Numerical methods for piecewise deterministic Markov processes with boundary

TL;DR

This paper introduces a numerical method for analyzing piecewise deterministic Markov processes with boundaries, establishing theoretical properties and demonstrating practical simulations of transmission control protocols.

Contribution

It presents a new numerical scheme for such processes, proves solution uniqueness, and applies it to simulate TCP window-size models.

Findings

Proved uniqueness of solutions to the generalized Kolmogorov equation.

Established existence and uniqueness of the finite volume scheme solution.

Simulated TCP models to validate the numerical method's effectiveness.

Abstract

In this paper is described the general aspect of a numerical method for piecewise determin-istic Markov processes with boundary. Under very natural hypotheses, a crucial result about uniqueness of solution of a generalized Kolmogorov equation with respect to a test function space is proved. Next we prove the existence and uniqueness of a positive solution to the finite volume scheme without result about convergence. Finally different models of transmission control protocol window-size processes are simulated to illustrate the efficiency of the numerical method for describing the evolution of the density of a piecewise deterministic Markov process with boundary. Obviously some technical aspects have been skipped for reader convenience but the full theory will be exposed in a forthcoming paper in collaboration with C.