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 bm, which is linear on m intervals. The mean squared error for the boundary bm is of order O(m/N), leading to a mean squared error for the boundary b order O(1/Nβ) with β = 2α/(2α + 1), if the difference of the (exact) BCP's for b and bm is O(1/mα). Let bk 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 bk is the control variable. Iterated it improves the convergence rate of the MC estimator to O(1/N1−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.