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?
Best Answer
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
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.