Linear Variance Bounds for Particle Approximations of Time-Homogeneous Feynman-Kac Formulae

TL;DR

This paper proves that under certain conditions, the variance of particle approximations for time-homogeneous Feynman-Kac formulae grows linearly over time, improving understanding of their stability in various applications.

Contribution

It establishes sufficient conditions involving a multiplicative drift condition for linear-in-time variance bounds in particle approximations of Feynman-Kac models.

Findings

Variance grows linearly under specified conditions

Conditions are flexible for non-compact state spaces

Applicable to extreme cases with unbounded potentials or non-ergodic kernels

Abstract

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance of particle approximations of time-homogeneous Feynman-Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman-Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: 1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and 2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.