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Let $(E, \mathcal{E})$ be a [[σ-algebra|measurable space]].
> [!definition] Definition. ([[stochastic process]])
>
> An indexed family of [[random variable|random variables]] $(X_{t}:\Omega \to E)_{t \in T}$ on a single [[probability|probability space]] $(\Omega, \mathcal{F}, \mathbb{P})$ is called a **stochastic process**.
>
$E$ is called the **state space** of $(X_{t})_{t \in T}$. An **increment** is the [[random variable]] corresponding to the difference of the $(X_{t})$ over two index values (assuming subtraction in $E$ is well-defined).
>
For fixed $\omega \in \Omega$, the map $T \to E$, $t \mapsto X_{t}(\omega)$, is called a **sample path** or **realization** of $\{ X_{t} \}_{t \in T}$ corresponding to $\omega$.
>
A process $\big(Y_{t}:(\Omega, \mathcal{F}, \mathbb{P}) \to E\big)_{t \in T}$ is called a **modification** of $(X_{t})$ if $\mathbb{P}[X_{t}=Y_{t}]=1$
for all $t \in T$.
> [!basicexample]
> - [[random walk]]
> - [[branching processes]]
> - [[Markov process]] (and [[Feller property]])
> - [[Gaussian process]]
> - [[Lévy process]] ([[Poisson process]], [[Brownian motion]])
>
^basic-example
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####
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#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```