stochastic processes conditional on other stochastic processes

2012-02-27T17:34:30Z

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.

Answer by Joris Bierkens for stochastic processes conditional on other stochastic processes

2012-03-15T14:05:21Z

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

stochastic processes conditional on other stochastic processes - MathOverflow

stochastic processes conditional on other stochastic processes

Answer by Joris Bierkens for stochastic processes conditional on other stochastic processes