Feynman-Kac particle integration with geometric interacting jumps

TL;DR

This paper develops and analyzes new discrete-time Feynman-Kac particle models with geometric interacting jumps, providing non-asymptotic bias and variance bounds for both continuous and discrete reference processes.

Contribution

It introduces novel approximation models with geometric interacting jumps and provides the first non-asymptotic bias and variance theorems for this class of models.

Findings

Derived non-asymptotic bias bounds.

Established variance estimates for particle systems.

Provided uniform convergence results over time.

Abstract

This article is concerned with the design and analysis of discrete time Feynman-Kac particle integration models with geometric interacting jump processes. We analyze two general types of model, corresponding to whether the reference process is in continuous or discrete time. For the former, we consider discrete generation particle models defined by arbitrarily fine time mesh approximations of the Feynman-Kac models with continuous time path integrals. For the latter, we assume that the discrete process is observed at integer times and we design new approximation models with geometric interacting jumps in terms of a sequence of intermediate time steps between the integers. In both situations, we provide non asymptotic bias and variance theorems w.r.t. the time step and the size of the system, yielding what appear to be the first results of this type for this class of Feynman-Kac particle integration models. We also discuss uniform convergence estimates w.r.t. the time horizon. Our approach is based on an original semigroup analysis with first order decompositions of the fluctuation errors.