# 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 $\{\vv X(t)\}_{t\geq 0}$ and importance function $S$ over state space $\cc X$. We will assume for the moment that $\cc X=\bb R^d.$

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

Further, take, wlog, $S(\vv X(0))=0.$

We assume that for any threshold $\gamma\gt 0$ that the entry times to the sets $\{S(\vv X(t))\geq \gamma\}$ and $\{S(\vv X(t))\lt 0\}$ are well-defined stopping times.

We wish to estimate the probability $\bb P[\tau_\gamma\lt\tau_0]=\bb P[E_{\gamma}]$ where $E_{\gamma}$ is the event $E_{\gamma}=\{\tau_\gamma \lt \tau_0\}.$

Now consider a sequence of thresholds $\gamma_0\leq\gamma_1\leq\dots\gamma_L=\gamma.$ The process must reach $\gamma_i$ to reach $\gamma_j$ for $i, and so we have a nested series of events $E_{\gamma_0}\supseteq E_{\gamma_0}\supseteq\dots E_{\gamma_L},$ and thus

where $c_i=P(E_{\gamma_i}|E_{\gamma_{y-1}}).$

## Splitting for static problems

We have a density $f$ on $\bb R^d$ and a quasi-monotone importance function $S: \bb R^d\rightarrow\bb R.$ We wish to simulate from a conditional

where $\ell(\gamma)=\bb P[S(\vv X)\geq\gamma)],\quad \vv X \sim f.$