Subsolutions of an Isaacs equation and efficient schemes for importance sampling

It was established in Dupuis and Wang [Dupuis, P., H. Wang. 2004. Importance sampling, large deviations, and differential games. Stoch. Stoch. Rep. 76 481-508, Dupuis, P., H. Wang. 2005. Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15 1-38] that importance sampling algorithms for estimating rare-event probabilities are intimately connected with two-person zero-sum differential games and the associated Isaacs equation. This game interpretation shows that dynamic or state-dependent schemes are needed in order to attain asymptotic optimality in a general setting. The purpose of the present paper is to show that classical subsolutions of the Isaacs equation can be used as a basic and flexible tool for the construction. and analysis of efficient dynamic importance sampling schemes. There are two main contributions. The first is a basic theoretical result characterizing the asymptotic performance of importance sampling estimators based on subsolutions. The second is an explicit method for constructing classical subsolutions as a mollification of piecewise affine functions. Numerical examples are included for illustration and to demonstrate that simple, nearly asymptotically optimal importance sampling schemes can be obtained for a variety of problems via the subsolution approach.