stochastic process

Let $(\mathrm{\Omega },ℱ,\text{𝐏})$ be a probability space. Astochastic process is a collection

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

of random variables $X_t$ defined on$(\mathrm{\Omega },ℱ,\text{𝐏})$, where $T$ is a set, called theindex set of the process $\{X_t\mid t\in T\}$. $T$ isusually (but not always) a subset of $ℝ$.$X$ is sometimes known as a random function.

Given any $t$, the possible values of $X_t$ are called thestates 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 adiscrete-time process. If $T$ is an interval of $ℝ$,then $\{X_t\mid t\in T\}$ is a continuous-timeprocess. If $T$ can be linearly ordered, then $t$ is also known asthe 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 $\omega \in \mathrm{\Omega }$,

  $$X:T\times \mathrm{\Omega }\to S,(t,\omega )\mapsto X_t(\omega ).$$

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

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

  $$T\to S,t\mapsto X_t(\omega ).$$

  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 andimportant 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 arandom process, although a stochastic process is generallylinked to any “time” dependent process. In a random process, theindex set may not be linearly ordered, as in the case of a randomfield, where the index set may be, for example, the unit sphere $S^2\subseteq {ℝ}^3$.

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