Particle exchange Monte Carlo methods for eigenfunction and related nonlinear problems

TL;DR

This paper introduces a new particle exchange Monte Carlo method for eigenfunction problems where the eigenvalue is unknown, involving a pair of processes evolving forward and backward in time, with applications to quasistationary distributions and stochastic control.

Contribution

It develops a novel particle exchange Monte Carlo approach for eigenfunction problems with unknown eigenvalues, extending existing methods to new problem classes.

Findings

Effective particle exchange rule for unknown eigenvalues

Application to quasistationary distributions demonstrated

Potential for stochastic control problems explored

Abstract

We introduce and develop a novel particle exchange Monte Carlo method. Whereas existing methods apply to eigenfunction problems where the eigenvalue is known (e.g., integrals with respect to a Gibbs measure, which can be interpreted as corresponding to eigenvalue zero), here the focus is on problems where the eigenvalue is not known a priori. To obtain an appropriate particle exchange rule we must consider a pair of processes, with one evolving forward in time and the other backward. Applications to eigenfunction problems corresponding to quasistationary distributions and ergodic stochastic control are discussed.