I am interested in getting a firmer grip on filtrations in the context of Stochastic processes, and specifically the adaptation (is this the correct tense) of random variables to their (natural) filtration.
Consider a discrete time Stochastic Process $\{X_n\}_{n \in \mathbb{N}}$, and the natural filtration, $\mathscr{F}_n = \sigma (X_1, \cdots , X_n)$, where $\sigma$ here is a stand-in for the term "sigma-algebra generated by". We say then that $X_n$ is $\mathscr{F}_n$ adapted, if we have that $X_n$ is $\mathscr{F}_n$ is for every $n \in \mathbb{N}$.
Now, I know that if $X_n $ is $\mathscr{F}_n$ adapted, this is the same as saying:
$$
X_n = f(X_1, \cdots , X_n)
$$
where $f$ is a Borel-measurable, deterministic function. This follows from the approximation of a $\mathbb{F}$ measurable function by simple functions, for any $\sigma$-algebra $\mathscr{F}$.
This however, doesn't really seem to make much sense to me as a statement. It seems to be the case that any random variable $X$ is always $\sigma (X)$ measurable (by definition of $\sigma(X)$), and moreover $\sigma(X_n) \subset \sigma(X_1, \cdots , X_n)$. So I decided to look up processes with the property that each $X_n$ is $\mathscr{F}_{n-1}$ measurable, and these are called predictable, and with good, reason: $X_n$ is a deterministic function of $X_{1}, \cdots X_{n-1}$. This is however not what the natural filtration seems to want to capture. I have seen an interpretation for the natural filtration for the process being that the process cannot look forward into the future. How does this definition encapsulate that, and why is the condition not just a triviality?
Best Answer
The concept of filtration is required to give a formal definition of conditional expectation. In particular, conditional expectation is a random variable because of the sigma algebra of the conditioning variable.
The filtration is a way to encode the information contained in the history of a stochastic process. If a process is adapted then all information about the process is contained in the filtration.
Of couse, any process is trivially adapted to its natural filtration; more interesting is the case if a process $\{Y_n\}$ is adapted to the natural filtration of $X_n$. In this case, $$Y_n = f(X_1,\ldots, X_n).$$
The natural setting for filtrations, however, is martingale theory. A martingale is a process whose expected value in the future is the same as it's value now:
$$\mathbf E[M_{n+1}|\mathcal F_n] = M_n;$$
Hence, a martingale is a process whose increment is zero mean and independent from the history.