Let
$$dX_t = \mu(X_t,t) dt + \sigma dB_t$$
$$dY_t = h(X_t,Y_t,t) dt + \eta dW_t$$
where $B$ and $W$ are independent standard Brownian motions, $\eta, \sigma$ are positive real numbers.
What equation does the law/density of $E[X_t | \sigma(Y_{0 \rightarrow t})]$ satisfy ? This is supposed to be what is called the Kushner-Stratonovich equation, or filtering equation, right ? Depending on where I look, I get different equations, it is very confusing.
I found this paper: https://arxiv.org/pdf/1407.6043.pdf But it seems to treat a more general setting and I cannot make sense of it in the above case that interests me. A nice (modern) reference for the above case would be very welcome. The older papers of the "fathers" of filtering seem to not consider the case where the drifts depend on time.