Differential equation approximations for Markov chains

By R. Darling, James Norris. 2007

295 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5157a72bd6a1781f6a3f7cb1dd3f033674f4c99d

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In Mathematical Physics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Invertible bloom lookup tables (2011, background): “ as long as mis above the decoding threshold, the probability that the peeling process fails before reaching a core of size nfor any constant is asymptotically exponentially small in n. (See, e.g., [14].) By this argument the case of j> nadds a vanishing amount to the failure probability, complet”
Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions) (2012, background): “The convergence of the Markov chains with generators of type (2) to a deterministic evolution and various versions of this result are well known, see e.g. (Darling & Norris, 2008; Kolokoltsov, 2010; Benaı̈m & Le Boudec, 2008) and references therein.”
Limit theorems for discrete-time metapopulation models (2010, background): “Limit theorems of this kind are standard fare in the context of continuous-time Markovian models (see Kurtz [35] [36] [37] [38] [39] and Barbour [11] [12] [13] [14] and, more recently, Darling and Norris [20]); an approximating continuous-time deterministic trajectory is identified, and the fluctuat”
Moment Closure—A Brief Review (2015, background): “, to discrete-time and/or discrete-space stochastic processes; in fact, many discrete stochastic processes have natural ODE limits [9,27,89,90].”