Convergence rate of numerical scheme for SDEs with a distributional drift in Besov space

TL;DR

This paper develops and analyzes an Euler-Maruyama numerical scheme for one-dimensional SDEs with distributional drifts in Besov spaces, establishing convergence rates and demonstrating practical implementation.

Contribution

It introduces a novel numerical scheme for SDEs with distributional drifts in Besov spaces and proves its convergence rate.

Findings

Established strong $L^1$ convergence rate for the scheme

Implemented the scheme and discussed numerical results

Extended numerical analysis to SDEs with generalized distributional drifts

Abstract

This paper is concerned with numerical solutions of one-dimensional SDEs with the drift being a generalised function, in particular belonging to the H\"older-Zygmund space $C^{-\gamma}$ of negative order $-\gamma<0$ in the spatial variable. We design an Euler-Maruyama numerical scheme and prove its convergence, obtaining an upper bound for the strong $L^1$ convergence rate. We finally implement the scheme and discuss the results obtained.