# stochastic process

Let $(\Omega,\mathcal{F},\textbf{P})$ be a probability space. A stochastic process is a collection

 $\{X_{t}\mid t\in T\}$

of random variables $X_{t}$ defined on $(\Omega,\mathcal{F},\textbf{P})$, where $T$ is a set, called the index set of the process $\{X_{t}\mid t\in T\}$. $T$ is usually (but not always) a subset of $\mathbb{R}$. $X$ is sometimes known as a random function.

Given any $t$, the possible values of $X_{t}$ are called the states of the process at $t$. The set of all states (for all $t$) of a stochastic process is called its state space.

If $T$ is discrete, then the stochastic process is a discrete-time process. If $T$ is an interval of $\mathbb{R}$, then $\{X_{t}\mid t\in T\}$ is a continuous-time process. If $T$ can be linearly ordered, then $t$ is also known as the time.

A stochastic process $X$ with state space $S$ can be thought of in either of following three ways.

• As a collection of random variables, $X_{t}$, for each $t$ in the index set $T$.

• As a function in two variables $t\in T$ and $\omega\in\Omega$,

 $X\colon T\times\Omega\rightarrow S,\ (t,\omega)\mapsto X_{t}(\omega).$

The process is said to be measurable, or, jointly measurable if it is $\mathcal{B}(T)\otimes\mathcal{F}/\mathcal{B}(S)$-measurable. Here, $\mathcal{B}(T)$ and $\mathcal{B}(S)$ are the Borel $\sigma$-algebras on $T$ and $S$ respectively.

• In terms of the sample paths. Each $\omega\in\Omega$ maps to a function

 $T\rightarrow S,\ t\mapsto X_{t}(\omega).$

Many common examples of stochastic processes have sample paths which are either continuous or cadlag.

Examples. The following list is some of the most common and important stochastic processes:

1. 1.

a random walk, as well as its limiting case, a Brownian motion, or a Wiener process

2. 2.
3. 3.

Markov process; a Markov chain is a Markov process whose state space is discrete

4. 4.

renewal process

Remarks.

• Sometimes, a stochastic process is also called a random process, although a stochastic process is generally linked to any “time” dependent process. In a random process, the index set may not be linearly ordered, as in the case of a random field, where the index set may be, for example, the unit sphere $S^{2}\subseteq\mathbb{R}^{3}$.

• In statistics, a stochastic process is often known as a time series, where the index set is a finite (or at most countable) ordered sequence of real numbers.

 Title stochastic process Canonical name StochasticProcess Date of creation 2013-03-22 14:39:10 Last modified on 2013-03-22 14:39:10 Owner gel (22282) Last modified by gel (22282) Numerical id 14 Author gel (22282) Entry type Definition Classification msc 60G05 Classification msc 60G60 Synonym random process Related topic DistributionsOfAStochasticProcess Defines discrete-time process Defines continuous-time process Defines state Defines time series Defines state space Defines random function Defines jointly measurable