Categories: Mathematics, Measure theory, Stochastic analysis.

Stochastic process

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.

References

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

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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