Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T08:16:03.920Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ben-Ari, I. and Pinsky, R.G., Ergodic behavior of diffusions with random jumps from the boundary. Stoch. Proc. Appl. 119 (2009) 864–881. Google Scholar
M. Bieniek, K. Burdzy and S. Finch, Non-extinction of a Fleming−Viot particle model. Probab. Theory Relat. Fields (2011) 1–40.
Bieniek, M., Burdzy, K. and Pal, S., Extinction of Fleming-Viot-type particle systems with strong drift. Electron. J. Probab. 17 (2012) 1–15. Google Scholar
Burdzy, K., Holyst, R., Ingerman, D. and March, P., Configurational transition in a fleming-viot-type model and probabilistic interpretation of laplacian eigenfunctions. J. Phys. A 29 (1996) 2633–2642. Google Scholar
Burdzy, K., Holyst, R. and March, P., A Fleming−Viot particle representation of the Dirichlet Laplacian. Commun. Math. Phys. 214 (200) 679–703.
Del Moral, P. and Miclo, L., Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171–208. Google Scholar
Delarue, F., Hitting time of a corner for a reflected diffusion in the square. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 946–961. Google Scholar
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).
Ferrari, P.A. and Marić, N., Quasi stationary distributions and Fleming−Viot processes in countable spaces. Electron. J. Probab. 12 (2007) 684–702. Google Scholar
Friedman, A., Nonattainability of a set by a diffusion process. Trans. Amer. Math. Soc. 197 (1974) 245–271. Google Scholar
Grigorescu, I. and Kang, M., Hydrodynamic limit for a Fleming−Viot type system. Stoch. Proc. Appl. 110 (2004) 111–143. Google Scholar
Grigorescu, I. and Kang, M., Ergodic properties of multidimensional Brownian motion with rebirth. Electron. J. Probab. 12 (2007) 1299–1322. Google Scholar
I. Grigorescu and M. Kang, Immortal particle for a catalytic branching process. Probab. Theory Relat. Fields (2011) 1–29.
Kolb, M. and Steinsaltz, D., Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Probab. 40 (2012) 162–212. Google Scholar
Kolb, M. and Wübker, A., On the Spectral Gap of Brownian Motion with Jump Boundary. Electron. J. Probab. 16 1214–1237.
Kolb, M. and Wübker, A., Spectral Analysis of Diffusions with Jump Boundary. J. Funct. Anal. 261 1992–2012.
Lambert, A., Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007) 420–446. Google Scholar
Löbus, J.-U., A stationary Fleming−Viot type Brownian particle system. Math. Z. 263 (2009) 541–581. Google Scholar
Méléard, S. and Villemonais, D., Quasi-stationary distributions and population processes. Probab. Surveys 9 (2012) 340–410. Google Scholar
P. Pollett, Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/˜pkp/papers/qsds/qsds.pdf
Ramasubramanian, S., Hitting of submanifolds by diffusions. Probab. Theory Relat. Fields 78 (1988) 149–163. Google Scholar
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).
Rousset, M., On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824–844. Google Scholar
Villemonais, D., Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electron. J. Probab. 16 (2011) 1663–1692. Google Scholar
Zhen, W. and Hua, X., Multi-dimensional reflected backward stochastic differential equations and the comparison theorem. Acta Math. Sci. 30 (2010) 1819–1836. Google Scholar