Section: New Results

Marginalization in rare event simulation for switching diffusions

Participant : François Le Gland.

This is a collaboration with Anindya Goswami (IISER, Poone).

Switching diffusions are continuous–time Markov processes with a hybrid continuous / finite state space. A rare but critical event (such as a scalar function of the continuous component of the state exceeding a given threshold) can occur for several reasons:

• the process can remain in nominal mode, where the critical event is very unlikely to occur,

• or the process can switch in some degraded mode, where the critical event is much more likely to occur, but the switching itself is very unlikely to occur.

Not only is it important to accurately estimate the (very small) probability that the critical event occurs before some fixed final time, but it is also important to have an accurate account on the reason why it occured, or in other words to estimate the probability of the different modes. A classical implementation of the multilevel splitting would not be efficient. Indeed, as soon as (even a few) samples paths switch to a degraded mode, these sample paths will dominate and it will not be possible to estimate the contribution of samples paths in the nominal mode. Moreover, sampling the finite component of the state is not efficient to accurately estimate the (very small) probability of rare but critical modes. A more efficient implementation is based on marginalization, i.e. in sampling jointly the continuous component and the probability distribution of the finite component given the past continuous component [18] . The latter is a probability vector, known as the Wonham filter, that satisfies a deterministic equation.