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Some Apealing Classical and Stochastic Evolution Equations (pp.153-164) $100.00
Authors:  (Shambhu N. Sharma, Department of Electrical Engineering, National Institute of Technology, Surat, India)
Abstract:
The dynamical system theory in combination with the notion of feedback formalizes
the concept of mathematical control theory. The structure of dynamical systems is
adjudged on the basis of dimensionality, time-invariance/time varying as well as driving
input. The stochastic differential equation resulting from the phase space analysis of the
fluctuation equation describing noisy dynamical systems has received remarkable success
in different branches of sciences. Consider the stochastic differential system of the form
g( ) , t t t t x􀀅 = f (x ) + x ξ
where the terms t t t f (x ), g(x )ξ have interpretations as the system non-linearity and
the random forcing term respectively. For the input noise process , t t B
􀀅
ξ = where t B is
the Brownian motion, the resulting stochastic differential system would be the Itô
stochastic differential system in which the state vector is Markovian. More over, the
Ornstein-Uhlenbeck (OU) process-driven stochastic differential system can be recast as
the Itô stochastic differential system by writing down the evolution equation of the OU
process, i.e. , t t t dξ = −βξ dt + β dB β > 0. The contribution to the stochastic
evolution of the state vector comes from the system non-linearity as well as the
dispersion matrix, which introduces slight modifications into the motion of the dynamical
system. This book chapter encompasses five stochastic evolution equations as well as
their special cases, i.e. classical evolution equations, which are regarded as the
cornerstone formalism for analysing and accomplishing filtering for stochastic
differential systems. The stochastic evolution equations of the chapter are about the
filtering density evolution, the exact stochastic evolution of conditional moment as well
as its approximate counterparts, the stochastic evolution of the scalar function of the
multi-dimensional state vector. This chapter discusses evolution equations for the non-
Markovian stochastic differential system as well, which are not widely available in
literature.
The evolution equations of the chapter will be of interest to Researchers looking for
understanding the theory of noisy dynamical systems in Markovian as well as non-
Markovian setting. Rigorous mathematical treatments about stochastic differential
systems can be found in authoritative books, i.e. Multi-dimensional diffusion processes
authored by DS Strook and SRS Varadhan, as well as Brownian motion and stochastic
calculus authored by I Karatzas and Steven E. Shreve. For a greater detail about
stochastic filtering, the Author’s International Journal of Control contribution (Sharma, S.
N.; Parthasarathy, H. and Gupta, J.R.P. Third-order approximate Kushner filter for a nonlinear
dynamical system. International Journal of Control, 79(9), 1096-1106, 2006) can
be consulted as well. 


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Some Apealing Classical and Stochastic Evolution Equations (pp.153-164)