Interacting particle processes and approximation of Markov processes conditioned to not be killed

Details Interacting particle processes and approximation of Markov processes conditioned to not be killed Interacting particle processes and approximation of Markov processes conditioned to not be killed

2011-06-05

arxiv.org/abs/1106.0878v1

Mathematics

Probability

Scientific paper

We prove an approximation method for general strong Markov processes conditioned to not be killed. The method is based on a Fleming-Viot type interacting particle system, whose particles evolve as independent copies of the original strong Markov process and jump onto each others instead of being killed. We only assume that the number of jumps of the Fleming-Viot type system doesn't explode in finite time almost surely, and that the survival probability at fixed time of the original process is positive. We also give a speed of convergence for the approximation method. A criterion for the non-explosion of the number of jumps is then given for general systems of time and environment dependent diffusion particles, which includes the case of the Fleming-Viot type system of the approximation method. The proof of the criterion uses an original non-attainability of (0,0) result for a pair of non-negative semi-martingales with positive jumps.

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