Convergence of the stochastic Euler scheme for locally Lipschitz coefficients

TL;DR

This paper proves the convergence of the Monte Carlo Euler method for simulating one-dimensional stochastic differential equations with superlinear polynomial growth coefficients, addressing a longstanding open problem.

Contribution

It establishes convergence results for the Euler scheme in cases with superlinearly growing coefficients, extending beyond globally Lipschitz conditions.

Findings

Convergence proven for a broad class of SDEs with polynomial growth coefficients.

Addresses the failure of weak convergence in superlinear cases.

Provides theoretical foundation for numerical simulation of complex SDEs.

Abstract

Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.