$L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs

TL;DR

This paper develops a unified approach to determine the strong $L^p$-convergence rates of backward Euler schemes for monotone stochastic differential equations, including SDEs and SPDEs with polynomial growth coefficients.

Contribution

It introduces a general method for analyzing convergence rates of backward Euler schemes for a broad class of monotone SDEs and SPDEs, extending previous results.

Findings

Derived convergence rates for backward Euler schemes in $L^p$-norm for $p \,\geq 4$.

Applied results to SODEs with polynomial growth coefficients.

Extended analysis to Galerkin-based backward Euler schemes for SPDEs.

Abstract

We give a unified method to derive the strong convergence rate of the backward Euler scheme for monotone SDEs in $L^p(\Omega)$-norm, with general $p \ge 4$. The results are applied to the backward Euler scheme of SODEs with polynomial growth coefficients. We also generalize the argument to the Galerkin-based backward Euler scheme of SPDEs with polynomial growth coefficients driven by multiplicative trace-class noise.