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International Journal of Stochastic Analysis
Volume 2010 (2010), Article ID 217372, 7 pages
Research Article

Stochastic Integration in Abstract Spaces

Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA

Received 2 June 2010; Accepted 7 July 2010

Academic Editor: Andrew Rosalsky

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.


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 Γ— 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 ( ∫ 𝐻 𝑑 𝑋 ) 𝑑 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.