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We consider a strong Markov process with killing and prove an approximationmethod for the distribution of the process conditioned not to be killed when itis observed. The method is based on a Fleming-Viot type particle system withrebirths, whose particles evolve as independent copies of the original strongMarkov process and jump onto each others instead of being killed. Our onlyassumption is that the number of rebirths of the Fleming-Viot type systemdoesn't explode in finite time almost surely and that the survival probabilityof the original process remains positive in finite time. The approximationmethod generalizes previous results and comes with a speed of convergence. Acriterion for the non-explosion of the number of rebirths is also provided forgeneral systems of time and environment dependent diffusion particles. Thisincludes, but is not limited to, the case of the Fleming-Viot type system ofthe approximation method. The proof of the non-explosion criterion uses anoriginal non-attainability of $(0,0)$ result for pair of non-negativesemi-martingales with positive jumps.

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