Stochastic Process

Definition

Let

• $X(t)$ : random variable indexed by parameter $t$
• $T$ : index set (typically time)

A Stochastic Process $\{X(t),t\in T\}$ is a collection of random variables $X(t)$ indexed by $t$.

Interpretation

“Stochastic” means probabilistic, “Process” means depends on time. A stochastic process is a sequence (implies ordering) of random variables indexed by time.

Classification

A stochastic process is classified by its state space ($S$) and index set ($T$):

Type	Index Set $T$	State Space $S$	Example
Discrete-time, Discrete	$\{0,1,2,...\}$	$\{0,1,2,...\}$	Markov Chain
Continuous-time, Discrete	$[0,\infty )$	$\{0,1,2,...\}$	Poisson Process
Discrete-time, Continuous	$\{0,1,2,...\}$	$\mathbb{R}$	—
Continuous-time, Continuous	$[0,\infty )$	$\mathbb{R}$	Brownian Motion

Related

• State
• Markov Property
• Markov Process