Linear Noise Approximation for a Class of Piecewise Deterministic Markov Processes

Abstract

Piecewise deterministic Markov processes (PDMP) are a class of mathematical models that integrates discrete stochastic events with continuous deterministic dynamics. The exact solution of PDMP is usually not available due to nonlinearities present in the physical systems at which are modeled via PDMP. Not surprisingly researchers use a variety of approximations to obtain analytical results of statistical moments of PDMP. In this paper we aim to extend the Linear Noise Approximation (LNA) method to PDMP. The LNA method is obtained by small noise approximation of the probability distribution solution of the master equation, and is widely used in discrete-state continuous-time models. We prove that LNA is only directly applicable to a small sub-class of PDMP, and we show that for this sub-class, LNA is equivalent to calculating moments directly by linearizing nonlinearities of the system. Finally for the systems where direct application of LNA via omega expansion fails to give meaningful results, we provide a novel method for approximating moments.