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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   .

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