Asymptotic behavior for multi-scale PDMP's

Vlad Bally (1, 2) , Victor Rabiet (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 MATHRISK - Mathematical Risk handling

We study the asymptotic behaviour of a sequence of Piecewise Constant Markov Processes (in short PDMP) in which three different scales are at work: a rapid, a medium and a slow one. At the limit the rapid scale gives rise to a diffusion part (this is a CLT type regime), the medium scale produces a drift part (this is the law of large numbers type regime) and the slaw rate gives a finite variation jump process. So at the limit we obtain a stochastic differential equation which is similar to the P DM P evolution but now, in-between two jumps the equation evolutes as a general diffusion process including a Brownian part and moreover, an infinity of jumps occur in each finite time interval. This type of equations seems to be new in the literature and our first goal is to prove existence and uniqueness of the solution for them. Afterwords we study the regularity of the semigroup and we use it in order to prove the convergence result mentioned in the beginning.

Mots clés: Regularity of the Markov Semigroup; Averaging Phenomenon; Multi-scale Piecewise Deterministic Markov Processes; Stochastic Differential Equations with jumps

Domaines: Mathématiques [math]

Vlad Bally, Victor Rabiet. Asymptotic behavior for multi-scale PDMP's. 2015. ⟨hal-01144107⟩