Set-parametered martingales and multiple stochastic integration
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.
KeywordsSimple Function Wiener Process Atomic Function Borel Subset Stochastic Integral
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- 3.Hajek, B. E.: Stochastic Integration, Markov Property and Measure Transformation of Random Fields. Ph.D. dissertation, Berkeley, 1979.Google Scholar
- 7.Mitter, S. K., Ocone, D.: Multiple integral expansion for nonlinear filtering. Proc. 18th IEEE Conference on Decision and Control, 1979.Google Scholar