Definition of a stochastic process

Question

My understanding of a stochastic process is that it is a collection of random variables $\{X_t\}_{t \geq 0}$ that take a value $w$ from a sample space $\Omega$. It is said that we can see a stochastic process as a function of two variables $$(ω,t) \mapsto X_t(ω).$$ I feel that my understanding of this definition is somewhat confused, because I don't see why a stochastic process is a function of $w$? Isn't $w$ just a value that a given random variable indexed at time $t$ takes?

Given this definition, the notation: $$X : \Omega \times \mathbb{R}_+ \mapsto \mathbb{R}$$ is often used in describing a stochastic process. Could someone also explain how the '$\times$' symbol is used here and what it means?

Finally, it is said that it can more advantageous to consider $X$ as the (random) function $α \mapsto X(α, ω)$ where $\alpha \in \mathbb{R}_+$ which is called the sample path (or trajectory) of $X$ at $ω$ (and is also written $X(ω)$). Can someone explain how this definition differes intuitively from the last?

Sorry in advance for a potentially rudimentary, and random question.

Answer 1

Hope the following ramblings are somewhat useful to you ;)

• A normal random variable $Y$ is modelled as a map $Y:\Omega\to\mathbb{R}$, where $\Omega$ is called the "sample space". You should think of $\Omega$ as the set of all possible events that could happen. (usually this set will be increadibly huge, often more than countably-infinite). For each $\omega\in\Omega$, the value $Y(\omega)$ simply is the concrete value that $Y$ takes in the event $\omega$ happens.

• Now a "stochastic process" is simply a collection of many such variables, usually labeled by non-negative real numbers $t$. So $X_t$ is a random variable, and $X_t(\omega)$ is an actual number. This means that $X$ as a whole depends on two parameters. So $ X(t,\omega)$ and $X_t(\omega)$ mean exactly the same. its a real function of two parameters (one parameter is a real number $t$, the other parameter is an event $\omega$). Thats what is meant by $$X:\Omega \times \mathbb{R}_+ \to \mathbb{R}.$$ The $\times$ is just a "cartesian prodcut", which is always used if a function has more than one parameter.

• As with any function of two paramters, you can do two different things with it:

  • Fix $t$ and leave $\omega$ undefined. This is written as $X_t$ which is than simply a single stochastic variable. In this way, $X$ can be thought of as mapping $t$ to a random variable $X_t$.
  • Fix $\omega$ and leave $t$ open. This might be written as $X(\omega)$, or as $\alpha\mapsto X_\alpha(\omega)$, where $\omega$ is a fixed event. This object is a function from $\mathbb{R}_+$ to $\mathbb{R}$, so you can draw it as a path/trajectory. In this way, $X$ can be though of mapping an event $\omega$ to a path $X(\omega)$.

The difference between these two ways of thinking about it is really just in which order to plug in the two parameters $t$ and $\omega$.