On continuum approximations of continuous-time discrete-state stochastic processes of large system size

Davin Lunz (1, 2)

1 COMMANDS - Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires
2 InBio - InBio - Méthodes Expérimentales et Computationnelles pour la Modélisation des Processus Cellulaires / Experimental and Computational Methods for Modeling Cellular Processes

Discrete stochastic processes are an important class of models employed broadly across the sciences. When the system size becomes large, standard approaches can become intractable to solve exactly or even to simulate numerically. It is common to employ continuum approximations that may be more readily solved, and are presumed to converge in the limit as the system size tends to infinity. For example, an expansion of the master equation truncated at second order yields the Fokker-Planck equation, a widely used continuum approximation. Surprisingly, in [Doering et. al. Multiscale Model. Sim. 2005 3:2, p.283--299] it is shown that, for birth-death processes, the mean extinction times predicted by the Fokker-Planck approximation may exhibit exponentially large errors, even in the infinite system-size limit. The authors provide "a heuristic argument for the quantitative failure", and conclude that the Fokker-Planck approximation can only be accurate when restrictively tight constraints are placed on the birth and death rates, offering "a warning about the subtlety of the relationship between discrete and continuum approaches". Crucially, however, the underlying source of the inaccuracy has not been addressed. In this paper, we establish a novel quantitative argument characterising how the exponentially large error stems from the finite truncation order. We further argue that this inaccuracy is isolated to a subclass of problems. Thus, the continuum limit is not "delicate" as first thought, nor is the inaccuracy related, in any way, to the birth and death rates. Instead, the continuum limit is robust, the Fokker-Planck approximation is simply a low-order member of a family of superexponentially-converging approximations. Therefore the previously held belief that the continuum approach is only relevant in very special circumstances is to be relaxed: it is uniformly valid and quantifiably accurate. Retaining only a few more terms in the expansion affords very accurate approximations, and should alleviate much of the concern regarding the continuum approach. In establishing the utility of higher-order truncations, this work also contributes to the extensive discussion in the literature regarding the justification of the second-order truncation. While the second-order truncation has appealing features and is sufficiently accurate for many problems, in certain cases it may be necessary to proceed to higher order if additional exponential accuracy is required.

Domaines: Mathématiques générales [math.GM]

Davin Lunz. On continuum approximations of continuous-time discrete-state stochastic processes of large system size. 2020. ⟨hal-02560743v1⟩