Sampling per mode simulation for switching diffusions

Abstract

We consider the problem of rare event estimation in switching diffusions using an Interacting Particle Systems (IPS) based Monte Carlo simulation approach \cite{DelMoral}. While in theory the IPS approach is virtually applicable to any strong Markov process, in practice the straightforward application of this approach to switching diffusions may fail to produce reasonable estimates within a reasonable amount of simulation time. The reason is that there may be few if no particles in modes with small probabilities (i.e.\ "light" modes). This happens because each resampling step tends to sample more "heavy" particles from modes with higher probabilities, thus, "light" particles in the "light" modes tend to be discarded. This badly affects IPS estimation performance. By increasing the number of particles the IPS estimates should improve but only at the cost of substantially increased simulation time which makes the performance of IPS approach in switching diffusions similar to one of the standard Monte Carlo. To avoid this, a conditional "sampling per mode" algorithm has been proposed in \cite{Krystul}; instead of starting the algorithm with N particles randomly distributed, we draw in each mode j, a fixed number Nj particles and at each resampling step, the same number of particles is sampled for each visited mode. Using the techniques introduced in \cite{LeGland}, we recently established a Law of Large Number theorem as well as a Central Limit Theorem for the estimate of the rare event probability