Weak Convergence in the Prokhorov Metric of Methods for Stochastic Differential Equations

TL;DR

This paper investigates the weak convergence of numerical methods for stochastic differential equations using the Prokhorov metric, providing bounds on convergence rates and linking weak and strong convergence through re-embedding techniques.

Contribution

It introduces an alternative formulation of weak convergence via the Prokhorov metric and establishes explicit bounds on convergence rates for a broad class of methods.

Findings

Bounds on convergence rates depend on test function smoothness

Numerical solutions can be re-embedded to converge strongly in probability space

Wasserstein distance convergence and rates are established

Abstract

We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of weak convergence in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of methods, we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of weak convergence and show explicitly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen - Dudley theorem to show that the numerical approximation and the true solution to the system of SDEs can be re-embedded in a probability space in such a way that the method converges there in a strong sense. One corollary of this last result is that the method converges in the Wasserstein distance, another metric on spaces of random variables. Another corollary establishes rates of convergence for expected values of test functions assuming only local Lipschitz continuity. We conclude with a review of the existing results for pathwise convergence of weakly converging methods and the corresponding strong results available under re-embedding.