Categories: Mathematics, Measure theory, Stochastic analysis.

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

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

The **filtration** \(\{ \mathcal{F}_t : t \ge 0 \}\) is a time-indexed set of \(\sigma\)-algebras on \(\Omega\), which contains at least all the information generated by \(X_t\) up to the current time \(t\), and is a subset of \(\mathcal{F}_t\):

\[\begin{aligned} \mathcal{F} \supseteq \mathcal{F}_t \supseteq \sigma(X_s : 0 \le s \le t) \end{aligned}\]

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

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

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

© Marcus R.A. Newman, a.k.a. "Prefetch".
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