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+---
+title: "Stochastic process"
+firstLetter: "S"
+publishDate: 2021-11-07
+categories:
+- Mathematics
+
+date: 2021-11-07T18:45:42+01:00
+draft: false
+markup: pandoc
+---
+
+# Stochastic process
+
+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.