Regularity and Stability for the Semigroup of Jump Diffusions with State-Dependent Intensity

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

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

We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc). First, we give sufficient conditions guaranteeing that the semigroup associated with such an equation preserves regularity by mapping the space of the of k-times differentiable bounded functions into itself. Furthermore, we give an explicit estimate of the operator norm. This is the key-ingredient in a quantitative Trotter-Kato-type stability result: it allows us to give an explicit estimate of the distance between two semigroups associated with different sets of coefficients in terms of the difference between the corresponding infinitesimal operators. As an application, we present a method allowing to replace " small jumps " by a Brownian motion or by a drift component. The example of the 2D Boltzmann equation is also treated in all detail.

Mots clés: Piecewise diffusive jumps processes; Regularity of semigroups of operators; PDMP; Weak error; Trajectory-dependent jump intensity; Boltzmann equation

Domaines: Probabilités [math.PR]

Vlad Bally, Dan Goreac, Victor Rabiet. Regularity and Stability for the Semigroup of Jump Diffusions with State-Dependent Intensity. The Annals of Applied Probability, 2018, 28 (5), pp.3028 - 3074. ⟨10.1214/18-AAP1382⟩. ⟨hal-01558741v2⟩