Title:Compactness Approaches for Importance Sampling.

Abstract: Importance sampling (IS, i.e. change of probability measure) is a powerful tool that can provide large efficiency gains to Monte Carlo simulations. However, this approach can easily backfire and lead to estimators that under-perform even crude Monte Carlo. Theoretical guarantees of efficiency are difficult to prove outside of relatively simple settings. We develop a novel technique for proving the optimality properties of IS estimators for complex stochastic systems. The approach allows for a broad class of importance measures (relative to standard exponential “twisting”) and thus may be applied in very general settings. A key tool in the development is the large deviations (LD) counterpart to Prohorov’s (relative compactness) theorem from weak convergence theory. It facilitates the analysis of an estimator over subsequences along which the LD principle is guaranteed to exist and the estimator exhibits its worst-case performance. This is distinct from the standard program where the full LD principle for the system must be established. We illustrate the new technique on a challenging application in portfolio credit risk.