Construction de l’opérateur de Malliavin sur l’espace de Poisson. (Construction of the Malliavin operator on the Poisson space). (French) Zbl 0659.60079
Sémin. probabilités XXI, Lect. Notes Math. 1247, 100-113 (1987).
[For the entire collection see Zbl 0606.00022.]
The approach of D. Surgailis [Theory and application of random fields, Proc. IFIP-WG 7/1 Working Conf., Bangalore/India 1982, Lect. Notes Control Inf. Sci. 49, 233-248 (1983; Zbl 0511.60047)] is used for the construction of the Markovian Wiener-Poisson process and the associated symmetric diffusion semigroup by D. Stroock [J. Funct. Anal. 44, 212-257 (1981; Zbl 0475.60060)]. The main result says: the generator L of this process is exactly the Malliavin operator, introduced by K. Bichteler, J. B. Gravereaux and J. Jacod [Malliavin calculus for processes with jumps (to appear); see also C. R. Acad. Sci., Paris, Ser. I 300, 81-84 (1985; Zbl 0575.60058) and Sémin. de probabilités XVII, Proc. 1981/82, Lect. Notes Math. 986, 132-157 (1983; Zbl 0525.60067)]. It justifies the Malliavin calculus of variations for processes with jumps.
The approach of D. Surgailis [Theory and application of random fields, Proc. IFIP-WG 7/1 Working Conf., Bangalore/India 1982, Lect. Notes Control Inf. Sci. 49, 233-248 (1983; Zbl 0511.60047)] is used for the construction of the Markovian Wiener-Poisson process and the associated symmetric diffusion semigroup by D. Stroock [J. Funct. Anal. 44, 212-257 (1981; Zbl 0475.60060)]. The main result says: the generator L of this process is exactly the Malliavin operator, introduced by K. Bichteler, J. B. Gravereaux and J. Jacod [Malliavin calculus for processes with jumps (to appear); see also C. R. Acad. Sci., Paris, Ser. I 300, 81-84 (1985; Zbl 0575.60058) and Sémin. de probabilités XVII, Proc. 1981/82, Lect. Notes Math. 986, 132-157 (1983; Zbl 0525.60067)]. It justifies the Malliavin calculus of variations for processes with jumps.
Reviewer: E.I.Trofimov
