Importance sampling in path space for diffusion processes
Wei Zhang, Carsten Hartmann, Markus Weber, Christof Schütte · arXiv (Cornell University) · 2013
Abstract. Importance sampling is a widely used technique to reduce the variance of the Monte Carlo method. It uses the idea of change of measure to design efficient Monte Carlo estimators. In this work, we study the importance sampling method in the framework of diffusion process and consider the change of measures which can be realized by adding a control force to the original dynamics. For certain exponential-type expectations, the corresponding control force of the optimal change of measure leads to a zero-variance estimator that is related to the solution of a Hamilton-Jacobi-Bellmann equation. We first show that, for a general suboptimal control force, the variance of the resulting estimator is bounded by the L ∞ distance between the suboptimal and the optimal control force. We consider three situations in which we can approximate this optimal control force, thus obtaining efficient estimators with small variance. Numerical examples show the effectiveness of these approximation strategies. The asymptotic optimality of the change of measure approach is proved.