Improving the Monte Carlo estimation of boundary crossing probabilities by control variables

Abstract.

We propose an efficient Monte Carlo approach to compute boundary crossing probabilities (BCP) for Brownian motion and a large class of diffusion processes, the method of adaptive control variables. For the Brownian motion the boundary b (or the boundaries in case of two-sided boundary crossing probabilities) is approximated by a piecewise linear boundary , which is linear on m intervals. Monte Carlo estimators of the corresponding BCP are based on an m-dimensional Gaussian distribution. Let N denote the number of (univariate) Gaussian variables used. The mean squared error for the boundary is of order , leading to a mean squared error for the boundary b of order with , if the difference of the (exact) BCP's for b and is . Typically, for infinite-dimensional Monte Carlo methods, the convergence rate is less than the finite-dimensional .

Let be a further approximating boundary which is linear on k intervals. If k is small compared to m, the corresponding BCP may be estimated with high accuracy. The BCP for as control variable improves the convergence rate of the Monte Carlo estimator to with . The constant depends on the correlation of the estimators for and . We show that this method of adaptive control variable improves the convergence rate considerably. Iterating control variables leads to a rate of convergence (of the mean squared error) of order , reducing the problem of estimating the BCP to an essentially finite-dimensional problem.