How to interpret notation $X: [0,T] \times \Omega \rightarrow R^d $ for a stochastic process?

Question

How to interpret notation $X: [0,T] \times \Omega \rightarrow R^d $ for a stochastic process, where $T$ is time, $\Omega$ is a set of outcomes and $R^d$ denotes real numbers in $d$ dimensions? Is $\times$ to be interpreted as a cross-product, if so how?

Answer 1

https://en.wikipedia.org/wiki/Cartesian_product

$X$ is a function which takes as input an ordered pair of the form $(t, \omega)$, where $t$ is a real number between $0$ and $T$, and $\omega$ is an element of the sample space $\Omega$. It yields as output an element of $R^d$, which if you like is an ordered $d$-tuple of real numbers.

As a general tip, you ought to be quite familiar with basic set theory, real analysis, and rigorous probability theory, before taking up the study of stochastic processes.

Answer 2

A stochastic process indexed by $[0,T]$ is a collection of random variables $\{X_t:t\in [0,T]\}$. We can think of the process as a single function $X:[0,T] \times \Omega \to \mathbb R$ defined by $X(t,\omega)=X_t(\omega)$. This function may not be measurable w.r.t. the product sigma algebra on $[0,T] \times \Omega$ (by which I mean the product of the Borel sigma algebra on $[0,T]$ with the given sigma algebra $\mathcal F$ on $\Omega$) but if it is, we say that the original process is a measurable process. In many situations it is convenient to think of the process as a single map.