Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling

Mateusz B. Majka

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Abstract

By using the mirror coupling for solutions of SDEs driven by pure jump Lévy processes, we extend some transportation and concentration inequalities, which were previously known only in the case where the coefficients in the equation satisfy a global dissipativity condition. Furthermore, by using the mirror coupling for the jump part and the coupling by reflection for the Brownian part, we extend analogous results for jump diffusions. To this end, we improve some previous results concerning such couplings and show how to combine the jump and the Brownian case. As a crucial step in our proof, we develop a novel method of bounding Malliavin derivatives of solutions of SDEs with both jump and Gaussian noise, which involves the coupling technique and which might be of independent interest. The bounds we obtain are new even in the case of diffusions without jumps.

Résumé

En utilisant le couplage miroir pour les solutions d’EDS dirigées par un processus de Lévy de saut pur, nous généralisons des inégalités de transport et de concentration, qui étaient précédemment connues seulement dans le cas où les coefficients de l’équation satisfont une condition dissipative globale. De plus, en utilisant un couplage miroir pour la partie de sauts et le couplage par réflexion pour la partie Brownienne, nous étendons des résultats analogues pour les diffusions à sauts. A cette fin, nous améliorons des résultats précédents concernant le couplage et montrons comment combiner les cas à sauts et le cas brownien. Dans une étape cruciale de la preuve, nous développons une méthode nouvelle pour borner la dérivée de Malliavin des solutions d’EDS avec à la fois sauts et bruit Gaussien, ce qui utilise le couplage et peut être d’un intérêt indépendant. Les bornes que nous obtenons sont nouvelles même dans le cas de diffusions sans saut.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 2019-2057.

Dates
Received: 24 November 2016
Revised: 11 April 2018
Accepted: 28 September 2018
First available in Project Euclid: 8 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1573203622

Digital Object Identifier
doi:10.1214/18-AIHP941

Mathematical Reviews number (MathSciNet)
MR4029147

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60H10: Stochastic ordinary differential equations [See also 34F05] 60H07: Stochastic calculus of variations and the Malliavin calculus 60E15: Inequalities; stochastic orderings

Keywords
Stochastic differential equations Lévy processes Transportation inequalities Couplings Wasserstein distances Malliavin calculus

Citation

Majka, Mateusz B. Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 2019--2057. doi:10.1214/18-AIHP941. https://projecteuclid.org/euclid.aihp/1573203622


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