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.