The Annals of Applied Probability

Regularity and stability for the semigroup of jump diffusions with state-dependent intensity

Vlad Bally, Dan Goreac, and Victor Rabiet

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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.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 3028-3074.

Dates
Received: July 2017
Revised: December 2017
First available in Project Euclid: 28 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1535443241

Digital Object Identifier
doi:10.1214/18-AAP1382

Mathematical Reviews number (MathSciNet)
MR3847980

Zentralblatt MATH identifier
06974772

Subjects
Primary: 60J75: Jump processes 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}
Secondary: 35Q21

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

Citation

Bally, Vlad; Goreac, Dan; Rabiet, Victor. Regularity and stability for the semigroup of jump diffusions with state-dependent intensity. Ann. Appl. Probab. 28 (2018), no. 5, 3028--3074. doi:10.1214/18-AAP1382. https://projecteuclid.org/euclid.aoap/1535443241


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