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

Abstract

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