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

Seiichiro Kusuoka; Carlo Marinelli

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1347-1370
  • ISSN: 0246-0203

Abstract

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We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in 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 ∘ 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 ( 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 .

How to cite

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Kusuoka, Seiichiro, and Marinelli, Carlo. "On smoothing properties of transition semigroups associated to a class of SDEs with jumps." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1347-1370. <http://eudml.org/doc/272097>.

@article{Kusuoka2014,
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 $T$.},
author = {Kusuoka, Seiichiro, Marinelli, Carlo},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Lévy processes; subordination; transition semigroups; non-local operators; Malliavin calculus; stochastic differential equations; pure-jump Lévy processes},
language = {eng},
number = {4},
pages = {1347-1370},
publisher = {Gauthier-Villars},
title = {On smoothing properties of transition semigroups associated to a class of SDEs with jumps},
url = {http://eudml.org/doc/272097},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Kusuoka, Seiichiro
AU - Marinelli, Carlo
TI - On smoothing properties of transition semigroups associated to a class of SDEs with jumps
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1347
EP - 1370
AB - 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 $T$.
LA - eng
KW - Lévy processes; subordination; transition semigroups; non-local operators; Malliavin calculus; stochastic differential equations; pure-jump Lévy processes
UR - http://eudml.org/doc/272097
ER -

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