A stability vs. Monte-Carlo integration problem for SDEs
A talk in the Oberseminar Numerik series by
Evelyn Buckwar from JKU Linz
| Abstract: | In this talk 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.
This is joint work with Markus Ableidinger and Andreas Thalhammer. |