Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On smoothing properties of transition semigroups associated to a class of SDEs with jumps

Seiichiro Kusuoka and Carlo Marinelli

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Abstract

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in $\mathbb{R} ^{d}$ driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y=W\circ T$, where $T$ is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process $W$) and of an arbitrary Lévy process independent of $Y$, that the drift coefficient is continuous (but not necessarily Lipschitz continuous) and grows not faster than a polynomial, and that the SDE admits a Feller weak solution. By a combination of probabilistic and analytic methods, we provide sufficient conditions for the Markovian semigroup associated to the SDE to be strong Feller and to map $L_{p}(\mathbb{R} ^{d})$ to continuous bounded functions. A key intermediate step is the study of regularizing properties of the transition semigroup associated to $Y$ in terms of negative moments of the subordinator $T$.

Résumé

Nous établissons des propriétés de lissage de semi-groupes de transition non locaux associés à une classe d’équations différentielles stochastiques dans $\mathbb{R} ^{d}$ dirigées par un bruit additif de Lévy sans partie continue. En particulier, nous supposons que le processus de Lévy est la somme d’un processus de Wiener subordonné $Y$ (i.e. $Y=W\circ T$, où $T$ est un processus croissant de Lévy sans partie continue, avec $T_{0}=0$, indépendant du processus de Wiener $W$) et d’un processus de Lévy arbitraire indépendant de $Y$; que le coefficient de dérive est continu (mais pas nécessairement lipschitzien) et à croissance polynomiale; et que la EDS admet une solution faible fellerienne. Par une combinaison de méthodes probabilistes et analytiques, nous fournissons des conditions suffisantes pour le semi-groupe markovien associé à l’EDS soit fortement fellérien et envoye $L_{p}(\mathbb{R} ^{d})$ dans les fonctions continues bornées. Une étape intermédiaire essentielle est l’étude de certaines propriétés régularisantes du semi-groupe de transition associé à $Y$ qui dépendent de moments négatifs du subordinateur $T$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 4 (2014), 1347-1370.

Dates
First available in Project Euclid: 17 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1413555503

Digital Object Identifier
doi:10.1214/13-AIHP559

Mathematical Reviews number (MathSciNet)
MR3269997

Zentralblatt MATH identifier
1319.60127

Subjects
Primary: 60G30: Continuity and singularity of induced measures 60G51: Processes with independent increments; Lévy processes 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Keywords
Lévy processes Subordination Transition semigroups Non-local operators Malliavin calculus

Citation

Kusuoka, Seiichiro; Marinelli, Carlo. On smoothing properties of transition semigroups associated to a class of SDEs with jumps. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 4, 1347--1370. doi:10.1214/13-AIHP559. https://projecteuclid.org/euclid.aihp/1413555503


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