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.
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.
Revised:
Accepted:
Published online:
Keywords: Hörmander-type criterion, Malliavin calculus, Delayed stochastic differential equation, Rough path integration
@article{AHL_2020__3__1023_0,
author = {Chhaibi, Reda and Ekren, Ibrahim},
title = {A {H\"ormander} condition for delayed stochastic differential equations},
journal = {Annales Henri Lebesgue},
pages = {1023--1048},
publisher = {\'ENS Rennes},
volume = {3},
year = {2020},
doi = {10.5802/ahl.53},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/ahl.53/}
}
TY - JOUR AU - Chhaibi, Reda AU - Ekren, Ibrahim TI - A Hörmander condition for delayed stochastic differential equations JO - Annales Henri Lebesgue PY - 2020 SP - 1023 EP - 1048 VL - 3 PB - ÉNS Rennes UR - http://archive.numdam.org/articles/10.5802/ahl.53/ DO - 10.5802/ahl.53 LA - en ID - AHL_2020__3__1023_0 ER -
Chhaibi, Reda; Ekren, Ibrahim. A Hörmander condition for delayed stochastic differential equations. Annales Henri Lebesgue, Volume 3 (2020), pp. 1023-1048. doi : 10.5802/ahl.53. http://archive.numdam.org/articles/10.5802/ahl.53/
[Bel04] Stochastic differential equations and hypoelliptic operators, Real and stochastic analysis (Trends in Mathematics), Birkhäuser, 2004, pp. 9-42 | DOI | MR | Zbl
[BM91] The Malliavin calculus and stochastic delay equations, J. Funct. Anal., Volume 99 (1991) no. 1, pp. 75-99 | DOI | MR | Zbl
[BM95] Smooth densities for degenerate stochastic delay equations with hereditary drift, Ann. Probab., Volume 23 (1995) no. 4, pp. 1875-1894 | DOI | MR | Zbl
[CF10] Densities for rough differential equations under Hörmander’s condition, Ann. Math., Volume 171 (2010) no. 3, pp. 2115-2141 | DOI | Zbl
[CHLT15] Smoothness of the density for solutions to Gaussian rough differential equations, Ann. Probab., Volume 43 (2015) no. 1, pp. 188-239 | DOI | MR | Zbl
[Con16] Pathwise calculus for non-anticipative functionals, Stochastic Integration by Parts and Functional Itô Calculus (Utzet, Frederic; Vives, Josep, eds.) (Advanced Courses in Mathematics – CRM Barcelona), Springer, 2016, pp. 125-152 | DOI | Zbl
[FH14] A course on rough paths. With an introduction to regularity structures, Universitext, Springer, 2014 | DOI | Zbl
[Hai11] On Malliavin’s proof of Hörmander’s theorem, Bull. Sci. Math., Volume 135 (2011) no. 6-7, pp. 650-666 | DOI | MR | Zbl
[HP13] Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths, Ann. Probab., Volume 41 (2013) no. 4, pp. 2544-2598 | DOI | MR | Zbl
[Hsu02] Stochastic analysis on manifolds, Graduate Studies in Mathematics, 38, American Mathematical Society, 2002 | DOI | MR | Zbl
[Hör67] Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967) no. 1, pp. 147-171 | DOI | MR | Zbl
[KM97] The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, 1997 | MR | Zbl
[KS84] Applications of the Malliavin calculus, Part I, Stochastic analysis (Katata/Kyoto, 1982) (North-Holland Mathematical Library), Volume 32, North-Holland, 1984, pp. 271-306 | Zbl
[Kun84] Stochastic differential equations and stochastic flows of diffeomorphisms, École d’été de probabilités de Saint-Flour, XII—1982 (Hennequin, P. L., ed.) (Lecture Notes in Mathematics), Volume 1097, Springer, 1984, pp. 143-303 | DOI | MR
[Kun97] Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, 1997 | MR | Zbl
[Lej12] Global solutions to rough differential equations with unbounded vector fields, Séminaire de probabilités XLIV, Springer, 2012, pp. 215-246 | DOI | Zbl
[LS01] Statistics of random processes. I. General theory, Applications of Mathematics, 5, Springer, 2001 (translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability) | Zbl
[NNT08] Delay equations driven by rough paths, Electron. J. Probab., Volume 13 (2008) no. 67, pp. 2031-2068 | DOI | MR | Zbl
[NP88] Stochastic calculus with anticipating integrands, Probab. Theory Relat. Fields, Volume 78 (1988) no. 4, pp. 535-581 | DOI | MR | Zbl
[Nua95] The Malliavin calculus and related topics, Probability and Its Applications, Springer, 1995 | Zbl
[OP89] A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 25 (1989) no. 1, pp. 39-71 | Numdam | Zbl
[Sto98] Regular delay Langevin equation with degenerate diffusion coefficient, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. (1998), pp. 289-294 | Zbl
[Str83] Some applications of stochastic calculus to partial differential equations, École d’Été de Probabilités de Saint-Flour XI—1981 (Lecture Notes in Mathematics), Volume 976, Springer, 1983, pp. 267-382 | MR | Zbl
[Tak07] Malliavin calculus for degenerate stochastic functional differential equations, Acta Appl. Math., Volume 97 (2007) no. 1-3, pp. 281-295 | DOI | MR | Zbl
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