Analysis of Timed Properties Using the Jump-Diffusion Approximation

Abstract

Density dependent Markov chains (DDMCs) describe the interaction of groups of identical objects. In case of large numbers of objects a DDMC can be approximated efficiently by means of either a set of ordinary differential equations (ODEs) or by a set of stochastic differential equations (SDEs). While with the ODE approximation the chain stochasticity is not maintained, the SDE approximation, also known as the diffusion approximation, can capture specific stochastic phenomena (e.g., bi-modality) and has also better convergence characteristics. In this paper we introduce a method for assessing temporal properties, specified in terms of a timed automaton, of a DDMC through a jump diffusion approximation. The added value is in terms of runtime: the costly simulation of a very large DDMC model can be replaced through much faster simulation of the corresponding jump diffusion model. We show the efficacy of the framework through the analysis of a biological oscillator.