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

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.