Copyright © 2010 J. K. Brooks and J. T. Kozinski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.