Importance sampling in path space for diffusion processes with slow-fast variables
Monte Carlo simulations can be improved by a change of measure that insures that the ``important'' region in the state space is sampled more frequently. This technique is called importance sampling. In this paper, this technique is studied for the path sampling problem for diffusion processes. The samples are drawn from a modified SDE to which a control force is added that drives the dynamics to the important regions in state space. The authors restrict their attention to certain exponential path functionals, for which an optimal change of measure exists (which can be found by solving a Hamilton-Jacobi-Bellmann (HJB) equation) that admits an importance sampling estimator with zero variance. They consider diffusions with both slow and fast variables and their main results is an upper bound for the relative error of the importance sampling estimators with control obtained from the limiting dynamics. Under certain assumptions and for small \(\varepsilon\), the relative error is bounded by \(C\varepsilon^{1/8}\), see Theorem 3.1. A numerical example is provided and some algorithmic issues related to the calculation of the suboptimal control force are discussion. As a secondary contribution, the authors also obtain several error estimates for approximation of the original process by the limiting process.
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