On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient

Abstract

We survey recent developments in the field of complexity of pathwise approximation in $p$-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least $1/2$ in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.