A stochastic process is a time-indexed random variable, , i.e. a set of (usually correlated) random variables, each labelled with a unique timestamp .
Whereas “ordinary” random variables are defined on a probability space , stochastic process are defined on a filtered probability space . As before, is the sample space, is the event space, and is the probability measure.
The filtration is a time-indexed set of -algebras on , which contains at least all the information generated by up to the current time , and is a subset of :
In other words, is the “accumulated” -algebra of all information extractable from , and hence grows with time: for . Given , all values for can be computed, i.e. if you know , then the present and past of can be reconstructed.
Given any filtration , a stochastic process is said to be ”-adapted” if ’s own filtration , meaning contains enough information to determine the current and past values of . Clearly, is always adapted to its own filtration.
Filtration and their adaptations are very useful for working with stochastic processes, most notably for calculating conditional expectations.
- U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.