The author establishes a relation between the theory of multiple
stochastic integration and the theory of Banach space valued measures.
To briefly state the results, let Y(t,o) be an L -stochastic process
with independent increments. The process Y(t,u)) defines a finitely
additive L -valued measure on the field of all elementary subsets of
]R via the formula Y(I,ui) = Y(t,a)) - Y(s,aj) whenever I = [s,t). Under
mild conditions, this measure has an extension to a (norm) countably
additive L -valued measure defined on the a-field of all Borel subsets
of ]R . This measure Y(I,u)) can be used to define a finitely additive
L -valued measure on the product space It n2. One of the main results
of this thesis is that this product measure has a unique extension to a
countably additive L -valued measure defined on the a-field of all Borel
+ n
subsets of ]R . Once this is established it is easy to prove the theorems
of K. Ito connecting multiple Wiener integrals and Hermite polynomials.
AMS (MOS) subject classification (1980). Primary 60H05; secondary 28A35,
28B05, 28C20.
Key words and phrases: homogeneous chaos, discrete chaos, Brownian
motion, Poisson process, Hermite polynomials, Poisson-Charlier
polynomials, Banach space valued measures, partitions:
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