Suppose I have signal process $\lambda_t$ following the dynamics \begin{equation} \begin{aligned} \zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\ \xi_t&=\mu^{\xi}(t,\xi_t)dt+\sigma^{\xi}(t,{\xi}_t)dW^{\xi}_t\\ \frac{\partial p_t}{\partial t}&= A p_t\\ \end{aligned} \end{equation} where $p_t^1\triangleq \mathbb{P}\left(\lambda_t={\zeta}\right)$, and $p_t^2\triangleq \mathbb{P}\left(\lambda_t={\xi}\right)$, are the states of the continuous-time finite state Markov process $\lambda_t$ (with state space $\mathbb{S}\triangleq \{\zeta_t,\xi_t\}$).

Moreover, my observation process follows the dynamics $$ dY_t = a(t,\lambda_t,Y_t)dt + b(t,\lambda_t,Y_t)d\tilde{W}_t, $$ where $\tilde{W}_t$ and $W_t$ are correlated Brownian motions. I know there are explicit solutions to the "The Kushner–Stratonovich Equation" for the general semi-martingale case (see citation below) as well as the particular KS Equation for finite-state continous-time Markov process. My question is how can I (is it known) how to combine these to obtain the explicit KS equation for the filtering problem I have stated. (Can I just combine the general KS solutions for both with the KS equations for the Markov process goverened by $A$?

*Cohen, Samuel N.; Elliott, Robert J.*, **Stochastic calculus and applications**, Probability and Its Applications. New York, NY: Birkhäuser/Springer (ISBN 978-1-4939-2866-8/hbk; 978-1-4939-2867-5/ebook). xxiii, 666 p. (2015). ZBL1338.60001.)