zbMATH — the first resource for mathematics

Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion. (English) Zbl 1180.93095
Summary: 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.

93E03 Stochastic systems in control theory (general)
93B18 Linearizations
93E12 Identification in stochastic control theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Arnold, L.: Random Dynamical Systems. Springer, Berlin (1997)
[2] Caraballo, T., Kloeden, P.E.: The pathwise numerical approximation of stationary solutions of semilinear stochastic evolution equations. Appl. Math. Optim. 54, 401–415 (2006) · Zbl 1113.60064 · doi:10.1007/s00245-006-0876-z
[3] Crauel, H.: Random point attractors versus random set attractors. J. Lond. Math. Soc. II. Ser. 63, 413–427 (2001) · Zbl 1011.37032 · doi:10.1017/S0024610700001915
[4] Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9, 307–341 (1997) · Zbl 0884.58064 · doi:10.1007/BF02219225
[5] Hüsler, J., Piterbarg, V., Seleznjev, O.: On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13, 1615–1653 (2003) · Zbl 1038.60040 · doi:10.1214/aoap/1069786514
[6] Kloeden, P.E., Keller, H., Schmalfuss, B.: Towards a theory of random numerical dynamics. In: Crauel, H., Gundlach, V.M. (eds.) Stochastic Dynamics, pp. 259–282. Springer, Berlin (1999) · Zbl 0930.93071
[7] Maslowski, B., Schmalfuss, B.: Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stoch. Anal. Appl. 22, 1577–1607 (2004) · Zbl 1062.60060 · doi:10.1081/SAP-200029498
[8] Mishura, Y., Shevchenko, G.: The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Working paper (2008) · Zbl 1154.60046
[9] Neuenkirch, A.: Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion. Stoch. Process. Appl. 118, 2294–2333 (2008) · Zbl 1154.60338 · doi:10.1016/j.spa.2008.01.002
[10] Neuenkirch, A., Nourdin, I.: Exact rate of convergence of some approximation schemes associated to SDEs driven by a fBm. J. Theor. Probab. 20, 871–899 (2007) · Zbl 1141.60043 · doi:10.1007/s10959-007-0083-0
[11] Robinson, J.C.: Stability of random attractors for a backwards Euler scheme. Stoch. Dyn. 4, 175–184 (2004) · Zbl 1082.37056 · doi:10.1142/S0219493704000997
[12] Schmalfuss, B.: Backward cocycles and attractors of stochastic differential equations. In: Reitmann, V., Riedrich, T., Koksch, N. (eds.) International Seminar on Applied Mathematics–Nonlinear Dynamics: Attractor Approximation and Global Behaviour, pp. 185–192. TU Dresden, Dresden (1992)
[13] Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996) · Zbl 0869.65043
[14] Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. Uppsala 67, 251–282 (1936) · Zbl 0016.10404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.