stochastic processes conditional on other stochastic processes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:50:23Z http://mathoverflow.net/feeds/question/89684 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89684/stochastic-processes-conditional-on-other-stochastic-processes stochastic processes conditional on other stochastic processes Tomas 2012-02-27T17:34:30Z 2012-03-15T14:05:21Z <p>Problem: <br/>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:<br/> $P\left [ X_{t}=k | Y_{t}=i \right ]=\pi _{k}\left ( i \right )$; <br/> 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. <br/></p> <p>My question: <br/></p> <p>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 <strong>stochastic conditional processes</strong>. I suppose it is not so trivial, because for continuouse stochastic process statements like $Y_{t}=i$ are meaningless.</p> http://mathoverflow.net/questions/89684/stochastic-processes-conditional-on-other-stochastic-processes/91287#91287 Answer by Joris Bierkens for stochastic processes conditional on other stochastic processes Joris Bierkens 2012-03-15T14:05:21Z 2012-03-15T14:05:21Z <p>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</p> <p>Explicit solutions exist for the linear case (Kalman-Bucy filter). For the nonlinear case the situation is more complicated. See also the wikipedia page <a href="http://en.wikipedia.org/wiki/Kalman_filter#Kalman.E2.80.93Bucy_filter" rel="nofollow">http://en.wikipedia.org/wiki/Kalman_filter#Kalman.E2.80.93Bucy_filter</a></p>