An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and p-th moment unstable equilibrium
Sprache des Vortragstitels:
12th German Probability and Statistics Days 2016 - Bochumer Stochastik-Tage
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In this work we investigate the influence of almost sure and p-th moment stability for (linear) SDEs on Monte Carlo methods for estimating the p-th moment of the solution process.
In the situation where the zero solution of the SDE is asymptotically stable in the almost sure sense but unstable in p-th moment sense, the latter 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 $p$-th moment 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 a combined approach: we first employ an appropriately tuned importance sampling technique to deal with the rare event simulations and afterwards apply further variance reduction techniques, such as multilevel Monte Carlo, to control the variance of the modified Monte Carlo estimator. As an illustrative example we discuss the numerical treatment of the stochastic heat equation and present simulation results.