Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 

 

Malliavin calculus for Lévy processes with arbitrary Lévy measures


Author: A. M. Kulik
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 72 (2005).
Journal: Theor. Probability and Math. Statist. 72 (2006), 75-92
MSC (2000): Primary 60H07; Secondary 60G51
Published electronically: August 18, 2006
MathSciNet review: 2168138
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new method is proposed to prove the absolute continuity of distributions of solutions of stochastic differential equations with jumps. The method is based on the differentiation in time in the space of functionals of the Poisson point measure. In contrast to the known Bismut and Picard methods, our approach can be applied to point measures with arbitrary Lévy measures. We obtain sufficient conditions for the absolute continuity of the solutions expressed in terms of the coefficients of the equation; the conditions do not involve assumptions on properties of the Lévy measure.


References [Enhancements On Off] (What's this?)

  • 1. Jean-Michel Bismut, Calcul des variations stochastique et processus de sauts, Z. Wahrsch. Verw. Gebiete 63 (1983), no. 2, 147–235 (French, with English summary). MR 701527, 10.1007/BF00538963
  • 2. Klaus Bichteler, Jean-Bernard Gravereaux, and Jean Jacod, Malliavin calculus for processes with jumps, Stochastics Monographs, vol. 2, Gordon and Breach Science Publishers, New York, 1987. MR 1008471
  • 3. James Norris, Simplified Malliavin calculus, Séminaire de Probabilités, XX, 1984/85, Lecture Notes in Math., vol. 1204, Springer, Berlin, 1986, pp. 101–130. MR 942019, 10.1007/BFb0075716
  • 4. Takashi Komatsu and Atsushi Takeuchi, On the smoothness of PDF of solutions to SDE of jump type, Int. J. Differ. Equ. Appl. 2 (2001), no. 2, 141–197. MR 1930241
  • 5. Robert J. Elliott and Allanus H. Tsoi, Integration by parts for Poisson processes, J. Multivariate Anal. 44 (1993), no. 2, 179–190. MR 1219202, 10.1006/jmva.1993.1010
  • 6. Alexey M. Kulik, Admissible transformations and Malliavin calculus for compound Poisson processes, Proceedings of the Third Ukrainian-Scandinavian Conference in Probability Theory and Mathematical Statistics (Kiev, 1999), 1999, pp. 120–126. MR 2018406
  • 7. Nicolas Privault, Connections and curvature in the Riemannian geometry of configuration spaces, J. Funct. Anal. 185 (2001), no. 2, 367–403. MR 1856271, 10.1006/jfan.2001.3768
  • 8. Nicolas Privault, Hypothesis testing and Skorokhod stochastic integration, J. Appl. Probab. 37 (2000), no. 2, 560–574. MR 1781013
  • 9. A. M. Kulik, Some remarks on time-stretching differentiation for general Lévy processes, Theory Stoch. Process. 7(23) (2001), no. 3-4, 50-63.
  • 10. Alexey M. Kulik, Anticipating linear stochastic differential equations by Poisson random measures, Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv, 2002), 2002, pp. 232–241. MR 2027395
  • 11. A. V. Skorohod, Random processes with independent increments, Mathematics and its Applications (Soviet Series), vol. 47, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the second Russian edition by P. V. Malyshev. MR 1155400
  • 12. Yu. A. Davydov, M. A. Lifshits, and N. V. Smorodina, Local properties of distributions of stochastic functionals, Translations of Mathematical Monographs, vol. 173, American Mathematical Society, Providence, RI, 1998. Translated from the 1995 Russian original by V. E. Nazaĭkinskiĭ and M. A. Shishkova. MR 1604537
  • 13. A. Yu. Pilipenko, Properties of stochastic differential operators in non-Gaussian case, Theory Stoch. Process. 2(18) (1996), no. 3-4, 153-161.
  • 14. Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • 15. Jean Picard, On the existence of smooth densities for jump processes, Probab. Theory Related Fields 105 (1996), no. 4, 481–511. MR 1402654, 10.1007/BF01191910

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60H07, 60G51

Retrieve articles in all journals with MSC (2000): 60H07, 60G51


Additional Information

A. M. Kulik
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovskaya 3, Kiev 01601, Ukraine
Email: kulik@imath.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-06-00666-1
Keywords: Malliavin calculus, L\'evy process, stratification method, admissible time-stretching transformations
Received by editor(s): March 19, 2004
Published electronically: August 18, 2006
Additional Notes: Supported by the Ministry of Science and Education of Ukraine, Project # 01.07/103.
Article copyright: © Copyright 2006 American Mathematical Society