Differential equation approximations for Markov chains

TL;DR

This paper establishes conditions for approximating Markov chains with differential equations, providing quantifiable error bounds, and illustrates the theory through examples in epidemics, population models, and hypergraph algorithms.

Contribution

It introduces a general framework for differential equation approximation of Markov chains with error estimates, emphasizing coordinate function choices.

Findings

Approximation conditions with error bounds are derived.

Applications include stochastic epidemic models and hypergraph algorithms.

The approach is demonstrated through three diverse examples.

Abstract

We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.