state space of stochastic process

Question

According to the definition of stochastic process in Wikipedia:

Given a probability space $(\Omega, \mathcal{F}, P)$, a stochastic process (or random process) with state space $X$ is a collection of $X$-valued random variables indexed by a set $T$ ("time"). That is, a stochastic process $F$ is a collection

$\{ F_t : t \in T \}$

where each $F_t$ is an $X$-valued random variable.

Does that mean for all values of the index $t$, the state space $X$ and the $\sigma$-algebra on it must be same? For a stochastic process, Can it be allowed that the state space and the $\sigma$-algebra on it is different for different value of the index?

Is it correct that, as a stochastic process, "a random walk on a graph" seems to have different state spaces at different stages/times?

Thanks and regards!

Answer 1

Yes, the measurable space $(X, \mathcal{M})$ ($\mathcal{M} \subset 2^X$ being a $\sigma$-algebra) should be the same for all $t$ in the index set. If you want the process to occupy different sets $X_t$ at different times, then the right things to do is to take $X$ to be the disjoint union of all the $X_t$. There is no requirement that $F_t$ be surjective, so there is no harm if some of the state space is not used at some times.

For a random walk on a graph, the state space $X$ is just the vertex set of the graph, typically with the discrete $\sigma$-algebra. Then for each $t \in T$, $F_t$ is some random element of $X$, i.e. a random vertex of the graph. Formally, if your underlying probability space is $(\Omega, \mathcal{F}, P)$, then $F$ is a map from $\Omega \times T$ into $X$, with the property that for each $t \in T$, the map $\Omega \ni \omega \mapsto F(\omega, t) \in X$, denoted $F_t$, is measurable.

Answer 2

It's strange that it's mentioned the same space. In fact random process is a collection $(X_t|t\in \mathcal{T})$ where $X_t\in \mathbb{X}_t$ and state spaces can be different. As I remember sometimes notion "stochastic process" for the cases when all state spaces are the same. You can find a lot of literature on the random process with changing state spaces.