Numerical methods for piecewise deterministic Markov processes with boundary
This work addresses the need for numerical methods for piecewise deterministic Markov processes with boundary, but the lack of convergence results and reliance on a forthcoming paper suggest incremental progress.
The paper develops a numerical method for piecewise deterministic Markov processes with boundary, proving uniqueness of solutions to a generalized Kolmogorov equation and existence/uniqueness of a positive solution to a finite volume scheme. Simulations of TCP window-size processes demonstrate the method's efficiency.
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.