$\sigma$-algebra generated by a stochastic process

Question

Let $x_t:\Omega \to \mathbb{R}$ a random variable and $\{x_t\}:\Omega \to \mathbb{R}^\mathbb{Z}$ a stochastic process. Can someone explain to me the notion of the $\sigma$-algebra generated by a stochastic process? I understand that of the $\sigma$-algebra generated by the random variable $x_t$ but I am struggling to extend it to stochastic processes. An example (maybe with discrete $\Omega$) would help too.

Answer 1

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

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