The Annals of Applied Probability

Stochastic approximation of quasi-stationary distributions on compact spaces and applications

Michel Benaim, Bertrand Cloez, and Fabien Panloup

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Abstract

As a continuation of a recent paper, dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ensuring the convergence of this measure to the quasi-stationary distribution of the chain. We then apply this method to the numerical approximation of the quasi-stationary distribution of a diffusion process killed on the boundary of a compact set. Finally, the sharpness of the assumptions is illustrated through the study of the algorithm in a nonirreducible setting.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2370-2416.

Dates
Received: December 2016
Revised: September 2017
First available in Project Euclid: 9 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1533780276

Digital Object Identifier
doi:10.1214/17-AAP1360

Mathematical Reviews number (MathSciNet)
MR3843832

Zentralblatt MATH identifier
06974754

Subjects
Primary: 65C20: Models, numerical methods [See also 68U20] 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03] 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 60J60: Diffusion processes [See also 58J65]

Keywords
Quasi-stationary distributions stochastic approximation reinforced random walks random perturbations of dynamical systems extinction rate Euler scheme

Citation

Benaim, Michel; Cloez, Bertrand; Panloup, Fabien. Stochastic approximation of quasi-stationary distributions on compact spaces and applications. Ann. Appl. Probab. 28 (2018), no. 4, 2370--2416. doi:10.1214/17-AAP1360. https://projecteuclid.org/euclid.aoap/1533780276


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