Lower Error Bounds for Strong Approximation of Scalar SDEs with non-Lipschitzian Coefficients

TL;DR

This paper establishes fundamental lower bounds on the error of approximating scalar SDEs with non-Lipschitz coefficients using finitely many Brownian motion observations, applicable to many complex models.

Contribution

It extends lower error bounds to SDEs with locally regular coefficients, including non-globally Lipschitz cases, providing sharp bounds for a wide class of equations.

Findings

Lower bounds apply to equations with locally regular coefficients.

Bounds are sharp for many non-Lipschitz SDEs.

Results hold for Cox-Ingersoll-Ross and superlinearly growing coefficient equations.

Abstract

We study pathwise approximation of scalar stochastic differential equations at a single time point or globally in time by means of methods that are based on finitely many observations of the driving Brownian motion. We prove lower error bounds in terms of the average number of evaluations of the driving Brownian motion that hold for every such method under rather mild assumptions on the coefficients of the equation. The underlying simple idea of our analysis is as follows: the lower error bounds known for equations with coefficients that have sufficient regularity globally in space should still apply in the case of coefficients that have this regularity in space only locally, in a small neighborhood of the initial value. Our results apply to a huge variety of equations with coefficients that are not globally Lipschitz continuous in space including Cox-Ingersoll-Ross processes, equations with superlinearly growing coefficients, and equations with discontinuous coefficients. In many of these cases the resulting lower error bounds even turn out to be sharp.