Error bounds for one-dimensional constrained Langevin approximations for nearly density dependent Markov chains
F. A. CAMPOS and R. J. WILLIAMS
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 analysed exactly. Additionally, simulation can be
computationally intensive. An approach to address these challenges is to
consider a more tractable diffusion approximation. Leite and Williams (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. (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.
Accepted for Advances in Applied Probability, September 2024.
A pdf of a preprint is available here (for personal, scientific, non-commercial use only).