Adapted process

In the study of , an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". 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. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process.

Let

The process ${\displaystyle X}$ is said to be adapted to the filtration ${\displaystyle \left({\mathcal {F}}_{i}\right)_{i\in I}}$ if the random variable ${\displaystyle X_{i}:\Omega \to S}$ is a ${\displaystyle ({\mathcal {F}}_{i},\Sigma )}$-measurable function for each ${\displaystyle i\in I}$.

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