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Ukrainian Mathematical Journal

, Volume 54, Issue 2, pp 266–279 | Cite as

Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes

  • A. M. Kulik
Article
  • 41 Downloads

Abstract

We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process.

Keywords

Distribution Density Formal Differentiation Local Time Probability Space Stable Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    P. Malliavin, “Stochastic calculus of variation and hypoelliptic operators,” in: Proc. Int. Symp. SDE Kyoto, Kinokunia, Tokyo (1976), pp. 195–263.Google Scholar
  2. 2.
    K. Bichteler, J.-B. Gravereaux, and J. Jacod, Malliavin Calculus for Processes with Jumps, Gordon and Breach, New York (1987).Google Scholar
  3. 3.
    T. Komatsu and A. Takeuchi, “On the smoothness of part of solutions to SDE with jumps,” J. Different. Equat. Appl., 2, 141–197 (2001).Google Scholar
  4. 4.
    A. M. Kulik, “Admissible transformations and Malliavin calculus of compound Poisson processes,” Theory Stochast. Process., 5(21), No. 3-4, 120–126 (1999).Google Scholar
  5. 5.
    Yu. A. Davydov, M. A. Lifshitz, and N. V. Smorodina, Local Properties of Distributions of Stochastic Functionals [in Russian], Nauka, Moscow (1995).Google Scholar
  6. 6.
    Yu. A. Davydov and M. A. Lifshitz, “Stratification method in some probability problems,” Probab. Theory, Math. Statist., Theor. Cybernetics, 22, 61–137 (1984).Google Scholar
  7. 7.
    A. Yu. Pilipenko, “Properties of stochastic differential operators in the non-Gaussian case,” Theory Stochast. Process., 2(18), No. 3-4, 153–161 (1996).Google Scholar
  8. 8.
    A. A. Dorogovtsev, Stochastic Equations with Anticipation [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).Google Scholar
  9. 9.
    A. Benassi, “Calcul stochastique anticipatif: martingales hierarchiques,” Comt. Rendus L'Acad. Sci. Ser. 1, 311, No. 7, 457–460 (1990).Google Scholar
  10. 10.
    N. I. Portenko, “Integral equations and some limit theorems for additive functionals of Markov processes,” Teor. Ver. Primen., 12, No. 3, 551–559 (1967).Google Scholar
  11. 11.
    P. Billingsley, Convergence of Probability Measures, New York (1975).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • A. M. Kulik
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev

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