Problem:   
I'm working in reliability field and have seen papers written on the topic like process of failures when systems are functioning under unobservable (or observable) Markov-like environment, i.e. probability to fail is dependent on the state of environment. This state is described as discrete-state discrete-time homogeneous Markov chain. In mathematical notation it would be like this:  

$ P\left [ X\_{t}=k | Y\_{t}=i \right ]=\pi \_{k}\left ( i \right ) $;   

where 
$ X \_{t} $ is a binary random failure process with possible states 0 (failed) and 1 (working). And $Y \_{t}$ is a Markov process at the moment $t$ being in the $i ^{th}$ state.   



My question:   



Is it possible (or even reasonable) to extend mentioned model? For example, from discrete-state to continuous-state Markov model? Is there any literature (I havent found yet) about continuous **stochastic conditional processes**. I suppose it is not so trivial, because for continuouse stochastic process statements like $Y\_{t}=i$ are meaningless.



Reference on continuous-time finite state filtering
This question is related to the topic of stochastic filtering theory. See e.g. the following monographs
\* Bucy, Joseph - Filtering for stochastic processes with applications to guidance
\* Bain, Crisan - Fundamentals of stochastic filtering
\* Kallianpur - Stochastic filtering theory


Explicit solutions exist for the linear case (Kalman-Bucy filter). For the nonlinear case the situation is more complicated. See also the wikipedia page 
<http://en.wikipedia.org/wiki/Kalman_filter#Kalman.E2.80.93Bucy_filter>


