Klaus Poetzelberger

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* = 2*α*/(2*α* + 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.