Linear Variance Bounds for Particle Approximations of Time-Homogeneous
Feynman-Kac Formulae
Feynman-Kac Formulae
File(s)
Accepted version
Author(s)
Whiteley, N
Kantas, N
Jasra, A
Type
Journal Article
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.
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.
Date Issued
2012-02-05
Citation
Stochastic Processes and their Applications, 2012
ISSN
0304-4149
Publisher
ELSEVIER SCIENCE BV
Start Page
1840
End Page
1865
Journal / Book Title
Stochastic Processes and their Applications
Volume
122
Issue
4
Copyright Statement
Copyright © 2012 Elsevier Ltd. All rights reserved. NOTICE: this is the author’s version of a work that was accepted for publication in Stochastic Processes and Their Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Stochastic Processes and Their Applications, 122(4), 2012. DOI:10.1016/j.spa.2012.02.002
Identifier
http://arxiv.org/abs/1108.3988v2
Publication Status
Published