Adapted process

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In the study of stochastic processes, an adapted process (or non-anticipating process) is one that cannot "see into the future". It is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process.

[edit] Definition

Let

$(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space;
$I$ be an index set with a total order $\leq$ and a least element $0 \in I$ (often, $I = \mathbb{N}_{0}$ , $[0, T]$ or $[0, \infty)$ );
$\{ \mathcal{F}_{i} | i \in I \}$ be a filtration of the sigma algebra $\mathcal{F}$ ;
$(\mathbb{X}, \mathcal{A})$ be a measurable space, the state space;
$X : I \times \Omega \to \mathbb{X}$ be a stochastic process.

The process $X$ is said to be adapted to the filtration if $X_{i} : \Omega \to \mathbb{X}$ is a $(\mathcal{F}_{i}, \mathcal{A})$ -measurable function for each $i \in I$ .

[edit] Examples

Consider a stochastic process $X : [0, T] \times \Omega \to \mathbb{R}$ , and equip the real line $\mathbb{R}$ with its usual Borel sigma algebra generated by the open sets.

If we take the natural filtration $\mathcal{F}_{*}^{X}$ , where $\mathcal{F}_{t}^{X}$ is the sigma algebra generated by the pre-images $X_{s}^{-1} (B)$ for Borel sets $B \subseteq \mathbb{R}$ and times $0 \leq s \leq t$ , then $X$ is automatically $\mathcal{F}_{*}^{X}$ -adapted. Intuitively, the filtration $\mathcal{F}_{t}^{X}$ contains "total information" about the behaviour of $X$ up to time $t$ .

This offers a simple example of a non-adapted process: set $\mathcal{F}_{t}$ to be the trivial sigma algebra $\{ \emptyset, \Omega \}$ for times $0 \leq t < T / 2$ , and $\mathcal{F}_{t} = \mathcal{F}_{t}^{X}$ for times $T /2 \leq t \leq T$ . Since the only way that a function can be measurable with respect to the trivial sigma algebra is to be constant, any process $X$ that is non-constant on $[0, T / 2]$ will fail to be adapted. The non-constant nature of such a process "uses information" from the more refined "future" sigma algebras $\mathcal{F}_{t}$ , $T /2 \leq t \leq T$ .