The Kalman-Bucy filter gives the best estimates for a partially observable linear system ( here I show a simplified version for exposition)
State $\theta_t $ of system: $d\theta_t = ( a_1\theta_t)dt + b_1dW_t$
Observation $S_t $ of system: $dS_t = ( A_1\theta_t)dt + B_2dZ_t$
where $dW_t$ and $dZ_t$ are independent random walks
the best estimate $m_t = E[\theta_t | \mathcal{F}^{S_t}$] is given by
$$dm_t = a_1m_tdt + \gamma_tA_1(dS_t-A_1m_tdt) \tag 1$$
where $\gamma_t = E[(\theta_t-m_t)^2]$ solves the following Riccati equation
$$\frac{d\gamma}{dt} = (a_1\gamma_t)^2 + b_1^2 - \frac{(\gamma_tA_1)^2}{B_2^2} \tag 2$$
My question: is there a way(maybe via linearizatio via Taylor or other method) to obtain a similar solution to the following non-linear problem
State $\theta_t $ of system: $d\theta_t = ( a_1\color{red}{f(\theta_t)})dt + b_1dW_t$
Observation $S_t $ of system: $dS_t = ( A_1\theta_t)dt + B_2dZ_t$
in which the evolution of
$$\gamma_t = E[(\theta_t-m_t)^2] \tag 3$$
is still given by an explicit equation that can be solved analytically?
Thank you!
