How does $\Omega$ figure in stochastic processes?

Question

So I read this page for clarification on trajectories and $X(\omega, \cdot): T\to \mathbb R$ maps while going through lectures on stochastic processes. I still have doubts which are described as follows.

The lecturer gives various examples of discrete-parameter-discrete-state stochastic processes, such as the collection of random variables $X_n$ which denote the number of customers waiting in queue after the $n$th customer has left a shop, and continuous-parameter-continuous-state stochastic processes, such as the collection of random variables $X_t$ which denote the volume of water in a dam after time $t$, starting from an arbitrary time $t=0$.

My confusion is how the sample space $\Omega$ figures in these stochastic processes; that is, how exactly does some $\omega$ come into the picture so that I can visualise the map $X(\omega, \cdot): T\to \mathbb R$ as dependent on a fixed value of $\omega$ while the other parameter, here time, changes? What even is $\omega$ in the above cases?

$\Omega$ is a very large space having enough degrees of freedom to produce trajectories $t\mapsto X(\omega,t),.$ In the construction of Brownian motion one starts with $\Omega$ being the space of all functions from $\mathbb R_+$ to $\mathbb R,.$ I guess all Borel measurable functions but these are technicalities. — Kurt G., Nov 28 '23 at 15:11
Slow down please, I only have an introductory probability theory course behind me and would love a more dumbed-down explanation — insipidintegrator, Nov 28 '23 at 15:15
If you ask what $\omega$ is I don't think this can be dumbed down. For now: You can view $\Omega$ as a large enough space that gives you all the trajectories when you plug its $\omega,$s into $X,.$ — Kurt G., Nov 28 '23 at 15:21
Hi @KurtG., could you please give examples of $\omega$ for the example processes I've mentioned in the post? — insipidintegrator, Nov 28 '23 at 15:23
Lack of context. Presumably you mean $T={1,2}$ and $X(\omega,t)$ taking values in $Z={\text{heads},\text{tails}},?$ If so: OP in that post has listed all maps from $T$ to $Z$ (there are not that many). That set of maps is your $\Omega,.$ — Kurt G., Nov 28 '23 at 15:27
Aha. "The lecturer gives various examples ...". General principle (applied to all my answers so far). Know what your $T$ is, know what your state space $Z$ is. Then the prime example of $\Omega$ is the set of all maps from $T$ to $Z,.$ — Kurt G., Nov 28 '23 at 15:36

score 1 · Accepted Answer · edited Nov 28 '23 at 15:37

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Since in your linked post there are examples for $\Omega$ in discrete time, this answer will focus on stochastic processes on a continuum $[0,T]$ mapping into $\mathbb{R}$.

Assume we have a probability space $(\Omega,\Sigma,P)$. If you fix $\omega \in \Omega$ you get the map $X(\omega,\cdot):[0,T]\to \mathbb{R}$. Per definition this is nothing more than a function, called trajectory or sample path. So you want $\Omega$ to be a space of functions (e.g. $C([0,T],\mathbb{R})$) that at least contains all the sample paths of the stochastic process. So a real random variable behaves to a real stochastic process like a real number to a real function. Basically your $\omega$ is a random function.
For some stochastic processes the construction of the probability space is not easy, for example the Wiener process. You can use $\Omega=C([0,T],\mathbb{R})$. Finding the measure $P$ is harder. In case of Wiener process this was explicitly done by Wiener (Link).

edited Nov 28 '23 at 15:37

insipidintegrator

6,642

answered Nov 28 '23 at 15:18

Keine_Maschine

319

Can you explain "If you want so a real random variable behaves to a real stochastic process like a real number to a real function" a bit more? – insipidintegrator Nov 28 '23 at 15:24
A real random variable $Y$ maps $Y:\Omega_1 \to \mathbb{R}$, which means $Y(\omega)$ is a real number . A real stochastic process $X$ maps $X:[0,T] \times \Omega_2 \to \mathbb{R}$, which means $X(\omega,\cdot)$ is a real function. The step from $\mathbb{R}$ to a function space, is a step from finite dimension to infinite dimension. Similiar is the step from $\Omega_1$ to $\Omega_2$. – Keine_Maschine Nov 28 '23 at 15:35
And if I may ask further, could you clarify what the dot in the function notation actually means? – insipidintegrator Nov 28 '23 at 15:41
$X(\omega,\cdot):[0,T]\to \mathbb{R}, t\to X_{\omega}(t)$ – Keine_Maschine Nov 28 '23 at 15:47
@insipidintegrator: I think of it in the following dumbed down way. Your input is an $\omega$ and the outcome is a full trajectory. Your input is a different $\omega$ and the out come is a different full trajectory. And so on and so forth. – mark leeds Nov 28 '23 at 16:02
@markleeds could you give a practical example of $\omega$ please? – insipidintegrator Nov 28 '23 at 16:08
@Keine_Maschine thank you for your time and patience, but one last question. I do not quite understand "a step from finite dimension to infinite dimension". – insipidintegrator Nov 28 '23 at 18:09
The dimension of typical function spaces (e.g. the space of all continous functions) is of infinite dimension. The behaviour of finite and infinite spaces differs greatly, therefore they are studied in different branches of mathematics (analysis and functional analysis). – Keine_Maschine Nov 28 '23 at 19:20
@insipidintegrator : As Keine_Maschine explained it, think of a particular $\omega$ as being one of the random outcomes of flipping a coin 50 times. you'll get a sequence of 50 H and T but that sequence can be thought of as one random variable just as set of $X_{t_{1}}, X_{t_{2}}, \ldots ,X_{t_{n}}$ can be thought of as one trajectory. So, think of $\omega$ as mapping to a function which outputs 50 H and T. – mark leeds Nov 29 '23 at 17:30
Thanks @markleeds and Keine, that clears up things massively! – insipidintegrator Nov 30 '23 at 11:26
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I'm glad it helped. There's a saying ( I forget creator ), "if you can't explain it simply, then you don't understand it". – mark leeds Dec 01 '23 at 16:37

How does $\Omega$ figure in stochastic processes?

1 Answers1