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Applications of the Malliavin calculus. I. (English) Zbl 0546.60056
Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 271-306 (1984).
[For the entire collection see Zbl 0538.00017.]
The authors first review the results and techniques for the Malliavin calculus, developed by using functional analysis methods by the second author in J. Funct. Anal. 44, 212-257 (1981; Zbl 0475.60060). Shigekawa’s results on regularity are then obtained by functional analysis methods.
The second section establishes that solutions of Itô stochastic integral equations are smooth functions, in the sense of Malliavin’s calculus, even when the coefficients are not necessarily Markov and may depend on the past of the process.
The principal result of the final section is that the distribution of the solution of a general Itô equation has, roughly speaking, the same regularity properties as the solution of a diffusion, as long as the coefficients of the white noise terms are non-degenerate.
Reviewer: R.J.Elliot

MSC:
60Hxx Stochastic analysis
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60H25 Random operators and equations (aspects of stochastic analysis)