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Dynamic Splitting simulation

Splitting for Markov processes

A particular method for simulation-based estimation for rare events about which I am learning. The original Dynamic splitting algorithm (KaHa51) works for rare states in Markov chains.

Consider a Markov process and importance function over state space . We will assume for the moment that

Suppose is quasi-monotone in its vector argument – that is, that

Further, take, wlog,

We assume that for any threshold that the entry times to the sets and are well-defined stopping times.

We wish to estimate the probability where is the event

Now consider a sequence of thresholds The process must reach to reach for , and so we have a nested series of events and thus

where

Splitting for static problems

We have a density on and a quasi-monotone importance function We wish to simulate from a conditional

where

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