Measure Theory Q's wrt Stochastic Processes

[X89codered89X]

Hello there.

The stochastic calc book I'm going through ( and others I've seen ) uses the phrase "$\mathscr{F}-measurable$" random variable $Y$ in the section on measure theory. What does this mean? I'm aware that $\mathscr{F}$ is a $\sigma-field$ over all possible values for the possible values of $Y$, but the language throws me.

A tougher thing that confuses is me is in the stochastic context it defines a filtration as the increasing family of sigma algebras indexed by $t$, like this $\mathscr{F}_t = \sigma{\lbrace X_s; s \leq t \rbrace }$ for a stochastic process $X_t$. I take it $\sigma$ is an operator here? I realize this is a theoretical construct, but is it possible to see an example (not given in the book). From what I understand $\mathscr{F}_s \subset \mathscr{F}_t \forall s < t$, but again what do either of these look like. Or a simpler, yet nontrivial example of a filtration? Maybe it would be easier in discrete stochastic process case. I can understand how most of this works simply by the fact that no definitions seem conflict with each other and everything is logically consistent but I'm just wondering if there is some more practical ground here ...

 

[jbunniii]

Hello there.

The stochastic calc book I'm going through ( and others I've seen ) uses the phrase "$\mathscr{F}-measurable$" random variable $Y$ in the section on measure theory. What does this mean? I'm aware that $\mathscr{F}$ is a $\sigma-field$ over all possible values for the possible values of $Y$, but the language throws me.

I'm very surprised if they use that terminology without defining it. A measurable function ("random variable" in probability theory) is simply a function $f$ for which $f^{-1}(A)$ is measurable whenever $A$ is measurable.

Your random variable $Y$ is defined on some space, let's call it $S$, and it is equipped with a sigma algebra $\mathcal{F}$. You didn't specify the target space, so let's call it $T$ and assume it is also a measure space with some sigma algebra $\mathcal{G}$. Then $Y : S \rightarrow T$, and "$Y$ is measurable" means that for any set $A \in \mathcal{G}$, we have $f^{-1}(A) \in \mathcal{F}$.

 

[camillio]

For an example, let's have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a stochastic process (coin tossing) $X$ adapted to a filtration $(\mathcal{F}_t)_{t=1,2}, \mathcal{F}_t\subset \mathcal{F}$.

Then, in the beginning, one knows nothing but possible results,
$\mathcal{F}_0 = \{\emptyset, \Omega\}$ where $\Omega=\{[HH], [HT], [TH], [TT]\}$.

After the first toss, the division becomes finer, one can say more about the history,
$\mathcal{F}_1 = \{\emptyset, \Omega, [HH, HT], [TH, TT]\} \supset \mathcal{F}_0$,

After the second toss,
$\mathcal{F}_2 = \{\emptyset, \Omega, [HH, HT], [TH, TT], [HH], [HT], [TH], [TT]\} \supset{F}_1$.

You can exchange 2 tosses of a coin by $m$ tosses of a dice, $\Omega=\{1,...,6\}^m$, or even to move to continuous cases.

 

[X89codered89X]

Both of your answers were incredibly helpful. Thank you.

 

[blue_raver22]

Hello there,

Firstly, let's define some terms to better understand the language used in stochastic processes and measure theory.

A stochastic process is a collection of random variables indexed by time, such as X_t where t is a time parameter. These random variables can represent, for example, the stock price of a company at different points in time.

A sigma-field or sigma-algebra, denoted by \mathscr{F}, is a collection of subsets of a sample space (a set of all possible outcomes) that satisfies certain properties. It is used to define the probability of an event occurring in a given sample space.

Now, to answer your question about \mathscr{F}-measurable random variables, this simply means that the random variable Y is measurable with respect to the sigma-field \mathscr{F}. In other words, the values of Y can be assigned probabilities based on the subsets of the sample space in \mathscr{F}.

Moving on to the definition of a filtration, it is indeed an increasing family of sigma-algebras indexed by time. This means that as time progresses, the sigma-algebras \mathscr{F}_t become larger and include more subsets of the sample space. This is because as more information is revealed over time, the probability of certain events occurring may change.

An example of a filtration in the context of a stock price could be the information available at different points in time, such as the company's financial reports, news articles, or market trends. As more information becomes available, the sigma-algebras become larger and more subsets are included in the probability calculations.

In the discrete case, a filtration can be represented by a sequence of sigma-algebras, where each one contains all the information available at a specific time point. For example, in a coin toss scenario, the filtration at time t could include all the possible outcomes up until that point (e.g. heads or tails), while at time t+1 it would include all the possible outcomes up until that point plus the new outcome of the coin toss.

I hope this helps clarify the concepts of \mathscr{F}-measurable random variables and filtrations in the context of stochastic processes. It is important to understand these concepts in order to properly analyze and model stochastic processes in various fields of science and mathematics.