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Formal Definition

Wikipedia gives the following definition of a process adapted to a filtration:

  • Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space;
  • $I$ be an index set with total order $\leq$ (often, $I$ is $\mathbb{N}$, $\mathbb{N_0}$, or $[0, T]$, or $[0, +\infty)$);
  • $\mathbb{F} = (\mathcal{F}_i)_{i \in I}$ be a filtration of the sigma algebra $\mathcal{F}$'
  • $(S, \Sigma)$ be a measurable space, the state space;
  • $X : I \times \Omega \mapsto S$ be a stochastic process. The process $X$ is said to be adapted to the filtration $(\mathcal{F}_i)_{i \in I}$ if the random variable $X_i : \Omega \mapsto S$ is a $(\mathcal{F}_i, \Sigma)$-measurable function for each $i \in I$.

Informal Definitions

Wikipedia gives there informal definitions, or intuitions, of what adaptive processes are like.

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[A]n adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future".

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An informal interpretation is that $X$ is adapted if and only if, for every realisation and every $n$, $X_n$ is known at time $n$.

Question

When I read the formal definition I get the impression that the filtrations are of key importance in providing some kind of constraint on what we can know. Filtrations have a defining property that $\mathcal{F}_k \subseteq \mathcal{F}_{\ell}$ whenever $k \leq \ell$. My speculation is the filtrations are imposing that if time $a$, then the function $X_{a+n}$ for $n>0$ is not measurable. So we're not saying that $X_a$ is independent of $X_{a+n}$, but rather that it won't measurable until the time is $\geq a+n$. At least, that is my attempt to understand it anyway.

How does the formal definition relate to the informal definitions (or intuitions, if you like)?

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    $\begingroup$ The binomial options pricing model I illustrate at stats.stackexchange.com/a/123754/919 provides a nice example of this that requires only a minimal amount of measure theory to understand. $\endgroup$
    – whuber
    Oct 24, 2022 at 16:22
  • $\begingroup$ One can say that the filtration at time $t$ is the set of events observed up to time $t$: it is possible to say at time $t$ whether such an event has been realized. $\endgroup$ Apr 30, 2023 at 10:17

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