  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.
