On Continuum Approximations of Discrete-State Markov Processes of Large System Size

Discrete-state continuous-time Markov processes are an important class of models employed broadly across the sciences. When the system size becomes large, standard approaches can become intractable to exact solution and numerical simulation. Approximations posed on a continuous state space are often more tractable 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 equipped with an underlying process of continuous state. Surprisingly, in [Doering \textit{et. al.} Multiscale Model. Sim. 2005 3:2, p.283--299] it is shown that the Fokker--Planck approximation may exhibit exponentially large errors, even in the infinite system-size limit. Crucially, the source of this inaccuracy has not been addressed. In this paper, we focus on the family of continuous-state approximations obtained by arbitrary-order truncations. We uncover how the exponentially large error stems from the truncation by quantifying the rapid error decay with increasing truncation order. Furthermore, we explain why this discrepancy only comes to light in a subset of problems. The approximations produced by finite truncation beyond second order lack underlying stochastic processes. Nevertheless, they retain valuable information that explains the previously observed discrepancy by bridging the gap between the continuous and discrete processes. The insight conferred by this broader notion of ``continuum approximation'', where we do not require an underlying stochastic process, prompts us to revisit previously expressed doubts regarding continuum approximations. In establishing the utility of higher-order truncations, this approach also contributes to the extensive discussion in the literature regarding the second-order truncation: while recognising the appealing features of an associated stochastic process, in certain cases it may be advantageous to dispense of the process in exchange for the increased approximation accuracy guaranteed by higher-order truncations.