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Stochastic Wiener filter in the white noise space

Alpay Daniel, Pinhas Ariel

Opuscula Mathematica

2020

Vol. 40, no. 3

323--339

. In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

Alpay D., Kipnis A.

Opuscula Mathematica

2013

Vol. 33, no. 3

395--417

Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

White noise based stochastic calculus associated with a class of Gaussian processes

Alpay D., Attia H., Levanony D.

Opuscula Mathematica

2012

Vol. 32, no. 3

401-422

Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula.