An approximation scheme for quasi-stationary distributions of killed diffusions

In this paper we study the asymptotic behavior of the normalized weighted empirical occupation measures of a diffusion process on a compact manifold which is killed at a smooth rate and then regenerated at a random location, distributed according to the weighted empirical occupation measure. We show that the weighted occupation measures almost surely comprise an asymptotic pseudo-trajectory for a certain deterministic measure-valued semiflow, after suitably rescaling the time, and that with probability one they converge to the quasi-stationary distribution of the killed diffusion. These results provide theoretical justification for a scalable quasi-stationary Monte Carlo method for sampling from Bayesian posterior distributions.Comment: v2: revised version, 29 pages, 1 figur

Stochastic Approximation with Averaging Innovation Applied to Finance

The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the "innovations" satisfy some "light" averaging properties in the presence of a pathwise Lyapunov function. These averaging assumptions allow us to unify apparently remote frameworks where the innovations are simulated (possibly deterministic like in Quasi-Monte Carlo simulation) or exogenous (like market data) with ergodic properties. We propose several fields of applications and illustrate our results on five examples mainly motivated by Finance