White noise stochastic integration

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White noise is often regarded as the informal nonexistent derivative B˙(t) of a Brownian motion B˙(t). Before K. Itô introduced the stochastic integral in 1944, white noise had been used as a random noise which is independent at different times and has large fluctuation. It was an innovative idea of Itô to consider the product of white noise B˙(t) and the time differential dt as a Brownian motion differential dB(t), a quantity to serve as an integrator in the Itô theory. In 1975 T. Hida introduced white noise theory which provides a rigorous mathematical definition of B˙(t) as a generalized function defined on the space of tempered distributions on the real line. The white noise B˙(t) can further be regarded as a multiplication operator and B˙(t) = ∂t + ∂*t; with ∂t being the white noise differentiation and ∂*t its adjoint. In this paper we will give a brief survey of Hida’s theory of white noise and its applications to stochastic integration. We will use the operator ∂*t to study stochastic integrals with anticipative integrands and stochastic differential equations with anticipative initial conditions and integrands. We will also point out and describe several perspectives for further applications of white noise theory to stochastic integration.

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Stochastic Analysis: Classical and Quantum: Perspectives of White Noise Theory

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