General approximation method for the distribution of Markov processes conditioned not to be killed
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 441-467.

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.

DOI : 10.1051/ps/2013045
Classification : 82C22, 65C50, 60K35, 60J60
Mots-clés : particle systems, conditional distributions
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     title = {General approximation method for the distribution of {Markov} processes conditioned not to be killed},
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     doi = {10.1051/ps/2013045},
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     url = {http://archive.numdam.org/articles/10.1051/ps/2013045/}
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Villemonais, Denis. General approximation method for the distribution of Markov processes conditioned not to be killed. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 441-467. doi : 10.1051/ps/2013045. http://archive.numdam.org/articles/10.1051/ps/2013045/

[1] I. Ben-Ari and R.G. Pinsky, Ergodic behavior of diffusions with random jumps from the boundary. Stoch. Proc. Appl. 119 (2009) 864-881. | MR | Zbl

[2] M. Bieniek, K. Burdzy and S. Finch, Non-extinction of a Fleming−Viot particle model. Probab. Theory Relat. Fields (2011) 1-40. | MR | Zbl

[3] M. Bieniek, K. Burdzy and S. Pal, Extinction of Fleming-Viot-type particle systems with strong drift. Electron. J. Probab. 17 (2012) 1-15. | MR | Zbl

[4] K. Burdzy, R. Holyst, D. Ingerman and P. March, Configurational transition in a fleming-viot-type model and probabilistic interpretation of laplacian eigenfunctions. J. Phys. A 29 (1996) 2633-2642. | Zbl

[5] K. Burdzy, R. Holyst and P. March, A Fleming−Viot particle representation of the Dirichlet Laplacian. Commun. Math. Phys. 214 (200) 679-703. | MR | Zbl

[6] P. Del Moral and L. Miclo, Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | MR | Zbl

[7] F. Delarue, Hitting time of a corner for a reflected diffusion in the square. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 946-961. | Numdam | MR | Zbl

[8] M.C. Delfour and J.-P. Zolésio, Shapes and geometries, Analysis, differential calculus, and optimization. Vol. 4, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001). | MR | Zbl

[9] P.A. Ferrari and N. Marić, Quasi stationary distributions and Fleming−Viot processes in countable spaces. Electron. J. Probab. 12 (2007) 684-702. | MR | Zbl

[10] A. Friedman, Nonattainability of a set by a diffusion process. Trans. Amer. Math. Soc. 197 (1974) 245-271. | MR | Zbl

[11] I. Grigorescu and M. Kang, Hydrodynamic limit for a Fleming−Viot type system. Stoch. Proc. Appl. 110 (2004) 111-143. | MR | Zbl

[12] I. Grigorescu and M. Kang, Ergodic properties of multidimensional Brownian motion with rebirth. Electron. J. Probab. 12 (2007) 1299-1322. | EuDML | MR | Zbl

[13] I. Grigorescu and M. Kang, Immortal particle for a catalytic branching process. Probab. Theory Relat. Fields (2011) 1-29. | MR | Zbl

[14] M. Kolb and D. Steinsaltz, Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Probab. 40 (2012) 162-212. | MR | Zbl

[15] M. Kolb and A. Wübker, On the Spectral Gap of Brownian Motion with Jump Boundary. Electron. J. Probab. 16 1214-1237. | MR | Zbl

[16] M. Kolb and A. Wübker, Spectral Analysis of Diffusions with Jump Boundary. J. Funct. Anal. 261 1992-2012. | MR | Zbl

[17] A. Lambert, Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007) 420-446. | EuDML | MR | Zbl

[18] J.-U. Löbus, A stationary Fleming−Viot type Brownian particle system. Math. Z. 263 (2009) 541-581. | MR | Zbl

[19] S. Méléard and D. Villemonais, Quasi-stationary distributions and population processes. Probab. Surveys 9 (2012) 340-410. | MR | Zbl

[20] P. Pollett, Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/˜pkp/papers/qsds/qsds.pdf

[21] S. Ramasubramanian, Hitting of submanifolds by diffusions. Probab. Theory Relat. Fields 78 (1988) 149-163. | MR | Zbl

[22] D. Revuz and M. Yor, Continuous martingales and Brownian motion, vol. 293, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 3rd edition (1999). | MR | Zbl

[23] M. Rousset, On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844. | MR | Zbl

[24] D. Villemonais, Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electron. J. Probab. 16 (2011) 1663-1692. | MR | Zbl

[25] W. Zhen and X. Hua, Multi-dimensional reflected backward stochastic differential equations and the comparison theorem. Acta Math. Sci. 30 (2010) 1819-1836. | MR | Zbl

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