A finite difference method for piecewise deterministic processes with memory

Abstract

In this paper the numerical approximation of solutions of Liouville‐Master Equation for time‐dependent distribution functions of Piecewise Deterministic Processes with memory is considered. These equations are linear hyperbolic PDEs with non‐constant coefficients, and boundary conditions that depend on integrals over the interior of the integration domain. We construct a finite difference method of the first order, by a combination of the upwind method, for PDEs, and by a direct quadrature, for the boundary condition. We analyse convergence of the numerical solution for distribution functions evolving towards an equilibrium. Numerical results for two problems, whose analytical solutions are known in closed form, illustrate the theoretical finding.

Key words:

• Piecewise‐deterministic process
• dichotomic noise
• random telegraph process
• binary noise
• upwind method
• conservative systems
• non local boundary conditions