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

Published online by Cambridge University Press:  08 October 2014

Denis Villemonais
Denis Villemonais*
Affiliation:
Institut Élie Cartan de Nancy, Université de Lorraine; TOSCA project-team, INRIA Nancy – Grand Est; IECN – UMR 7502, Université de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy cedex, France. denis.villemonais@univ-lorraine.fr
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Abstract

We consider a strong Markov process with killing and prove an approximation method forthe distribution of the process conditioned not to be killed when it is observed. Themethod is based on a Fleming−Viot type particle system with rebirths, whose particles evolve asindependent copies of the original strong Markov process and jump onto each others insteadof being killed. Our only assumption is that the number of rebirths of theFleming−Viot type systemdoesn’t explode in finite time almost surely and that the survival probability of theoriginal process remains positive in finite time. The approximation method generalizesprevious results and comes with a speed of convergence. A criterion for the non-explosionof the number of rebirths is also provided for general systems of time and environmentdependent diffusion particles. This includes, but is not limited to, the case of theFleming−Viot type system ofthe approximation method. The proof of the non-explosion criterion uses an originalnon-attainability of (0,0)result for pair of non-negative semi-martingales with positive jumps.

Keywords

Particle systemsconditional distributions
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 441 - 467
Copyright
© EDP Sciences, SMAI 2014

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