A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts
Kohatsu-Higa, Arturo ; Tanaka, Akihiro
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 871-883 / Harvested from Numdam
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On introduit une méthode générale qui permet l'utilisation du Calcul de Malliavin pour des fonctionnelles additives générées par des équations stochastiques avec une dérive irrégulière. Cette méthode utilise le théorème de Girsanov avec l'expansion d'Itô-Taylor pour obtenir la régularité de la densité. On applique cette méthodologie pour au cas de l'intégrale en temps d'une diffusion avec derive mesurable bornée.

We present a general method which allows to use Malliavin Calculus for additive functionals of stochastic equations with irregular drift. This method uses the Girsanov theorem combined with Itô-Taylor expansion in order to obtain regularity properties for this density. We apply the methodology to the case of the Lebesgue integral of a diffusion with bounded and measurable drift.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/11-AIHP418
Classification:  60H07,  60H10 
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     author = {Kohatsu-Higa, Arturo and Tanaka, Akihiro},
     title = {A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {48},
     year = {2012},
     pages = {871-883},
     doi = {10.1214/11-AIHP418},
     mrnumber = {2976567},
     zbl = {1248.60058},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_3_871_0}
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Kohatsu-Higa, Arturo; Tanaka, Akihiro. A Malliavin calculus method to study densities of additive functionals of SDE's with irregular drifts. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) pp. 871-883. doi : 10.1214/11-AIHP418. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_3_871_0/

[1] V. Bally. Lower bounds for the density of locally elliptic Itô processes. Ann. Probab. 34 (2006) 2406-2440. | MR 2294988 | Zbl 1123.60037

[2] R. F. Bass and E. Pardoux. Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 (1987) 557-572. | MR 917679 | Zbl 0617.60075

[3] F. Flandoli. Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations. Metrika 69 (2009) 101-123. | MR 2481917

[4] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence for SDEs with nonregular drift. Preprint, 2010. | MR 2810591 | Zbl 1221.60081

[5] A. Figalli. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254 (2008) 109-153. | MR 2375067 | Zbl 1169.60010

[6] I. Gyongy and T. Martinez. On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J. 51 (2001) 763-783. | MR 1864041 | Zbl 1001.60060

[7] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland, Kodansha, Amsterdam, 1989. | MR 1011252 | Zbl 0495.60005

[8] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer-Verlag, New York, 1991. | MR 1121940 | Zbl 0638.60065

[9] A. Kohatsu-Higa. Lower bounds for densities of uniformly elliptic non-homogeneous diffusions. Proceedings of the Stochastic Inequalities Conference in Barcelona. Progr. Probab. 56 (2003) 323-338. | MR 2073439 | Zbl 1040.60046

[10] A. N. Krylov. On weak uniqueness for some diffusions with discontinuous coefficients. Stochastic. Process. Appl. 113 (2004) 37-64. | MR 2078536 | Zbl 1073.60064

[11] N. V. Krylov and M. Rockner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 (2005) 691-708. | MR 2117951 | Zbl 1072.60050

[12] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus, Part I. Stochastic analysis. In Proceedings Taniguchi International Symposium Katata and Kyoto 1982 271-306. North Holland, Amsterdam, 1984. | MR 780762 | Zbl 0546.60056

[13] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus, Part II. J. Fac. Sci. Univ. Tokyo Sect IA Math. 32 (1985) 1-76. | MR 783181 | Zbl 0568.60059

[14] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus, Part III. J. Fac. Sci. Univ. Tokyo Sect IA Math. 34 (1987) 391-442. | MR 914028 | Zbl 0633.60078

[15] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'Ceva. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Amer. Math. Soc., Providence, RI, 1968. | Zbl 0174.15403

[16] C. Le Bris and P. L. Lions. Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Comm. Partial Differential Equations 33 (2008) 1272-1317. | MR 2450159 | Zbl 1157.35301

[17] C. Le Bris and P. L. Lions. Renormalized solutions of some transport equations with partially W 1,1 velocities and applications. Ann. Mat. Pura Appl. (4) 183 (2004) 97-130. | MR 2044334 | Zbl 1170.35364

[18] P. Mathieu. Dirichlet processes associated to diffusions. Stochastics Stochastics Rep. 71 (2001) 165-176. | MR 1922563 | Zbl 0983.60074

[19] D. Nualart. Analysis on Wiener Space and Anticipating Stochastic Calculus. In Lectures on Probability Theory and Statistics: Ecole d'Ete de Probabilites de Saint-Flour XXV 123-227. Lecture Notes in Math. 1690, 1998. | MR 1668111 | Zbl 0915.60062

[20] D. Nualart. The Malliavin Calculus and Related Topics. Springer-Verlag, Berlin, 2006. | MR 2200233 | Zbl 1099.60003

[21] D. Nualart. The Malliavin Calculus ans Its Applications. CBMS Regional Conference Series in Mathematics 110. Amer. Math. Soc., Providence, RI, 2009. | MR 2498953 | Zbl 1198.60006

[22] N. I. Portenko. Generalized Diffusion Processes. Translations of Mathematical Monographs 83. Amer. Math. Soc., Providence, RI, 1990. | MR 1104660 | Zbl 0727.60088

[23] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Springer-Verlag, New York, 2004. | MR 2020294 | Zbl 0694.60047

[24] D. Stroock. Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In Séminaire de probabilités de Strasbourg XXII 316-347. Springer, Berlin, 1988. | Numdam | MR 960535 | Zbl 0651.47031

[25] J. A. Verentennikov. On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sbornik 39 (1981) 387-403. | Zbl 0462.60063

[26] S. Watanabe. Fractional order Sobolev spaces on Wiener space. Probab. Theory Related Fields 95 (1993) 175-198. | MR 1214086 | Zbl 0792.60049

[27] G. G. Yin and C. Zhu. Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability 63. Springer, New York, 2010. | MR 2559912 | Zbl 1279.60007

[28] X. Zhang. Strong solutions of SDES with singular drift and Sobolev diffusion coefficients. Stochastic. Process. Appl. 115 (2005) 1805-1818. | MR 2172887 | Zbl 1078.60045