### Sample per mode simulation for switching diffusions

**Krystul, J; Le Gland, F; Lezaud, P ***(Twente; Rennes; Toulouse)*

Tuesday 22 June 2010, 10:15-10:40

Seminar Room 1, Newton Institute

#### 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 $N^j$ 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.
\bibliographystyle{plain}
%\bibliography{}
\begin{thebibliography}{3}
\bibitem[Del Moral \& Lezaud, 2006]{DelMoral}
Del Moral, P. and Lezaud, P.,
\newblock Branching and interacting particle interpretations of rare event probabilities. (2006)
\newblock {\em in Stochastic Hybrid Systems : Theory and Safety Critical Applications, Henk Blom and John Lygeros, editors, Lecture Notes in Control and Information Sciences 337, pp. 277--323, Springer, Berlin, 2006}
\bibitem[Krystul, 2006]{Krystul}
Krystul, J.,
\newblock Modelling of Stochastic Hybrid Systems with Applications to Accident
Risk Assessment (2006).
\newblock {\em PhD Dissertation: University of Twente The Netherlands}.
\bibitem[LeGland \& Oudjane, 2006]{LeGland}
LeGland, F. and Oudjane, N.,
\newblock A sequential particle algorithm that keeps the particle system alive. (2006)
\newblock {\em in Stochastic Hybrid Systems : Theory and Safety Critical Applications, Henk Blom and John Lygeros, editors, Lecture Notes in Control and Information Sciences 337, pp. 351--389, Springer, Berlin, 2006

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