A Hörmander condition for delayed stochastic differential equations

Chhaibi, Reda; Ekren, Ibrahim

Annales Henri Lebesgue, Tome 3 (2020), pp. 1023-1048.

Nous envisageons des équations différentielles stochastiques (EDS) avec une dépendance en la trajectoire à travers des retards. Dans ce contexte non-markovien, nous exhibons un critère de Hörmander pour la régularité des marginales des solutions. Notre critère s’exprime en effet grâce à des crochets de Lie de champs de vecteurs. Une nouveauté dans le cas avec retard est que le bruit peut « se propager depuis le passé » et donner lieu à de la régularité grâce à des demi-crochets.

La preuve suit dans les grandes lignes celle de Malliavin pour le cas markovien. Néanmoins, afin de traiter les intégrales anticipatives de façon trajectorielle ainsi que certains aspects non-markoviens dûs aux retards, nous invoquons la théorie des chemins rugueux de façon essentielle.

In this paper, we are interested in path-dependent stochastic differential equations (SDEs) which are controlled by Brownian motion and its delays. Within this non-Markovian context, we give a Hörmander-type criterion for the regularity of solutions. Indeed, our criterion is expressed as a spanning condition with brackets. A novelty in the case of delays is that noise can “flow from the past” and give additional smoothness thanks to semi-brackets.

The proof follows the general lines of Malliavin’s probabilistic proof, in the Markovian case. Nevertheless, in order to handle the non-Markovian aspects of this problem and to treat anticipative integrals in a path-wise fashion, we heavily invoke rough path integration.

Classification : 60G30, 60H07, 60H10
Mots clés : Hörmander-type criterion, Malliavin calculus, Delayed stochastic differential equation, Rough path integration

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