Let $\mathcal P=(\Omega ,\mathcal F,\mathbb P)$ a probability space and $(X_n)_n$ a stochastic process of $\mathcal P$. Since we know that $X_n$ are $\mathcal F-$measurable, why in exercise we always take filtration $(\mathcal F_n)$ ? In what they are useful ? I really have problem to understand this thing with $\sigma -$algebra. Could someone give me some explanation ? By the way, I really don't understand this thing that $\mathcal F$ is the available information, what does it mean ? When I do an exercise, I really consider they don't exist, and it always work very good, so what's the trick with them, they really looks un-useful.

For example, if I take $X(\omega )=2\omega^2$, why do I need $\mathcal F$ ? I never use it... People always specify a $\sigma -$algebra. Is it really necessary ? I have the impression that we make the problem more complicated than what it is with those filtration and $\sigma -$algebra !


1 Answer 1


Your question consists of multiple parts.

(i) The need for measure spaces.

Probability theory primarily deals with the study of probability distributions rather than the random variables themselves. Hence we study probability measures on measurable spaces. The need for $\sigma$-algebras arises when working on uncountable spaces, such as the real line $\mathbb{R}$. We want to assign probabilities to subsets of $\mathbb{R}$. It turns out doing this in a consistent way taking all possible subsets into account is troublesome. Restricting us to certain nice subsets, given by a suitable $\sigma$-algebra smaller than the set of all subsets, is necessary. See also this: https://stats.stackexchange.com/questions/199280/why-do-we-need-sigma-algebras-to-define-probability-spaces.

An example of a measurable space is $(\mathbb{R},\mathbb{B})$ with $\mathbb{B}$ being the Borel $\sigma$-algebra of $\mathbb{R}$. A probability distribution is now simple any measure $\mu$ on $\mathbb{B}$ with $\mu(\mathbb{R})=1$. For any event, i.e. set in $\mathbb{B}$ (any nice subset of $\mathbb{R}$), the probability distribution $\mu$ assigns a number between $0$ and $1$, i.e. a probability, to this event. The requirements that $\mu$ is a measure will ensure that the probability axioms of Kolmogorov hold.

(ii) Background spaces and random variables

In probability theory, emphasis is primarily on the study of probability distributions, but random variables are an extremely useful tool.

Say you are studying a distribution $\mu$ on $(\mathbb{R},\mathbb{B})$, this could e.g. be a standard normal distribution describing the error in some measurement. If we let $\Omega = \mathbb{R}$, $\mathbb{F}=\mathbb{B}$, and $\mathbb{P}=\mu$, it follows that the random variable $X : (\Omega,\mathbb{F}) \to (\mathbb{R},\mathbb{B})$ given by $X(\omega)=\omega$ has distribution $\mu$. This is an explicit decription of the background space $(\Omega,\mathbb{F},\mathbb{P})$ and the random variable -- and it is completely unnecessary as long as we are simply interested in studying the distribution of $X$, i.e. $\mu$. But often, we will introduce another source of randomness: the error of another measurement. Having to reformulate the whole setup ($\mathbb{R}^2$ instead of $\mathbb{R}$ etc.) is very unappealing. Describing it as two random variables, each with values in $\mathbb{R}$ and with common distribution $(X,Y)(\mathbb{P})$ is much simpler; it is convenient, and it the core idea behind probabilistic thinking.

It might be very useful for you to read (the first few sections) of these notes from the blog of Terence Tao: https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/. This is a good introduction to probabilistic thinking.

(iii) Filtrations

When studying stochastic processes, it is interesting to view them as just that: processes in time. If we want to work with subsequences of $X=(X_n)_{n\in\mathbb{N}}$ without changing probabilistic setting, specifying a collection of $\sigma$-algebras with a natural link to the distributions $(X_1,\ldots,X_n)$ is needed.

It is correct that $(X_1,\ldots,X_n)$ necessarily has to be measurable w.r.t. $\mathbb{F}$, but far less suffices. If we let $\mathcal{F}^X:=(\mathcal{F}_n^X)_{n\in\mathbb{N}}$ be the filtration generated by $X$, $(X_1,\ldots,X_n)$ is measurable w.r.t. just $\mathcal{F}_n^X$ rather than the larger $\sigma$-algebra $\mathbb{F}$. If $Y$ is some function of $(X_1,\ldots,X_5)$, $Y$ will be measurable w.r.t. $\mathcal{F}_5^X$. If we know $(X_1,\ldots,X_5)$, we also know $Y$. This is why we interpret filtrations as streams of information. $\mathcal{F}_n^X$ is 'all the information generated by $X$ up until and including time $n$'.


$\sigma$-algebras and measure theory is required for technical reasons. Background spaces and implicit random variables are at the core of probabilistic thinking -- allowing us to focus on the aspects that matter. Filtrations are a technical equivalent to $\sigma$-algebras when we need to also study the time dynamics of a random system.

I hope this answers some of your questions.


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