Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling

It was established in [6, 7] 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. ∗Research of this author supported in part by the National Science Foundation (NSFDMS-0306070 and NSF-DMS-0404806) and the Army Research Office (DAAD19-02-10425). †Research of this author supported in part by the National Science Foundation (NSFDMS-0103669 and NSF-DMS-0404806).