Stochastic Integration in Abstract Spaces

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let ð ¸ , ð ¹ , and ð º 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 ð ¸ Ã— ð ¹ into ð º . If ð » is an integrable, ð ¸ -valued predictable process and ð ‘‹ is an ð ¹ -valued square integrable martingale, then there exists a ð º -valued process ( ∫ ð » ð ‘‘ ð ‘‹ ) ð ‘¡ 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.