Some Problems About B-valued Generalized Functional Of White Noise

Vector-valued generalized functional of white noise play an important role in white noise theory and its application. In this paper, we discuss the analytic characterization of B-valued generalized functionals and some related problems. Firstly, we give a definition of S-transform to a B-valued generalized functional; we characterize such functionals by using their S-transforms and an analytic characterization theorem is obtained; moreover, based on the characterization theorem, we introduce a notion of Wick product of a scalar-valued generalized functional and a B-valued generalized functional. Secondly, we apply the results obtained to Pettis integral and Bochner integral of abstract functions valued in B-valued generalized functional space. Thirdly, we use the analytic characterization theorem to discuss the convergence of sequences of B-valued generalized functionals and the continuity of abstract functions valued in B-valued generalized functional space. Finally, we discuss a Wick-type stochastic differential equation in terms of B-valued generalizedfunctional of white noise:Existence and uniqueness of a solution are proved. Continuity and continuous dependence on initial values of the solutions are shown. A filtration of B-valued generalized functionals is defined and existence of a solution adapted to the filtration is also proved of the above equation.