I am a bit puzzled having no experience in this area on the following problem.
It is well know that if we consider a system as:
\begin{equation} \dot x = f(x(t)), x \in R \end{equation}
with $f$ of class $C^{\infty}$ it possible to linearize the dynamic around the equlibria (and to be honest along a trajectory with appropriate modifications) via the computation of the Jacobian:
\begin{equation} \delta \dot x = \dfrac{\partial f(x)}{\partial x}|_{x=x^*}, \quad \delta x = x-x^* \end{equation}
But how can a similar reasoning apply in the following stochastic setting? Consider for example:
\begin{equation} dX = f(X)dt + \sqrt{\beta \cdot X}dW, \quad x \in R,\beta \in R, dW \sim \mathcal{N}(0,\,dt)\,. \end{equation}
where $dW$ is a Wiener process.
I imagine that Jacobian remains the same for $f(X)$ part but how is the term $\sqrt{\beta \cdot X}$ handled?
I have found a solution to my specific problem however it definitely does not suite a formal treatment.
Specifically, I was investigating how to deal with the afore mentioned nonlinear diffusion terms in order to compute a linear model to be used by a Kalman filter.
It appears that it is a common practice - yet not mathematically formal - that the diffusion term is evaluated at the mean value or equilibria under considerations. See this book- Applied Stochastic Differential Equations Simo Särkkä and Arno Solin -, algorithm 9.4 p. 173.
This approximation results in having once again the Jacobian for the drift term $f(X)$ but the a diffusion term that varies with the equilibria/trajectory of the mean under consideration.