Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada–Watanabe principle

David R. Baños, Sindre Duedahl, Thilo Meyer-Brandis, and Frank Proske

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Abstract

In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart (C. R. Math. Acad. Sci. Paris 315 (1992) 1287–1291) for square integrable Brownian functionals to construct strong solutions of SDE’s under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut–Elworthy–Li formula for solutions of the Kolmogorov equation. We emphasise that our approach exhibits high flexibility to study a variety of other types of stochastic (partial) differential equations as e.g. stochastic differential equations driven by fractional Brownian motion.

Résumé

Dans cet article, nous cherchons à utiliser un critère de compacité de Da Prato, Malliavin, Nualart pour les fonctionnelles browniennes de carré intégrable pour construire des solutions fortes d’EDS sous une condition d’intégrabilité sur le coefficient de dérive. Les solutions obtenues se révèlent être Malliavin-différentiables et sont utilisées pour dériver une formule Bismut–Elworthy–Li pour des solutions de l’équation de Kolmogorov. Nous soulignons que notre approche présente une grande souplesse pour étudier une variété d’autres types d’équations différentielles stochastiques (aux dérivées partielles) comme par exemple des équations différentielles stochastiques conduites par un mouvement brownien fractionnaire.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1464-1491.

Dates
Received: 13 January 2016
Revised: 5 April 2017
Accepted: 15 May 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296026

Digital Object Identifier
doi:10.1214/17-AIHP845

Mathematical Reviews number (MathSciNet)
MR3825888

Zentralblatt MATH identifier
06976082

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H07: Stochastic calculus of variations and the Malliavin calculus 60H40: White noise theory 60J60: Diffusion processes [See also 58J65]

Keywords
Strong solutions of SDEs Malliavin calculus Kolmogorov equation Bismut–Elworthy–Li formula Singular drift coefficient

Citation

Baños, David R.; Duedahl, Sindre; Meyer-Brandis, Thilo; Proske, Frank. Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada–Watanabe principle. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1464--1491. doi:10.1214/17-AIHP845. https://projecteuclid.org/euclid.aihp/1531296026


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References

  • [1] D. R. Baños, T. Nilssen and F. Proske. Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE’s with singular drift. Available at arXiv:1511.02717.
  • [2] A. S. Chernyĭ. On strong and weak uniqueness for stochastic differential equations. Teor. Veroyatn. Primen. 46 (3) (2001) 483–497.
  • [3] G. Da Prato, F. Flandoli, E. Priola and M. Röckner. Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 (5) (2013) 3306–3344.
  • [4] G. Da Prato, P. Malliavin and D. Nualart. Compact families of Wiener functionals. C. R. Math. Acad. Sci. Paris 315 (Série I) (1992) 1287–1291.
  • [5] G. Di Nunno, B. Øksendal and F. Proske. Malliavin Calculus for Lévy Processes with Applications to Finance. Springer, Berlin, 2008.
  • [6] H. J. Engelbert. On the theorem of T. Yamada and S. Watanabe. Stoch. Stoch. Rep. 36 (3–4) (1991) 205–216.
  • [7] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence for SDE’s with non-regular drift. Stochastics 83 (3) (2011) 241–257.
  • [8] E. Fedrizzi and F. Flandoli. Hölder flow and differentiability for SDEs with nonregular drift. Stoch. Anal. Appl. 31 (4) (2013) 708–736.
  • [9] E. Fedrizzi and F. Flandoli. Noise prevents singularities in linear transport equations. J. Funct. Anal. 264 (6) (2013) 1329–1354.
  • [10] F. Flandoli. Random Pertubation of PDEs and Fluid Dynamic Models. École d’Été de Probabilitiés de Saint-Flour XL-2010. Lecture Notes in Mathematics. Springer, 2011.
  • [11] F. Flandoli, M. Gubinelli and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (1) (2010) 1–53.
  • [12] F. Flandoli, T. Nilssen and F. Proske. Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces. Preprint, 2015.
  • [13] I. Gyöngy and N. V. Krylov. Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Related Fields 105 (1996) 143–158.
  • [14] S. Haadem and F. Proske. On the construction and Malliavin differentiability of solutions of Lévy noise driven SDE’s with singular coefficients. J. Funct. Anal. 266 (8) (2014) 5321–5359.
  • [15] T. Hida, H.-H. Kuo, J. Potthoff and L. Streit. White Noise: An Infinite-Dimensional Calculus. Kluwer Academic Publishers, 1993.
  • [16] J. Jacod. Weak and strong solutions of stochastic differential equations. Stochastics 3 (3) (1980) 171–191.
  • [17] G. Kallianpur and J. Xiong. Stochastic Differential Equations in Infinite-Dimensional Spaces. Institute of Mathematical Statistics, Hayward, CA, 1995.
  • [18] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer-Verlag, New York, 1991.
  • [19] N. V. Krylov. Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces. J. Funct. Anal. 183 (1) (2001) 1–41.
  • [20] N. V. Krylov and M. Röckner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 (2) (2005) 154–196.
  • [21] H.-H. Kuo. White Noise Distribution Theory. Soch. Series, CRC Press, Boca Raton, FL, 1996.
  • [22] A. Lanconelli and F. Proske. On explicit strong solutions of Itô-SDE’s and the Donsker delta function of a diffusion. Anal. Quantum Probab. Relat. Top. 7 (3) (2004).
  • [23] P. Malliavin. Stochastic calculus of variations and hypoelliptic operators. In Proc. Inter. Symp. on Stoch. Diff. Equations (Kyoto, 1976) 195–263. Wiley, 1978.
  • [24] P. Malliavin. Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 1997.
  • [25] O. Menoukeu-Pamen, T. Meyer-Brandis, T. Nilssen, F. Proske and T. Zhang. A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s. Math. Ann. 357 (2) (2013) 761–799.
  • [26] T. Meyer-Brandis and F. Proske. On the existence and explicit representability of strong solutions of Lévy noise driven SDE’s with irregular coefficients. Communications in Mathematical Sciences 4 (1) (2006).
  • [27] T. Meyer-Brandis and F. Proske. Construction of strong solutions of SDE’s via Malliavin calculus. J. Funct. Anal. 258 (2010) 3922–3953.
  • [28] S.-E. A. Mohammed, T. Nilssen and F. Proske. Sobolev differentiable stochastic flows of SDE’s with measurable drift and applications. Ann. Probab. 43 (3) (2015) 1535–1576.
  • [29] T. Nilssen. One-dimensional SDE’s with discontinuous, unbounded drift and continuously differentiable solutions to the stochastic transport equation. Preprint.
  • [30] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer-Verlag, Berlin, 2006.
  • [31] N. Obata. White Noise Calculus and Fock Space. LNM 1577, Springer-Verlag, Berlin, 1994.
  • [32] J. Potthoff and L. Streit. A characterization of Hida distributions. J. Funct. Anal. 101 (1991) 212–229.
  • [33] F. Proske. Stochastic differential equations – some new ideas. Stochastics 79 (2007) 563–600.
  • [34] A. Y. Veretennikov. On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24 (1979) 354–366.
  • [35] T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. II (1971) 155–167.
  • [36] A. K. Zvonkin. A transformation of the state space of a diffusion process that removes the drift. Math. USSR, Sb. 22 (1974) 129–149.