized stochastic system, if the drift-implicit Euler method is used for the discretization. Moreover, the unique stationary solution of the drift-implicit Euler scheme converges to the unique stationary solution of the original system as the stepsize of the discretization decreases.

引用

页码：151 / 172

页数：22

共 14 条

[1]

[Anonymous], 1996, Dynamical systems and numerical analysis

[2]

Arnold L., 1997, RANDOM DYNAMICAL SYS

[3] The pathwise numerical approximation of stationary solutions of semilinear stochastic evolution equations [J].

Caraballo, T. ;

Kloeden, P. E. .

APPLIED MATHEMATICS AND OPTIMIZATION, 2006, 54 (03) :401-415

[4] Random attractors [J].

Crauel H. ;

Debussche A. ;

Flandoli F. .

Journal of Dynamics and Differential Equations, 1997, 9 (2) :307-341

[5] Random point attractors versus random set attractors [J].

Crauel, H .

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2001, 63 :413-427

[6]

Hüsler J, 2003, ANN APPL PROBAB, V13, P1615

[7] Towards a theory of random numerical dynamics [J].

Kloeden, PE ;

Keller, H ;

Schmalfuss, B .

STOCHASTIC DYNAMICS, 1999, :259-282

[8] Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion [J].

Maslowski, B ;

Schmalfuss, B .

STOCHASTIC ANALYSIS AND APPLICATIONS, 2004, 22 (06) :1577-1607

[9]

MISHURA Y, 2008, RATE CONVERGENCE EUL

[10] Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion [J].

Neuenkirch, Andreas ;

Nourdin, Ivan .

JOURNAL OF THEORETICAL PROBABILITY, 2007, 20 (04) :871-899

← 1 2 →