Definition of sigma-algebra of a continuous time stochastic process in a countable set

Question

Let $X_{t} : t \geq 0$ be a continuous time stochastic process taking values in a countable set $I$, namely $X_t$ is a function defined in a measure space, for any $t \geq 0$. How is the sigma-algebra of the stochastic process, $$ \sigma \Big ( X_{t} : t \geq 0 \big ) $$ defined?

Answer 1

It's the $\sigma -$algebra generated by cylinder.

Denote $C_{t_1,...,t_n}(B_1,...,B_n)=\{X_{t_1}\in B_1,...,X_{t_n}\in B_n\}$. Then

$$\sigma (X_t\mid t\geq 0)=\sigma \{C_{t_1,...,t_n}(B_1,...,B_n)\mid n\in\mathbb N, B_i\in \mathcal B(\mathbb R), 0\leq t_1<...<t_n\}.$$

Answer 2

Let $(X_t)_{t\geq 0}$ be a (real-valued) stochastic process on some measurable space $(\Omega,\mathcal{F})$, i.e., for all $t\geq 0$ $X_t:\Omega\to\mathbb{R}$ is a random variable.

The sigma algebra generated by the stochastic process $(X_t)_{t\geq 0}$ is the smallest sigma algebra such that $X_t$ is measurable for all $t\geq 0$, i.e., $$\sigma\left(X_{t}\colon t\geq 0\right)=\left\{A\in\mathcal{F}\mid \exists t\geq 0 \;\exists B ∈ \mathcal{B}(\mathbb{R}):\; A=X^{-1}_t(B)\right\},$$ where $\mathcal{B}(\mathbb{R})$ denote the Borel sets over the real line.