Mean-square numerical approximations to random periodic solutions of stochastic differential equations

Qingyi Zhan

doi:10.1186/s13662-015-0626-0

Abstract

This paper is devoted to the possibility of mean-square numerical approximations to random periodic solutions of dissipative stochastic differential equations. The existence and expression of random periodic solutions are established. We also prove that the random periodic solutions are mean-square uniformly asymptotically stable, which ensures the numerical approximations are feasible. The convergence of the numerical approximations by the random Romberg algorithm is also proved to be mean-square. A numerical example is presented to show the effectiveness of the proposed method.

Highlights

Stochastic differential equations (SDEs) have an important position in theory and application, for more details we refer the reader to [ ] and [ ]
Numerical approximations to random periodic solutions are an important method for studying their dynamic behavior
The main results we obtain are the numerical approximations to random periodic solutions of dissipative SDEs, and the proof of mean-square convergence

Summary

Introduction

Stochastic differential equations (SDEs) have an important position in theory and application, for more details we refer the reader to [ ] and [ ]. The main results we obtain are the numerical approximations to random periodic solutions of dissipative SDEs, and the proof of mean-square convergence. (ii) The random periodic solution Y (t, ω) of SDE ( ) is said to be mean-square uniformly stable if for any given > and every other random periodic solution Y (t, ω) of SDE ( ), there exists δ = δ( ) such that x – x ≤ δ implies the inequality Y (t, ω) – Y (t, ω) < holds for any t ≥ s, where s = t – mτ .

Results

Conclusion