A numerical scheme for stochastic differential equations with distributional drift

TL;DR

This paper introduces a novel numerical scheme for solving one-dimensional stochastic differential equations with distributional drifts in fractional Sobolev spaces, providing convergence rates and practical implementation.

Contribution

It is the first to develop and implement a numerical method for SDEs with distributional drifts, extending the scope of stochastic numerical analysis.

Findings

Established convergence rate in $L^1$-norm for the scheme.

Successfully implemented the numerical scheme.

Provided convergence estimates for drifts in $L^p$-spaces.

Abstract

In this paper we present a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a rate of convergence in a suitable $L^1$-norm and we implement the scheme numerically. To the best of our knowledge this is the first paper to study (and implement) numerical solutions of SDEs whose drift lives in a space of distributions. As a byproduct we also obtain an estimate of the convergence rate for a numerical scheme applied to SDEs with drift in $L^p$-spaces with $p\in(1,\infty)$.