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A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts. (English. French summary) Zbl 1248.60058
Summary: 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.
Reviewer: Reviewer (Berlin)

##### MSC:
 60H07 Stochastic calculus of variations and the Malliavin calculus 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
##### Keywords:
Malliavin calculus; non-smooth drift; density function
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##### References:
 [1] V. Bally. Lower bounds for the density of locally elliptic Itô processes. Ann. Probab. 34 (2006) 2406-2440. · Zbl 1123.60037 · doi:10.1214/009117906000000458 [2] R. F. Bass and E. Pardoux. Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 (1987) 557-572. · Zbl 0075.28002 · doi:10.1007/BF00960074 [3] F. Flandoli. Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations. Metrika 69 (2009) 101-123. · Zbl 1433.34079 · doi:10.1007/s00184-008-0210-7 [4] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence for SDEs with nonregular drift. Preprint, 2010. · 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. · Zbl 1169.60010 · doi:10.1016/j.jfa.2007.09.020 [6] I. Gyongy and T. Martinez. On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J. 51 (2001) 763-783. · Zbl 1001.60060 · doi:10.1023/A:1013764929351 [7] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes , 2nd edition. North-Holland, Kodansha, Amsterdam, 1989. · Zbl 0684.60040 [8] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus , 2nd edition. Springer-Verlag, New York, 1991. · Zbl 0734.60060 [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. · Zbl 1040.60046 · doi:10.1007/978-3-0348-8069-5_18 [10] A. N. Krylov. On weak uniqueness for some diffusions with discontinuous coefficients. Stochastic. Process. Appl. 113 (2004) 37-64. · Zbl 1073.60064 · doi:10.1016/j.spa.2004.03.012 [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. · Zbl 1072.60050 · doi:10.1007/s00440-004-0361-z [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. · 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. · 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. · 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. · Zbl 1157.35301 · doi:10.1080/03605300801970952 [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. · Zbl 1170.35364 · doi:10.1007/s10231-003-0082-4 [18] P. Mathieu. Dirichlet processes associated to diffusions. Stochastics Stochastics Rep. 71 (2001) 165-176. · Zbl 0983.60074 · doi:10.1080/17442500108834264 [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. · Zbl 0132.37901 · doi:10.1007/BFb0092538 [20] D. Nualart. The Malliavin Calculus and Related Topics . Springer-Verlag, Berlin, 2006. · 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. · Zbl 1198.60006 [22] N. I. Portenko. Generalized Diffusion Processes. Translations of Mathematical Monographs 83 . Amer. Math. Soc., Providence, RI, 1990. · Zbl 0915.60062 [23] P. E. Protter. Stochastic Integration and Differential Equations , 2nd edition. Springer-Verlag, New York, 2004. · Zbl 1041.60005 [24] D. Stroock. Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In Séminaire de probabilités de Strasbourg XXII 316-347. Springer, Berlin, 1988. · Zbl 0651.47031 · numdam:SPS_1988__22__316_0 · eudml:113641 [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 · doi:10.1070/SM1981v039n03ABEH001522 [26] S. Watanabe. Fractional order Sobolev spaces on Wiener space. Probab. Theory Related Fields 95 (1993) 175-198. · Zbl 0792.60049 · doi:10.1007/BF01192269 [27] G. G. Yin and C. Zhu. Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability 63 . Springer, New York, 2010. [28] X. Zhang. Strong solutions of SDES with singular drift and Sobolev diffusion coefficients. Stochastic. Process. Appl. 115 (2005) 1805-1818. · Zbl 1078.60045 · doi:10.1016/j.spa.2005.06.003
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