Nonlinear Filtering Theory for McKean--Vlasov Type Stochastic Differential Equations

2016 ◽  
Vol 54 (1) ◽  
pp. 153-174 ◽  
Author(s):  
Nevroz Şen ◽  
Peter E. Caines
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonlinear Filtering ◽  
Filtering Theory

scholarly journals On the machine swing dynamics: a perspective 

2021 ◽  
Author(s):  
Prashant G. Medewar ◽  
Shambhu N. Sharma
Keyword(s):  
Differential Equation ◽  
Markov Process ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Nonlinear Filtering ◽  
Applied Mathematics ◽  
Formal Approach ◽  
Augmented State ◽  
Swing Equation

Abstract A formal approach to rephrase nonlinear filtering of stochastic differential equations is the Kushner setting in applied mathematics and dynamical systems. Thanks to the ability of the Carleman linearization, the ‘nonlinear’ stochastic differential equation can be equivalently expressed as a finite system of ‘bilinear’ stochastic differential equations with the augmented state under the finite closure. Interestingly, the novelty of this paper is to embed the Carleman linearization into a stochastic evolution of the Markov process. To illustrate the Carleman linearization of the Markov process, this paper embeds the Carleman linearization into a nonlinear swing stochastic differential equation. Furthermore, we achieve the nonlinear swing equation filtering in the Carleman setting. Filtering in the Carleman setting has simplified algorithmic procedure. The concerning augmented state accounts for the nonlinearity as well as stochasticity. We show that filtering of the nonlinear stochastic swing equation in the Carleman framework is more refined as well as sharper in contrast to benchmark nonlinear EKF. This paper suggests the usefulness of the Carleman embedding into the stochastic differential equation to filter the concerning nonlinear stochastic differential system. This paper will be of interest to nonlinear stochastic dynamists exploring and unfolding linearization embedding techniques to their research.

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