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Calculus on Gaussian and Poisson white noises

Published online by Cambridge University Press:  22 January 2016

Yoshifusa Ito  and
Izumi Kubo
Yoshifusa Ito
Affiliation:
Nagoya University College of Medical Technology, Higashi-ku, Nagoya 461, Japan
Izumi Kubo
Affiliation:
Hiroshima University, Naka-ku, Hiroshima 730, Japan
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Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc. ([8], [9]), analogously to the works of T. Hida ([3], [4], [5]). Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf. [10], [11], [12], [13]). Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc. in a way completely parallel with the Gaussian case. The only one exceptional point, which is most significant, is that the multiplications are described by

for the Gaussian case,

for the Poisson case,

as will be stated in Section 5. Conversely, those formulae characterize the types of white noises.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 111 , September 1988 , pp. 41 - 84
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

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