Categories: Mathematics, Measure theory, Stochastic analysis.

Stochastic process

A stochastic process XtX_t is a time-indexed random variable, {Xt:t>0}\{ X_t : t > 0 \}, i.e. a set of (usually correlated) random variables, each labelled with a unique timestamp tt.

Whereas “ordinary” random variables are defined on a probability space (Ω,F,P)(\Omega, \mathcal{F}, P), stochastic process are defined on a filtered probability space (Ω,F,{Ft},P)(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P). As before, Ω\Omega is the sample space, F\mathcal{F} is the event space, and PP is the probability measure.

The filtration {Ft:t0}\{ \mathcal{F}_t : t \ge 0 \} is a time-indexed set of σ\sigma-algebras on Ω\Omega, which contains at least all the information generated by XtX_t up to the current time tt, and is a subset of Ft\mathcal{F}_t:

FFtσ(Xs:0st)\begin{aligned} \mathcal{F} \supseteq \mathcal{F}_t \supseteq \sigma(X_s : 0 \le s \le t) \end{aligned}

In other words, Ft\mathcal{F}_t is the “accumulated” σ\sigma-algebra of all information extractable from XtX_t, and hence grows with time: FsFt\mathcal{F}_s \subseteq \mathcal{F}_t for s<ts < t. Given Ft\mathcal{F}_t, all values XsX_s for sts \le t can be computed, i.e. if you know Ft\mathcal{F}_t, then the present and past of XtX_t can be reconstructed.

Given any filtration Ht\mathcal{H}_t, a stochastic process XtX_t is said to be Ht\mathcal{H}_t-adapted” if XtX_t’s own filtration σ(Xs:0st)Ht\sigma(X_s : 0 \le s \le t) \subseteq \mathcal{H}_t, meaning Ht\mathcal{H}_t contains enough information to determine the current and past values of XtX_t. Clearly, XtX_t 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.


  1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.