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On Markov chain approximations for computing boundary crossing probabilities of diffusion processes

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  11 May 2023

Vincent Liang  and
Konstantin Borovkov Open the ORCID record for Konstantin Borovkov [Opens in a new window]
Vincent Liang*
Affiliation:
The University of Melbourne
Konstantin Borovkov*
Affiliation:
The University of Melbourne
*
*Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville 3010, Australia
*Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville 3010, Australia
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Abstract

We propose a discrete-time discrete-space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the proposed approximation with the Brownian bridge correction is $O(n^{-2})$ in the case of $C^2$ boundaries and a uniform time grid with n steps.

Keywords

boundary crossing probabilitydiffusion processesMarkov chains

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 65C30: Stochastic differential and integral equations
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1386 - 1415
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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