A Variable Step Size Riemannian Sum for an Itô Integral

Abstract

We investigate the problem of using a Riemannian sum with random subintervals to approximate the iterated Itô integral w dw-or, equivalently, solving the corresponding stochastic differential equation by Euler's method with variable step sizes. In the past this task has been used as a counterexample to illustrate that variable step sizes must be used with extreme caution in stochastic numerical analysis. This article establishes a class of variable step size schemes which do work.