Numerical approximations to the stationary solutions of stochastic differential equations

Yevik, Andrei; Zhao, Huaizhong

Abstract

In this paper, we investigate the possibility of approximating the stationary solution of a stochastic differential equation (SDE). We start with the random dynamical system generated by the SDE with the multiplicative noise. We prove that the pullback flow has a stationary point. However, the stationary point is not constructible explicitly; therefore, we look at the numerical approximation. We prove that the discrete time random dynamical system also has a stationary point. Finally, we prove mean-square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes, as well as almost surely convergence when the time step is rational.

Citation

Yevik, A., & Zhao, H. (2011). Numerical approximations to the stationary solutions of stochastic differential equations. SIAM Journal on Numerical Analysis, 49(4), 1397 - 1416. https://doi.org/10.1137/100797886

• Journal Article Type: Article
• Acceptance Date: Apr 4, 2011
• Online Publication Date: Jul 7, 2011
• Publication Date: 2011-01
• Deposit Date: Oct 6, 2021
• Journal: SIAM journal on numerical analysis
• Print ISSN: 0036-1429
• Electronic ISSN: 1095-7170
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 49
• Issue: 4
• Pages: 1397 - 1416
• DOI: https://doi.org/10.1137/100797886
• Public URL: https://durham-repository.worktribe.com/output/1233975