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Master Equations in the Theory of Stochastic Processes (pp.419-432) $100.00
Authors:  (Nanasaheb S. Patil, Shambhu N. Sharma, Department of Electrical Engineering, National Institute of Technology, Surat, India, and others)
Abstract:
The stochastic processes, which are exploited to analyze the dynamical system in
noisy environments, are white noise process, the Brownian motion process, the Ornstein-
Uhlenbeck (OU) process. More notably, the OU process is a coloured noise process. In
1991, Karatzas and Shreve (I. Karatzas and S. E. Shreve, Brownian Motion and
Stochastic Calculus, Springer-Verlag, New York, 1991) published an authoritative book
discussing the Brownian motion process as well as the Brownian motion process-driven
stochastic differential system in the rigorous mathematical framework, especially in the
Itô setting. The estimation algorithm for the Itô stochastic differential equation (SDE) is
developed using the Fokker-Planck equation, the master equation for the Itô SDE setting.
The power spectral density of the stochastic process associated with the stochastic
problems in physics, digital wireless communications is the frequency-dependent that
justifies the usefulness of coloured noise processes in dynamical systems. The problem of
analyzing the coloured noise-driven stochastic differential system can be accomplished
by generalizing the Fokker-Planck equation. Master equations for coloured noise-driven
stochastic differential systems involve the more general structures in contrast to the
Fokker-Planck equation. For this reason, it seems quite worthwhile to summarize
available results in literature, which are widespread, about the master equations. This
paper encompasses six different master equations, which can be exploited to accomplish
the estimation procedures for stochastic differential systems, especially where
observations are not available. The structure of a master equation is dictated on the basis
of the noise process associated with the dynamical system. The proofs of master
equations of this paper can be accomplished using the functional calculus and
perturbation-theoretic approaches to the system non-linearity-process noise coupling
terms. Furthermore, this paper discusses briefly the general theory of two mathematical
problems related to the estimation theory for coloured noise processes. 


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Master Equations in the Theory of Stochastic Processes (pp.419-432)