Stochastic theta methods for random periodic solution of stochastic differential equations under non-globally Lipschitz conditions

TL;DR

The article addresses numerical approximation of random periodic solutions for stochastic differential equations under non-globally Lipschitz conditions. It analyzes stochastic theta methods with $\theta\in(1/2,1]$ to discretize the infinite-horizon pull-back problem and proves existence and uniqueness of random periodic solutions for both the SDE and its ST discretizations. The main results establish strong mean-square convergence of the ST approximations to the true random periodic solution, with convergence orders $1/2$ in the multiplicative-noise case and $1$ in the additive-noise case. Numerical experiments corroborate the theoretical rates and demonstrate the periodicity and initial-condition insensitivity of the random periodic solutions.

Abstract

This work focuses on the numerical approximations of random periodic solutions of stochastic differential equations (SDEs). Under non-globally Lipschitz conditions, we prove the existence and uniqueness of random periodic solutions for the considered equations and its numerical approximations generated by the stochastic theta (ST) methods with theta within (1/2,1]. It is shown that the random periodic solution of each ST method converges strongly in the mean square sense to that of SDEs for all step size. More precisely, the mean square convergence order is 1/2 for SDEs with multiplicative noise and 1 for SDEs with additive noise. Numerical results are finally reported to confirm these theoretical findings.

Figures (5)

• Figure 1: Random periodic solution does not dependent on the initial values
• Figure 2: Validate periodicity based on $\hat{X}_{t}^{*}(\Theta_{-\tau}\omega) = \hat{X}_{t-\tau}^{*}(\omega)$
• Figure 3: Validate periodicity via the periodicity of $\hat{X}^{0}(t,\Theta_{-t}\omega)$
• Figure 4: Mean square convergence orders for \ref{['expl:one']}
• Figure 5: Mean square convergence orders for \ref{['eq:exadditive']}

Theorems & Definitions (22)

• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 2.6
• proof