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October 2018 Regularity and stability for the semigroup of jump diffusions with state-dependent intensity
Vlad Bally, Dan Goreac, Victor Rabiet
Ann. Appl. Probab. 28(5): 3028-3074 (October 2018). DOI: 10.1214/18-AAP1382

Abstract

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 $k$-times differentiable bounded functions into itself. Furthermore, we give an upper estimate of the operator norm. This is the key-ingredient in a quantitative Trotter–Kato-type stability result: it allows us to give an upper 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.

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Vlad Bally. Dan Goreac. Victor Rabiet. "Regularity and stability for the semigroup of jump diffusions with state-dependent intensity." Ann. Appl. Probab. 28 (5) 3028 - 3074, October 2018. https://doi.org/10.1214/18-AAP1382

Information

Received: 1 July 2017; Revised: 1 December 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974772
MathSciNet: MR3847980
Digital Object Identifier: 10.1214/18-AAP1382

Subjects:
Primary: 47D07 , 60J75
Secondary: 35Q21

Keywords: Boltzmann equation , PDMP , Piecewise diffusive jumps processes , regularity of semigroups of operators , trajectory-dependent jump intensity , weak error

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October 2018
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