Strong convergence in the infinite horizon of numerical methods for stochastic differential equations

TL;DR

This paper proves the strong convergence of numerical methods for stochastic differential equations over an infinite time horizon, including applications to the backward Euler-Maruyama method with super-linear coefficients, supported by numerical simulations.

Contribution

It establishes the strong convergence for a broad class of numerical methods over infinite time, including a new result for the backward Euler-Maruyama method with super-linear coefficients.

Findings

Strong convergence in the infinite horizon for numerical SDE methods.

Convergence of numerical stationary distribution to the true distribution.

Validation through numerical simulations.

Abstract

The strong convergence of numerical methods for stochastic differential equations (SDEs) for $t\in[0,\infty)$ is proved. The result is applicable to any one-step numerical methods with Markov property that have the finite time strong convergence and the uniformly bounded moment. In addition, the convergence of the numerical stationary distribution to the underlying one can be derived from this result. To demonstrate the application of this result, the strong convergence in the infinite horizon of the backward Euler-Maruyama method in the $L^p$ sense for some small $p\in (0,1)$ is proved for SDEs with super-linear coefficients, which is also a a standalone new result. Numerical simulations are provided to illustrate the theoretical results.