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.