The Kurzweil-Henstock theory of stochastic integration

The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of...

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Bibliographic Details
Published inCzechoslovak mathematical journal Vol. 62; no. 3; pp. 829 - 848
Main Authors Toh, Tin-Lam, Chew, Tuan-Seng
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 01.09.2012
Subjects
Analysis
Convex and Discrete Geometry
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Kurzweil-Henstock
26A39
stochastic integral
60H05
convergence theorem
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Summary:The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive Henstock’s Lemmas, absolute continuity property of the primitive process, integrability of stochastic processes and convergence theorems for the Kurzweil-Henstock stochastic integrals. These properties are well-known in the classical (non-stochastic) integration theory but have not been explicitly derived in the classical stochastic integration.
ISSN:0011-4642
1572-9141
DOI:10.1007/s10587-012-0048-z