Integration by parts on Wiener space and the existence of occupation densities. (English) Zbl 0803.60049

Motivated by recent developments in the anticipative stochastic calculus [see D. Nualart and E. Pardoux, Stochastic differential systems, Proc. IFIP-WG 7/1 Work Conf., Eisenach/GDR 1986, Lect. Notes Control Inf. Sci. 96, 363-372 (1987; Zbl 0633.60075)] and, in particular, in the theory of Skorokhod’s stochastic integral (introduced by Skorokhod, 1975) which allows to integrate also non-adapted processes, the authors investigate conditions under which Skorokhod integral processes possess an occupation density or a local time. The paper generalizes earlier results of Imkeller about Skorokhod integral processes belonging to the second Wiener chaos, and it uses for this purpose a new general approach: the authors combine a sufficient criterion for the existence of an occupation density by D. Geman and J. Horowitz [Ann. Probab. 8, 1-67 (1980; Zbl 0499.60081)] with the technique of integration by parts on the Wiener space and deduce several sufficient criteria for the existence of occupation densities of Skorokhod integral processes.
In particular, the authors prove that, if \((W_ t)\) denotes the Wiener process and \(u\) a possibly anticipating process which belongs w.r.t. \((W_ t)\) to \(\mathbb{L}^{2,2}\), then the process \((W_ t+ \int_ 0^ t u_ s dW_ s)\) has a square integrable occupation density. If, moreover, for some suitable \(\gamma>0\), \(E[\int^ 1_ 0 | u_ t|^{-\gamma} dt] <+\infty\), then, under some additional smoothness assumptions and boundedness assumptions on the Malliavin derivatives of \(u\), also the process \((\int^ t_ 0 u_ s dW_ s)\) possesses a square integrable occupation density.
Reviewer: R.Buckdahn (Brest)


60H05 Stochastic integrals
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