Data Analysis and Related Applications, Volume 1

15Stochastic Runge–Kutta Solvers Based on Markov Jump Processes and Applications to Non-autonomous Systems of Differential Equations

We present a solver for non-autonomous systems of ordinary differential equations based on the approximation principle by Markov jump processes. Each jump occurs after an exponentially distributed random waiting time which is intrinsically adapted, being computed in dependence on the current state, and is scalable by a given factor which controls the precision. The step function computed by simulating the jump processes can serve as a predictor which is further improved by suitable correction steps, which can be described as Picard iterations followed by Runge–Kutta approximations. The correction steps are applied after every jump of the original process, and the final result is a high precision scheme with several layers, which starts from the crude approximation delivered by the standard jump process, and based on this data, it computes several steps in which the approximations are successively refined.

15.1. Introduction

The numerical method presented in this chapter is based on the connection between the infinitesimal generators of Markov jump processes and corresponding differential equations. The theoretical background for this property is presented in Ethier and Kurtz (1986).

The transitions X ↦ X’ of a Markov jump process X with state space (E, d) (Polish space, for example ℝⁿ or space of Radon measures) take place according to a transition ...