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I would like to gain some intuition or context for the study of Martingales. What I have seen so far seemed to be motivated more by measure theory that probability theory.

In the context of studying Martingales I came upon Filtrations. Let $(\Omega, \mathcal F,\mathbb P)$ be a probability space and $(\mathcal F_n)_n$ a filtration of $\mathcal F$. This means for all $n\in\mathbb N: \mathcal F_{n}\subseteq\mathcal F_{n+1}$. The canonical filtration is $\mathcal C_n = \sigma(X_1,\dots, X_n)$, where $X_n$ are $\mathcal C_n$-measurable functions. $(X_n)_n$ is called a Martingale, if $\mathbb E[X_{n+1}|\mathcal F_n]=X_n$.

How can I interpret a filtration $(\mathcal F_n)_n$ from conditional expectation context and why do we use the canonical filtration $(\mathcal C_n)_n$ aside from measure theoretic considerations?

I believe stopping times come from the study of stochastic processes but the examples I have seen have all been measure theoretical. $N$ is a stopping time, if $\forall n\in\mathbb N:\{N>n\}\in\mathcal F_n$. I know that $n_o\in\mathbb N_o$ is a stopping time and that if $N_1, N_2$ are stopping times that $N_1\land N_2$ and $N_1\lor N_2$ are stopping times. The term stopping time seems like it should be more tangible.

What is a practical interpretation of a stopping time and what are examples of proofs of stopping times that go beyond properties of $\sigma$-algebras?

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  • $\begingroup$ What do you mean by "examples of proofs of stopping times"? $\endgroup$ Jan 16, 2023 at 16:42
  • $\begingroup$ @user6247850 the proofs that show $n_0, N_1\land N_2$ and $N_1\lor N_2$ I have seen have all been based on manipulating the set and showing that all sets belong to the $\sigma$-algebra and therefore the set we started out with. $\endgroup$
    – HelloWorld
    Jan 16, 2023 at 17:31
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    $\begingroup$ Math formatting is not intended for italicizing normal text. As you can see, it produces the wrong spacing for that. You can get proper italic text by enclosing it in asterisks *like this*. $\endgroup$
    – joriki
    Jan 16, 2023 at 18:20

1 Answer 1

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The standard interpretation of a filtration $(\mathcal F_n)$ is that it represents information accumulated over time, i.e. $\mathcal F_n$ represents the information available at time $n$. In other words, if a set of potential outcomes $A \in \mathcal F_n$, then at time $n$ we know whether or not we are in the set $A$. The reason for modeling this with a $\sigma$-algebra is that the properties of a $\sigma$-algebra correspond quite naturally to properties we would like for an information set to have. Recall that $\mathcal F$ is a $\sigma$-algebra if

  1. $\Omega$ is in $\mathcal F$ (We always know that some outcome has happened)
  2. If $A \in \mathcal F$, then $A^c \in \mathcal F$ (If we know whether or not we are in $A$, then we also know whether or not we aren't in $A$)
  3. If $(A_n)_{n \in \mathbb{N}} \in \mathcal F$, then $\bigcup_{n \in \mathbb{N}} A_n \in \mathcal F$ (If we know whether or not $A_n$ happened for all $n$, then we know if any one of the $A_n$s happened).

The canonical filtration is simply the information we get from observing a process, i.e. $\mathcal C_n$ represents the information we gain from observing a process up until time $n$.

As for stopping times, an equivalent definition is that $N$ is a stopping time iff $\{N \le n\} \in \mathcal F_n$ for all $n$. Often we think of $N$ as the time that something happens, e.g. the first time that $X_n$ reaches a certain value. Then the property that $\{N \le n\} \in \mathcal F_n$ means that, at time $n$, we know whether or not that event has happened.

One important property of this definition is that it makes sense to stop a process at a stopping time, and doing so doesn't require any knowledge of the future. Stopping a process means to define the process $X_n^N := X_{\min(n,N)}$. If $(X_n)$ is adapted (i.e. the value of $X_n$ is known at time $n$) and $N$ is a stopping time, then $(X_n^N)$ is also adapted.

The proofs that a random variable is a stopping time will pretty much always involve $\sigma$-algebras because that is how they defined. However, you can often guess whether or not a random variable is a stopping time without thinking about the filtrations. For example, the first time that $X_n$ reaches a value $K$ is a stopping time because at time $n$ we know whether or not $X_n$ has ever reached $K$. On the other hand, the last time that $X_n$ reaches $K$ is not a stopping time because at time $n$ we don't know whether or not $X_n$ will reach the value $K$ again later.

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  • $\begingroup$ Thank you for taking the time to write such an elaborate answer! (I believe you forgot to write A_n in the 3rd property of $\sigma$-algebras.) $\endgroup$
    – HelloWorld
    Jan 16, 2023 at 20:09
  • $\begingroup$ @HelloWorld Indeed I did! Thank you for catching that! $\endgroup$ Jan 16, 2023 at 20:13
  • $\begingroup$ @user6247850, and what is the intuition for the Martingale then? $\endgroup$
    – edamondo
    Feb 20 at 15:39
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    $\begingroup$ @edamondo A martingale is a process whose value is known at time $n$, and on average neither increases nor decreases over time. It doesn't stay constant, but it doesn't tend upwards more than downwards over any amount of time periods. $\endgroup$ Feb 20 at 21:22

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