General approximation method for the distribution of Markov processes conditioned not to be killed

TL;DR

This paper introduces an approximation method using Fleming-Viot type particle systems to estimate the distribution of Markov processes conditioned on survival, with proven convergence speed and criteria for non-explosion.

Contribution

It generalizes previous approximation techniques for killed Markov processes and provides a new non-explosion criterion applicable to complex diffusion systems.

Findings

The method effectively approximates conditioned distributions.

Convergence speed of the approximation is established.

A non-explosion criterion for particle systems is proven.

Abstract

We consider a strong Markov process with killing and prove an approximation method for the distribution of the process conditioned not to be killed when it is observed. The method is based on a Fleming-Viot type particle system with rebirths, whose particles evolve as independent copies of the original strong Markov process and jump onto each others instead of being killed. Our only assumption is that the number of rebirths of the Fleming-Viot type system doesn't explode in finite time almost surely and that the survival probability of the original process remains positive in finite time. The approximation method generalizes previous results and comes with a speed of convergence. A criterion for the non-explosion of the number of rebirths is also provided for general systems of time and environment dependent diffusion particles. This includes, but is not limited to, the case of the Fleming-Viot type system of the approximation method. The proof of the non-explosion criterion uses an original non-attainability of $(0,0)$ result for pair of non-negative semi-martingales with positive jumps.