Minimal Errors for Strong and Weak Approximation of Stochastic Differential Equations

Abstract

We present a survey of results on minimal errors and optimality of algorithms for strong and weak approximation of systems of stochastic differential equations. For strong approximation, emphasis lies on the analysis of algorithms that are based on point evaluations of the driving Brownian motion and on the impact of non-commutativity, if present. Furthermore, we relate strong approximation to weighted integration and reconstruction of Brownian motion, and we demonstrate that the analysis of minimal errors leads to new algorithms that perform asymptotically optimal. In particular, these algorithms use a path-dependent step-size control. for weak approximation we consider the problem of computing the expected value of a functional of the solution, and we concentrate on recent results for a worst-case analysis either with respect to the functional or with respect to the coefficients of the system. Moreover, we relate weak approximation problems to average Kolmogorov widths and quantization numbers as well as to high-dimensional tensor product problems.