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February 2021 Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps
Martin Friesen, Peng Jin, Barbara Rüdiger
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Ann. Inst. H. Poincaré Probab. Statist. 57(1): 250-271 (February 2021). DOI: 10.1214/20-AIHP1077

Abstract

We study existence and Besov regularity of densities for solutions to stochastic differential equations with Hölder continuous coefficients driven by a d-dimensional Lévy process Z=(Z(t))t0, where, for t>0, the density function ft of Z(t) exists and satisfies, for some (αi)i=1,,d(0,2) and C>0,

lim supt0t1/αiRd|ft(z+eih)ft(z)|dzC|h|,hR,i=1,,d.

Here e1,,ed denote the canonical basis vectors in Rd. The latter condition covers anisotropic (α1,,αd)-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from (J. Funct. Anal. 264 (2013), 1757–1778).

Nous étudions le problème de l’existence et de l’appartenance à un espace de Besov pour les densités de solutions d’équations différentielles stochastiques à coefficients hölderiens, conduites par un processus de Lévy d-dimensionnel Z=(Z(t))t0, où, pour t>0, la densité ft de la loi de Z(t) existe et vérifie, pour un certain (αi)i=1,,d(0,2) et C>0,

lim supt0t1/αiRd|ft(z+eih)ft(z)|dzC|h|,hR,i=1,,d.

Ici, e1,,ed désignent les vecteurs de la base canonique de Rd. La précédente condition s’applique au cas de lois anisotropiques (α1,,αd)-stables, mais aussi à des cas particuliers de mouvements browniens subordonnés. Pour démontrer ces résultats, nous utilisons certaines idées de (J. Funct. Anal. 264 (2013), 1757–1778).

Acknowledgements

The authors would like to thank the anonymous referees for useful remarks and comments. Peng Jin is supported by the STU Scientific Research Foundation for Talents (No. NTF18023).

Citation

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Martin Friesen. Peng Jin. Barbara Rüdiger. "Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps." Ann. Inst. H. Poincaré Probab. Statist. 57 (1) 250 - 271, February 2021. https://doi.org/10.1214/20-AIHP1077

Information

Received: 17 October 2018; Revised: 1 October 2019; Accepted: 15 June 2020; Published: February 2021
First available in Project Euclid: 12 March 2021

Digital Object Identifier: 10.1214/20-AIHP1077

Subjects:
Primary: 60E07, 60G30, 60H10

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 1 • February 2021
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