Finite variance unbiased estimation of stochastic differential equations

Abstract

We develop a new unbiased estimation method for Lipschitz continuous functions of multi-dimensional stochastic differential equations with Lipschitz continuous coefficients. This method provides a finite variance estimator based on a probabilistic representation which is similar to the recent representations obtained through the parametrix method and recursive application of the automatic differentiation formula. Our approach relies on appropriate change of variables to carefully handle the singular integrands appearing in the iterated integrals of the probabilistic representation. It results in a scheme with randomized intermediate times where the number of intermediate times has a Pareto distribution.