Generalized stochastic integrals and the Malliavin calculus
The paper first reviews the Skorohod generalized stochastic integral with respect to the Wiener process over some general parameter space T and its relation to the Malliavin calculus as the adjoint of the Malliavin derivative. Some new results are derived and it is shown that every sufficiently smooth process \(\{u_ t\), \(t\in T\}\) can be decomposed into the sum of a Malliavin derivative of a Wiener functional, and a process whose generalized integral over T vanishes. Using the results on the generalized integral, the Bismut approach to the Malliavin calculus is generalized by allowing non adapted variations of the Wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions. Let \(e_ i\) be a non-random complete orthonormal system on T, the Ogawa integral \(\int u\overset\circ\delta W\) is defined as \(\sum_{i}(e_ i,u)\int e_ idW\) where the integrals are Wiener integrals. Conditions are given for the existence of an intrinsic Ogawa integral i.e. independent of the choice of the orthonormal system and results on its relation to the Skorohod integral are derived. The transformation of measures induced by \((W+\int ud\mu)\) with u non adapted is discussed and a Girsanov-type theorem under certain regularity conditions is derived.
Skorohod generalized stochastic integral
scientific article; zbMATH DE number 3969792
scientific article