Product Stochastic Measures, Multiple Stochastic Integrals and Their Extensions to Nuclear Space Valued Processes

摘要

A theory of L sub 2 valued product stochastic measures of non-identically distributed L sub 2 - independently scattered measures is developed using concepts of symmetric tensor product Hilbert spaces. Applying the theory of vector valued measures we construct multiple stochastic integrals with respect to the product stochastic measures. A clear relationship between the theories of vector valued measures and multiple stochastic integrals is established. This work is related to the work by D.D. Engel (1982) who gives a different approach to the construction of product stochastic measures. The two approaches are compared. The second part of the work deals with multiple Wiener integrals and nonlinear functionals of a phi - valued Wiener process W sub t where phi is the dual of a Countably Hilbert Nuclear Space. We obtain the Wiener decomposition of the space of phi-valued nonlinear functionals as an inductive limit of appropriate Hilbert spaces. It is shown that every phi - valued nonlinear functional admits an expansion in terms of multiple Wiener integrals in one of these Hilbert spaces and can be represented as an operator valued stochastic integral of the Ito type. (Author)