# 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.

## One

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

## Two

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)?

• 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.
– whuber
Oct 24, 2022 at 16:22
• 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. Apr 30, 2023 at 10:17