Brownian Meanders, Importance Sampling and Unbiased Simulation of Diffusion Extremes

Computing expected values of functions involving extreme values of diffusion processes can find wide ap-plications in financial engineering. Conventional discretization simulation schemes often converge slowly. We propose a Wiener-measure-decomposition based approach to construct unbiased Monte Carlo estimators. Combined with the importance sampling technique and the Williams path decomposition of Brownian motion, this approach transforms simulating extreme values of a general diffusion process to simulating two Brownian meanders. Numerical experiments show this estimator performs efficiently for diffusions with and without boundaries. Key words: stochastic differential equation; exact simulation; importance sampling; extreme values; Brownian meanders 1