The method of adaptive ≤ control variables is an efficient Monte Carlo approach to compute boundary crossing probabilities (BCP) for Brownian motion and a large class of diffusion processes. Let N denote the number of (univariate) Gaussian variables used for the MC estimation. Typically for infinite-dimensional MC methods, the convergence rate is less than the finite-dimensional O(1/N).

The boundary b (or the boundaries in case of two-sided boundary crossing probabilities) is approximated by a piecewise linear boundary b_m, which is linear on m intervals. The mean squared error for the boundary b_m is of order O(m/N), leading to a mean squared error for the boundary b order O(1/N^β) with β = 2α/(2α + 1), if the difference of the (exact) BCP's for b and b_m is O(1/m^α). Let b_k 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 b_k is the control variable. Iterated it improves the convergence rate of the MC estimator to O(1/N^1−ε) for all ε > 0, reducing the problem of estimating the BCP to an essentially finite-dimensional problem.

Keywords: Boundary crossing probability; First passage time; Adaptive control variable; First hitting time

Biography: Main research interests are boundary crossing probabilities, decision problems for high dimensional data, quantization.