# Definition of a stochastic process

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.

## 1 Answer

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 $$\omega$$.

• Great ramblings! Cheers. Clarifies a lot. Apr 28 at 12:13