Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients

TL;DR

This paper investigates the asymptotic distribution of errors in Euler scheme approximations for SDEs with locally Lipschitz coefficients, extending classical results to more realistic conditions.

Contribution

It establishes convergence rates and the asymptotic normalized error distribution for Euler schemes under locally Lipschitz conditions, filling key theoretical gaps.

Findings

Derived the rate of convergence for Euler approximations with locally Lipschitz coefficients.

Identified the asymptotic distribution of the normalized error process.

Provided conditions ensuring finite variance of the error.

Abstract

In traditional work on numerical schemes for solving stochastic differential equations (SDEs), it is usually assumed that the coefficients are globally Lipschitz. This assumption has been used to establish a powerful analysis of the numerical approximations of the solutions of stochastic differential equations. In practice, however, the globally Lipschitz assumption on the coefficients is on occasion too stringent a requirement to meet. Some Brownian motion driven SDEs used in applications have coefficients that are Lipschitz only on compact sets. Reflecting the importance of the locally Lipschitz case, it has been well studied in recent years, yet some simple to state, fundamental results remain unproved. We attempt to fill these gaps in this paper, establishing both a rate of convergence, but also we find the asymptotic normalized error process of the error process arising from a sequence of approximations. The result is analogous to the original result of this type, established in KT1991-2 back in 1991. This result was improved in 1998 in JJ1998, and recently(2009) it was partially extended in AN. As we indicate, the results in our paper provide the basis of a statistical analysis of the error; in this spirit we give conditions for a finite variance.