Variance Reduction for Simulated Diffusions

This article develops some variance reduction techniques for the Monte-Carlo integration of functionals of the solutions of Itô stochastic differential equations (sdes). The Monte-Carlo method for sdes offers a means of calculating solutions to certain types of parabolic partial differential equation and so has applications in various fields including stochastic control, particle physics and econometrics; it involves the representation of the required integrals as means of random variables defined on infinite-dimensional Wiener spaces, which cannot be simulated directly—sthey must at some stage be approximated by variables defined on high, but finite-dimensional spaces. The approach taken here is to construct variance reduced random variables on the infinite-dimensional spaces, which can subsequently be approximated by any of a number of known finite difference methods. The methods of control variates and importance sampling are developed. In both cases, a perfect variate (i.e., one which is unbiased and has zero variance) is first constructed by means of the Funke– Shevlyakov–Haussmann integral representation theorem for functionals of Itô processes. These involve terms which cannot be calculated exactly but which can be approximated to yield unbiased estimators of the desired integrals with reduced variances. A number of methods of approximating these terms are suggested and numerical experiments are performed for one of the methods. The experiments suggest that, at least in some cases, the method can yield variance reduction well worth the additional calculation required.