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Set-parametered martingales and multiple stochastic integration

  • Bruce Hajek
  • Eugene Wong
Papers Based On Main Talks And Courses
Part of the Lecture Notes in Mathematics book series (LNM, volume 851)

Abstract

The starting point of this paper is the problem of representing square-integrable functionals of a multiparameter Wiener process. By embedding the problem in that of representing set-parameter martingales, we show that multiple stochastic integrals of various order arise naturally. Such integrals are defined relative to a collection of sets that satisfies certain regularity conditions. The classic cases of multiple Wiener integral and Ito integral (as well as its generalization by Wong-Zakai-Yor) are recovered by specializing the collection of sets appropriately.

Using the multiple stochastic integrals, we obtain a martingale representation theorem of considerable generality. An exponential formula and its application to the representation of likelihood ratios are also studied.

Keywords

Simple Function Wiener Process Atomic Function Borel Subset Stochastic Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Bruce Hajek
    • 1
    • 2
  • Eugene Wong
    • 1
    • 2
  1. 1.Coordinated Sciences LaboratoryUniversity of Illinois at UrbanaUSA
  2. 2.Electronics Research LaboratoryUniversity of California at BerkeleyUSA

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