Strong convergence rate of Euler-Maruyama approximations in temporal-spatial H\"older-norms

TL;DR

This paper proves that Euler-Maruyama approximations of stochastic differential equations converge at a rate of 0.5 in temporal-spatial H"older-norms, extending classical results to a new norm setting under smoothness conditions.

Contribution

It establishes the strong convergence rate of Euler-Maruyama methods in temporal-spatial H"older-norms, a novel analysis compared to traditional L^p-error metrics.

Findings

Convergence rate of 0.5 in H"older-norms.

Requires bounded derivatives of first and second order for coefficients.

Extends classical convergence results to new norm setting.

Abstract

Classical approximation results for stochastic differential equations analyze the $L^p$-distance between the exact solution and its Euler-Maruyama approximations. In this article we measure the error with temporal-spatial H\"older-norms. Our motivation for this are multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate $0.5$ with respect to temporal-spatial H\"older-norms if the coefficient functions have bounded derivatives of first and second order.