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 $\mathcal{F}_t$:

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

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