Numerical methods for piecewise deterministic Markov processes with boundary

Christiane Cocozza-Thivent (1) , Robert Eymard (1) , Ludovic Goudenège (2) , Michel Roussignol (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 Analyse numérique, Processus Stochastiques

We study the approximation of the distribution of Piecewise Deterministic Markov Processes jumping when the process reaches some boundary of the domain. We first introduce an equation to which the marginal distributions of the process are solution, which generalizes Kolmogorov equations in this case. We then prove the uniqueness of this solution, and propose a finite volume numerical scheme for its approximation. This finite volume scheme enables the approximation of the asymptotic steady problem. We then prove the convergence of this numerical scheme to the marginal distributions of the process. We conclude this paper by some properties of the marginal distributions, directly resulting from the generalized Kolmogorov equation with boundary.

Domaines: Analyse numérique [math.NA]; Probabilités [math.PR]

Christiane Cocozza-Thivent, Robert Eymard, Ludovic Goudenège, Michel Roussignol. Numerical methods for piecewise deterministic Markov processes with boundary. IMA Journal of Numerical Analysis, 2017, 37 (1), pp.170-208. ⟨10.1093/imanum/drv069⟩. ⟨hal-00917203⟩