>[!info] Definition >**Collection of [[Random Variable|random variables]]** indexed by some [[Set|set]] (index set), usually time $t \in T$ **on a common [[Probability Space|probability space]]** $(\Omega, \mathcal{A}, \mu)$ with [[Sigma-Algebra|sample-space]] $\Omega$, [[Sigma-Algebra|Sigma-Algebra]] $\mathcal{A}$ and [[Measure|probability measure]] $\mu$. > >The collection of these random variables is denoted $\{X(t)\colon t \in T \}.$ Index set can be discrete timesteps or subsets of $\mathbb{R}$. --- #### Sampling A **single outcome** of a stochastic process is called a _sample function, a sample path_, a *trajectory* , a *path function* or, a _realization_. It is formed by **taking a single value of each [[Random Variable|random variable]] of the stochastic process**. More precisely, if $\{𝑋(𝑡):𝑡∈𝑇\}$ is a stochastic process, then for any point $𝜔∈Ω$, the [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|mapping]] $𝑋(⋅,𝜔):𝑇→𝑆,$is a sample function or sample path of the stochastic process $\{𝑋(𝑡,𝜔):𝑡∈𝑇\}$. --- #### Classification and Examples - [[Stationary Process]] - Statistical properties stay constant in time - **[[Bernoulli Process]]** - No dependence between the random variables - [[Wiener Process or Brownian Motion]], Brownian Motion Process - [[Poisson Process]] - ... - **[[Markov Chain and Kernel|Markov Process]]** - Dependence between the random variables - [[Ornstein-Uhlenbeck Process]] - [[Itô Diffusion]] (under additional assumptions) - Only carries forst and second moment information, higher moments require e.g. Levy Process - [[Lévy Process]] - ... >[!brainwaves] Intuition >Time flows forward for [[Stochastic Processes|stochastic processes]] (unless bernoulli), because each value is dependent on the previous realization, while for deterministic [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|functions]] everything is known