Let $(\Omega,\mathcal{F},{P})$ be a probability space. A stochastic process is a collection $$\lbrace X_t \mid t\in T \rbrace$$ of random variables $X_t$ defined on $(\Omega,\mathcal{F},{P})$ where $T$ is a set, called the index set of the process $\lbrace X_t \mid t\in T \rbrace$ $T$ 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 $\lbrace X_t \mid t\in T \rbrace$ 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 $\omega\in\Omega$ \begin{equation*} X\colon T\times\Omega\rightarrow S,\ (t,\omega)\mapsto X_t(\omega). \end{equation*}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 \begin{equation*} T\rightarrow S,\ t\mapsto X_t(\omega). \end{equation*}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. a random walk, as well as its limiting case, a Brownian motion, or a Wiener process
2. Poisson process
3. Markov process; a Markov chain is a Markov process whose state space is discrete
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.