A finite difference method for piecewise deterministic Markov processes

Abstract

An extension of non-deterministic processes driven by the random telegraph signal is introduced in the framework of "piecewise deterministic Markov processes" [Davis], including a broader category of random systems. The corresponding Liouville-Master Equation is established and the upwind method is applied to numerical calculation of the distribution function. The convergence of the numerical solution is proved under an appropriate Courant-Friedrichs-Lewy condition. The same condition preserve the non-decreasing property of the calculated distribution function. Some numerical tests are presented.