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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A summability criterion for stochastic integration


Authors: Nicolae Dinculeanu and Peter Gray
Journal: Proc. Amer. Math. Soc. 136 (2008), 4437-4444
MSC (2000): Primary 60G20; Secondary 60G44
Published electronically: July 30, 2008
MathSciNet review: 2431060
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Abstract: In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes $ X$ with values in a Banach space $ E$; namely, $ X$ is of class (LD), and the stochastic measure $ I_{X}$ is bounded and strongly additive in $ L_{E}^{p}$ (in particular, if $ I_{X}$ is bounded in $ L_{E}^{p}$ and $ c_{0}\nsubseteq E$) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with $ p$-integrable semivariation or $ p$-integrable variation, the conditions of $ I_{X}$ to be bounded and have bounded semivariation are superfluous. For processes with $ 1$-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.


References [Enhancements On Off] (What's this?)

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Additional Information

Nicolae Dinculeanu
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: nd@math.ufl.edu

Peter Gray
Affiliation: Department of Mathematics, Lake City Community College, Lake City, Florida 32025

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09497-5
PII: S 0002-9939(08)09497-5
Keywords: Stochastic processes, Stochastic integral, summable processes.
Received by editor(s): May 3, 2007
Received by editor(s) in revised form: June 7, 2007, and November 23, 2007
Published electronically: July 30, 2008
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.