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.