---
title: "Stochastic process"
date: 2021-11-07
categories:
- Mathematics
- Stochastic analysis
- Measure theory
layout: "concept"
---

A **stochastic process** $X_t$ is a time-indexed
[random variable](/know/concept/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](/know/concept/sigma-algebra/) 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](/know/concept/conditional-expectation/).



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