A finite difference method for piecewise deterministic Markov processes

TL;DR

This work studies continuous piecewise deterministic Markov processes (PDMPs) defined by $dX/dt = A_s(X)$ with random switching rates $\mu_s$ and a transition matrix $Q$, extending the random telegraph signal model. It derives the Liouville-Master equations $\partial_t F_s + A_s(x)\partial_x F_s = \sum_j Q_{sj} F_j$ for state-resolved distributions and forms the total distribution $\mathcal{F}(x,t)=\sum_s F_s(x,t)$, then introduces an explicit upwind finite-difference scheme to compute $\mathcal{F}$ on a grid. A CFL-based convergence analysis shows that the global error grows at most linearly in time and that the scheme preserves the non-decreasing property of the distributions, with $\|I+\Delta t Q\|_1=1$ due to the stochasticity of $q_{ij}$. Numerical tests compare results to Monte Carlo histograms and demonstrate convergence to an equilibrium density within a bounded domain, validating the method’s accuracy and robustness. The work provides a practical numerical framework for PDMPs with potential broad applications and suggests future directions including stiffness handling, discontinuities, higher-order methods, and extensions to non-Markovian or higher-dimensional settings.

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.

Figures (2)

• Figure 1: On the left: convergent solution $p(x_k,T)$ of Eq. (\ref{['ME']}) when the CFL condition (\ref{['CFL']}) is satisfied. On the right: non-convergent solution when that is not satisfied.
• Figure 2: Temporal evolution of the density probability distribution functions $p(x,t)$ at fixed time. $S=4$, $\Delta t=0.009$, $\Delta x = 0.04004$, $\mu_j =0.2$, $\gamma_j=0.1$. From top left to bottom right: $t=\{0,4,8,12,16,20\}$. Monte Carlo's histograms in dashed lines.