Stochastic Integration in Abstract Spaces

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let E, F, and G be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of E×F into G. If H is an integrable, E-valued predictable process and X is an F-valued square integrable martingale, then there exists a G-valued process (∫HdX)t called the stochastic integral. The Lebesgue space of these integrable processes is studied and convergence theorems are given. Extensions to general locally convex spaces are presented.