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Product of Twomultiple Stochastic Integrals With Respect to a Normal Martingale

By Francesco Russo, Pierre Vallois and Nancy Cedex


: Let M be a normal martingale #i.e. #M;M##t#=t#, we decompose the product of twomultiple stochastic integrals #with respect to M# I n #f#I m #g# as a sum of nm terms H k . H k is equal to the integral over R k + of the function t ! I n+m,2k #h k #t; :##, with respect to the k-tensor product of d#M;M#:; h k being an explicit function depending only on f and g. Our formula generalizes the well-known result concerning Brownian motion and compensated Poisson process and allows us to improve some results of Emery related to the chaos representation property of solution of the structure equation. Mathematics Subject Classi#cation: 60 G 44, 60 H 05, 60 H 07, 60 J 65. Key Words: normal martingale, chaos representation property. 1. Introduction 1# In his historical paper, Ito #1951#, proved that the Brownian motion B has the chaos representation property #CRP#. This means that every square integrable random variable Z measurable with respect to the #-algebra generated by the Brownian mot..

Topics: Key Words, normal martingale, chaos representation property
Year: 1998
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