An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and mean-square unstable equilibrium

In this work we investigate the interplay of almost sure and mean-square stability for linear SDEs and the Monte Carlo method for estimating the second moment of the solution process. In the situation where the zero solution of the SDE is asymptotically stable in the almost sure sense but asymptotically mean-square unstable, the latter property is determined by rarely occurring trajectories that are sufficiently far away from the origin. The standard Monte Carlo approach for estimating higher moments essentially computes a finite number of trajectories and is bound to miss those rare events. It thus fails to reproduce the correct mean-square dynamics (under reasonable cost). A straightforward application of variance reduction techniques will typically not resolve the situation unless these methods force the rare, exploding trajectories to happen more frequently. Here we propose an appropriately tuned importance sampling technique based on Girsanov's theorem to deal with the rare event simulation. In addition further variance reduction techniques, such as multilevel Monte Carlo, can be applied to control the variance of the modified Monte Carlo estimators. As an illustrative example we discuss the numerical treatment of the stochastic heat equation with multiplicative noise and present simulation results.