0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $\omega$.

$\endgroup$
1
  • $\begingroup$ Great ramblings! Cheers. Clarifies a lot. $\endgroup$
    – Semmah
    Apr 28 at 12:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.