Estimating Rare Event Probabilities in Large Scale Stochastic Hybrid Systems by Sequential Monte Carlo Simulation

We study the problem of estimating small reachability probabilities for large scale stochastic hybrid processes through Sequential Monte Carlo (SMC) simulation. Recently, [Cerou et al., 2002, 2005] developed an SMC approach for diffusion processes, and referred to the resulting SMC algorithm as an Interacting Particle System (IPS). In [Krystul&Blom, 2004, 2005] it was shown that this IPS approach works very well for a diffusion example, but has its limits when applied to a switching diffusion with large differences in discrete state (mode) probabilities or with rare mode switching. In order to cope with these problems, in [Krystul&Blom, 2004, 2005, 2006] the IPS approach has been extended to Hybrid IPS (HIPS) versions. Unfortunately, these HIPS versions may need impractically many particles when the space of the discrete state component is very large. Such situation typically occurs when the stochastic process considered is highly distributed and incorporates many local discrete valued switching processes. Then the vector of local discrete valued components has a state space the size of which is exponentially large. The aim of the current work is formulate the estimation of extremely small rare event probabilities in stochastic hybrid systems with a large state space for the discrete valued process component into one of a hierarchical estimation process, and to use this for the derivation of a Hierarchical HIPS version. The effectiveness of the approach is illustrated for evaluating the risk of collision between two aircraft in a scenario of the future.