A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts

Arturo Kohatsu-Higa; Akihiro Tanaka

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 3, page 871-883
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Kohatsu-Higa, Arturo, and 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 48.3 (2012): 871-883. <http://eudml.org/doc/272091>.

@article{Kohatsu2012,
abstract = {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.},
author = {Kohatsu-Higa, Arturo, Tanaka, Akihiro},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Malliavin calculus; non-smooth drift; density function},
language = {eng},
number = {3},
pages = {871-883},
publisher = {Gauthier-Villars},
title = {A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts},
url = {http://eudml.org/doc/272091},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Kohatsu-Higa, Arturo
AU - Tanaka, Akihiro
TI - A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 871
EP - 883
AB - 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.
LA - eng
KW - Malliavin calculus; non-smooth drift; density function
UR - http://eudml.org/doc/272091
ER -

References

top
  1. [1] V. Bally. Lower bounds for the density of locally elliptic Itô processes. Ann. Probab.34 (2006) 2406–2440. Zbl1123.60037MR2294988
  2. [2] R. F. Bass and E. Pardoux. Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields76 (1987) 557–572. Zbl0617.60075MR917679
  3. [3] F. Flandoli. Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations. Metrika69 (2009) 101–123. Zbl06493839MR2481917
  4. [4] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence for SDEs with nonregular drift. Preprint, 2010. Zbl1221.60081MR2810591
  5. [5] A. Figalli. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal.254 (2008) 109–153. Zbl1169.60010MR2375067
  6. [6] I. Gyongy and T. Martinez. On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J.51 (2001) 763–783. Zbl1001.60060MR1864041
  7. [7] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland, Kodansha, Amsterdam, 1989. Zbl0495.60005MR1011252
  8. [8] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer-Verlag, New York, 1991. Zbl0638.60065MR1121940
  9. [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. Zbl1040.60046MR2073439
  10. [10] A. N. Krylov. On weak uniqueness for some diffusions with discontinuous coefficients. Stochastic. Process. Appl.113 (2004) 37–64. Zbl1073.60064MR2078536
  11. [11] N. V. Krylov and M. Rockner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields131 (2005) 691–708. Zbl1072.60050MR2117951
  12. [12] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus, Part I. Stochastic analysis. In Proceedings Taniguchi International Symposium Katata and Kyoto 1982271–306. North Holland, Amsterdam, 1984. Zbl0546.60056MR780762
  13. [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. Zbl0568.60059MR783181
  14. [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. Zbl0633.60078MR914028
  15. [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. Zbl0174.15403
  16. [16] C. Le Bris and P. L. Lions. Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients. Comm. Partial Differential Equations33 (2008) 1272–1317. Zbl1157.35301MR2450159
  17. [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. Zbl1170.35364MR2044334
  18. [18] P. Mathieu. Dirichlet processes associated to diffusions. Stochastics Stochastics Rep.71 (2001) 165–176. Zbl0983.60074MR1922563
  19. [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. Zbl0915.60062MR1668111
  20. [20] D. Nualart. The Malliavin Calculus and Related Topics. Springer-Verlag, Berlin, 2006. Zbl1099.60003MR2200233
  21. [21] D. Nualart. The Malliavin Calculus ans Its Applications. CBMS Regional Conference Series in Mathematics 110. Amer. Math. Soc., Providence, RI, 2009. Zbl1198.60006MR2498953
  22. [22] N. I. Portenko. Generalized Diffusion Processes. Translations of Mathematical Monographs 83. Amer. Math. Soc., Providence, RI, 1990. Zbl0727.60088MR1104660
  23. [23] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Springer-Verlag, New York, 2004. Zbl0694.60047MR2020294
  24. [24] D. Stroock. Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In Séminaire de probabilités de Strasbourg XXII 316–347. Springer, Berlin, 1988. Zbl0651.47031MR960535
  25. [25] J. A. Verentennikov. On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sbornik39 (1981) 387–403. Zbl0462.60063
  26. [26] S. Watanabe. Fractional order Sobolev spaces on Wiener space. Probab. Theory Related Fields95 (1993) 175–198. Zbl0792.60049MR1214086
  27. [27] G. G. Yin and C. Zhu. Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability 63. Springer, New York, 2010. Zbl1279.60007MR2559912
  28. [28] X. Zhang. Strong solutions of SDES with singular drift and Sobolev diffusion coefficients. Stochastic. Process. Appl.115 (2005) 1805–1818. Zbl1078.60045MR2172887

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.