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Abstract:
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In this dissertation we study the main properties of the operators
of Malliavin calculus de ned on a set of singular generalized stochastic
processes, which admit chaos expansion representation form in terms of orthogonal
polynomial basis and having values in a certain weighted space of
stochastic distributions in white noise framework.
In the rst part of the dissertation we focus on white noise spaces and
introduce the fractional Poissonian white noise space. All four types of white
noise spaces obtained (Gaussian, Poissonian, fractional Gaussian and fractional
Poissonian) can be identi ed through unitary mappings.
As a contribution to the Malliavin di erential theory, theorems which
characterize the operators of Malliavin calculus, extended from the space
of square integrable random variables to the space of generalized stochastic
processes were obtained. Moreover the connections with the corresponding
fractional versions of these operators are emphasized and proved. Several examples of stochastic di erential equations involving the operators of the Malliavin calculus, solved by use of the chaos expansion
method, have found place in the last part of the dissertation. Particularly,
obtained results are applied to solving a generalized eigenvalue problem
with the Malliavin derivative and a stochastic Dirichlet problem with a
perturbation term driven by the Ornstein-Uhlenbeck operator. |