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Error bounds for one-dimensional constrained Langevin approximations for nearly density-dependent Markov chains

Part of: Physiological, cellular and medical topics Markov processes Stochastic analysis Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  26 July 2024

Felipe A. Campos  and
Ruth J. Williams
Felipe A. Campos*
Affiliation:
University of California, San Diego
Ruth J. Williams*
Affiliation:
University of California, San Diego
*
*Postal address: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA.
**Email address: fcamposv91@gmail.com
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Abstract

Continuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob. 29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al. (SIAM Multiscale Modeling Simul. 17, 2019).

In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one.

Keywords

Density-dependent Markov chainsdiffusion approximationconstrained Langevin approximationerror boundlinear noise approximationchemical reaction networksstochastic differential equation with oblique reflectionsystems biology

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J60: Diffusion processes 65C30: Stochastic differential and integral equations
Secondary: 60H10: Stochastic ordinary differential equations 92C42: Systems biology, networks 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 3 , September 2024 , pp. 825 - 867
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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