Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes
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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.
KeywordsDistribution Density Formal Differentiation Local Time Probability Space Stable Process
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- 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.K. Bichteler, J.-B. Gravereaux, and J. Jacod, Malliavin Calculus for Processes with Jumps, Gordon and Breach, New York (1987).Google Scholar
- 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.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.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.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.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.A. A. Dorogovtsev, Stochastic Equations with Anticipation [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).Google Scholar
- 9.A. Benassi, “Calcul stochastique anticipatif: martingales hierarchiques,” Comt. Rendus L'Acad. Sci. Ser. 1, 311, No. 7, 457–460 (1990).Google Scholar
- 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.P. Billingsley, Convergence of Probability Measures, New York (1975).Google Scholar