Jonathan Weare (CIMS)

Abstract:

I will discuss an importance sampling method for certain rare event problems involving small noise diffusions. Standard Monte Carlo schemes for these problems behave exponentially poorly in the small noise limit. Previous work in rare event simulation has focused on developing, in very specific situations, estimators with optimal exponential variance decay rates. This criterion still allows for exponential growth of the statistical relative error. I will introduce an estimator related to a deterministic control problem that not only has an optimal variance decay rate under certain conditions, but that can even have vanishingly small statistical relative error in the small noise limit. The method can be seen as the limit of a well known zero variance importance sampling scheme for diffusions which requires the solution of a second order partial differential equation. I will also give several numerical illustrations of our results.