Volume 18, 2014
|Page(s)||441 - 467|
|Published online||08 October 2014|
General approximation method for the distribution of Markov processes conditioned not to be killed
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
Revised: 1 April 2013
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.
Mathematics Subject Classification: 82C22 / 65C50 / 60K35 / 60J60
Key words: Particle systems / conditional distributions
© EDP Sciences, SMAI 2014
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