ON THE EXISTENCE OF CANONICAL FORM IN ALL SQUARE- INTEGRABLE MARTINGALE WITH RESPECT TO {????????(????),???? ∈ ????} IN THE WIENER FUNCTIONAL SPACE UNDER VERY GENERAL CONDITION ON ℳ

Abstract

The Wiener space is the collection of all continuous function on a given domain, taking values in a metric space, and the Wiener functional space has a canonical form of any square- integrable functional in terms of the integrals. This paper attempted to shown the existence of a canonical representation of all square-integrable Martingale with respect to {ℱ????(????), ???? ∈ ℳ} under very general condition on ℳ. The major key here is to define multiple stochastic integral of the form ∫ ????(ℎ1, ℎ2, … . , ℎ????)????(????ℎ) … ????(????ℎ????)
ℋ???? where ???? is (in general) a random integrand ????-adapted in a suitable sense. This was achieved by critically examining a formula for changing a multiple stochastic integrals onto L2 (η,ℱW ????) and adapting an iterated formula, which will be to obtain through the application of iterated integrals.