The Malliavin calculus
Let w denote the Wiener process \(\{w_ t,0\leq t\leq T\}\) and let F(w) be a square integrable functional of the Wiener process. The author gives another approach to the Malliavin calculus starting from the Wiener chaos decomposition by multiple Wiener-Itô integrals and not from the infinite dimensional Ornstein-Uhlenbeck process. Suppose F(w) is given by \[ F(w)=\sum^{\infty}_{m=0}\int^{T}_{0}\int^{t_ m}_{0}...\int^{t_ 2}_{0}f_ m(t_ 1,t_ 2,...,t_ m)dw_{t_ 1}...dw_{t_ m}. \] The author defines F(\(\lambda\) w) for \(| \lambda | <1\) as \[ F(\lambda w)=\sum^{\infty}_{m=0}\lambda^ m\int^{T}_{0}\int^{t_ m}_{0}...\int^{t_ 2}_{0}f_ m(t_ 1,t_ 2,...,t_ m)dw_{t_ 1}...dw_{t_ m} \] and the Malliavin derivative by the \(L^ 2\)-limit for \(\epsilon\to 0\) of \(\epsilon^{-1}(F(w)-F((1- \epsilon)w))\) which can also be interpreted as \(\partial F(\lambda w)/\partial \lambda |_{\lambda =1}\). He compares his approach for example with \textit{I. Shigekawa} [J. Math. Kyoto Univ., 20, 263-289 (1980; Zbl 0476.28008)] and \textit{D. W. Stroock} [Ecole d'été de probabilités de Saint Flour XI-1981, Lect. Notes Math. 976, 267-382 (1983; Zbl 0494.60060)] on the one hand side and with \textit{J. M. Bismut} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469-505 (1981; Zbl 0445.60049)] on the other side. It is shown, that the two methods are not equivalent and conditions are given under which both approaches can be considered as essentially the same.
scientific article; zbMATH DE number 3881602
scientific article