In this talk, we consider (multilevel) Monte Carlo-based methods for estimating E[f(X(T))], where X(T) denotes the solution of a stochastic differential equation at a given time $T$. It is a well-known result that the resulting Monte Carlo error can be controlled by either enlarging the number of realisations or by applying appropriate variance reduction methods. Obviously, a natural bound on the number of trajectories is imposed by the computational cost of the time integration method, which limits the possibility of increasing the number of numerical
trajectories for high dimensional SODEs - especially for systems arising from semi-discretised stochastic partial differential equations (SPDEs). For this reason, we want to reduce the underlying Monte Carlo error by variance reduction techniques based on importance sampling.
The first part of the talk consists of analysing finite-dimensional linear SODE systems. In particular, we show that in the presence of a strongly scaled, multiplicative noise, the standard and multilevel Monte Carlo estimators typically fail to reproduce the right dynamics of the second moment of the solution. The main reason for this failure is that the Monte Carlo estimators heavily depend on rare events. Thus, in order to reduce the variance, we propose an importance sampling - rare event simulation technique based on stability properties of the trivial solution such that occurring Monte Carlo error is significantly reduced.
In the second part, we show how importance sampling techniques can be used for approximations of E[f(X(T))] for mild solutions of SPDEs.