0
votes
0answers
43 views

Continuous time Markov chains, how would this definition be expanded from time-homogeneous to time-inhomogeneous.

Below I have a picture of how we can view a continuous time Markov chain that is time-homogeneous. Now, I am wondering what happens when we have a inhomogeneous continuous Markov chain. I have ...
3
votes
2answers
292 views

What does it mean for a stochastic process to be independent of a sigma algebra?

Does anybody know what it means for a stochastic process $ X = (X_t)_{t \geq 0} $ on a filtered probability space $ (\Omega, \mathcal{F}, \mathbb{F}, P) $ to be independent of a sigma-algebra $ ...
3
votes
2answers
674 views

Definition: transient random walk

What exactly does a "transient random walk on a graph/binary tree" mean? Does it mean that we never return to the origin (assuming there is one as for the tree) or just any vertex of the graph or ...
1
vote
2answers
274 views

Almost surely in definition of Martingale

$(X_n), n \in \mathbb{N}$ is a stochastic process. I saw in one definition of Martingale that $$E [X_{n+1} |X_0 , X_1 , . . . , X_n ] = X_n \quad a.s., \forall n \geq 0.$$ I understand what "almost ...