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Malliavin calculus for Lévy processes with arbitrary Lévy measures. (Ukrainian, English) Zbl 1123.60040

Teor. Jmovirn. Mat. Stat. 72, 67-83 (2005); translation in Theory Probab. Math. Stat. 72, 75-92 (2006).
The author proposes a method of investigation of the absolute continuity of distributions of solutions of stochastic differential equations with jumps. The method is based on J.-M. Bismut [Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 147–235 (1983; Zbl 0494.60082)] ideas. It uses differentiation in time in the space of functionals of the Poisson point measure. In contrast to J.-M. Bismut’s [loc. cit.] and J. Picard’s [Probab. Theory Relat. Fields 105, No. 4, 481–511 (1996; Zbl 0853.60064)] approaches, the proposed method can be applied to point measures with arbitrary Lévy measures. The sufficient conditions for the absolute continuity of solutions of stochastic differential equations with jumps expressed in terms of the coefficients of equations are derived. These conditions do not involve assumptions on properties of the Lévy measure. A specific example is proposed which illustrates the necessity of the conditions.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60G51 Processes with independent increments; Lévy processes