A single [random variable](/page/Random%20Variable) describes one uncertain observation. Many probabilistic questions are not about one observation, but about an evolving record: the position of a particle after each collision, the wealth of a gambler after each bet, the number of calls arriving before time $t$, or the price of an asset as information accumulates. In all these cases, the object of interest is not just the marginal law at one time. The order of the observations, their dependence, and the information revealed along the way are part of the mathematics.

The first warning is that the distribution at a fixed time can hide the event we care about. A fair gambler might have final wealth $0$ after ten plays, but the path could have visited bankruptcy earlier. A diffusion might have the same one-time Gaussian law as another process, while having different correlations and different hitting behaviour. A stochastic process is the language that keeps the whole random evolution visible.

[example: A Fair Random Walk Remembers Its Path] Let $Y_1,Y_2,\dots$ be i.i.d. random variables with $\mathbb P(Y_i=1)=\mathbb P(Y_i=-1)=1/2$, and define $S_0=0$ and $S_n=\sum_{i=1}^n Y_i$. At time $2$, \begin{align*} S_2=Y_1+Y_2. \end{align*} The four possible pairs $(Y_1,Y_2)$ are $(1,1)$, $(1,-1)$, $(-1,1)$, and $(-1,-1)$. By independence, each has probability $(1/2)(1/2)=1/4$. Therefore $S_2$ takes the values $2,0,-2$, and the value $0$ occurs exactly for the two pairs $(1,-1)$ and $(-1,1)$, so \begin{align*} \mathbb P(S_2=0)=\mathbb P((Y_1,Y_2)=(1,-1))+\mathbb P((Y_1,Y_2)=(-1,1))=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}. \end{align*} The terminal event $S_2=1$ is impossible, because $Y_1+Y_2$ is always one of $-2,0,2$; hence $\mathbb P(S_2=1)=0$. By contrast, the event that the walk has reached level $1$ by time $2$ is \begin{align*} \left\{\max\{S_0,S_1,S_2\}\ge1\right\}. \end{align*} Since $S_0=0$ and $S_1=Y_1$, if $Y_1=1$ then the maximum is already at least $1$. If $Y_1=-1$, then $S_1=-1$ and $S_2=-1+Y_2$ is either $0$ or $-2$, so the maximum among $S_0,S_1,S_2$ is $0$ and the walk has not hit $1$. Thus \begin{align*} \left\{\max\{S_0,S_1,S_2\}\ge1\right\}=\{Y_1=1\}, \end{align*} and therefore \begin{align*} \mathbb P\left(\max\{S_0,S_1,S_2\}\ge1\right)=\mathbb P(Y_1=1)=\frac{1}{2}. \end{align*} For example, the path with $Y_1=-1$ and $Y_2=1$ has $S_0=0$, $S_1=-1$, and $S_2=0$, so it ends at the same terminal value as the path $(1,-1)$ but never reaches $1$. This is why the random evolution must be treated as the whole family $(S_n)_{n\ge0}$, not just through the law of one terminal value. [/example]

The random walk already contains most of the conceptual ingredients of the subject. There is an index set, a state space, a [probability space](/page/Probability%20Space), a family of random variables, and a growing body of information. Changing any one of these changes the theory: discrete time behaves differently from continuous time, countable state spaces behave differently from path spaces, and processes observed with a filtration support questions that cannot even be stated for an isolated random variable.

## Definition

The definition packages the minimal data needed to talk about random evolution. The index set records the parameter along which the process evolves; it is often time, but the same language also handles random fields indexed by space, stochastic sequences indexed by $\mathbb N$, and families indexed by arbitrary sets.

[definition: Stochastic Process] Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, let $(E, \mathcal E)$ be a measurable space, and let $T$ be a non-empty index set. A stochastic process with index set $T$ and state space $(E,\mathcal E)$ is a family of random variables \begin{align*} X_t : (\Omega,\mathcal F) \to (E,\mathcal E), \qquad t \in T. \end{align*} It is denoted by $(X_t)_{t \in T}$. [/definition]

When $T=\mathbb N$ or $T=\{0,1,2,\dots\}$, the process is a discrete-time process. When $T=[0,\infty)$ or an interval in $\mathbb R$, it is a continuous-time process. The word continuous in continuous-time refers to the index set, not to the sample paths. The formal definition is deliberately lean; the next section separates the choices hidden inside it: time, state, and path behaviour.

## Time, State, and Sample Paths

### Discrete Time

The simplest classification asks what kind of time and what kind of state space are being used. This distinction is not cosmetic. A [Markov chain](/page/Markov%20Chain) on a [countable set](/page/Countable%20Set) is governed by transition probabilities between states; [Brownian motion](/page/Brownian%20Motion) in continuous time is governed by Gaussian increments and path regularity; a random field indexed by $\mathbb R^n$ is shaped by spatial dependence rather than temporal order.

A discrete-time process is a random sequence. Because the index set is countable, many events involving all times can be built from countable unions and intersections. This makes measurability less fragile than in continuous time.

[example: Bernoulli Process as Repeated Observation] Let $X_1,X_2,\dots$ be i.i.d. random variables with $X_n\sim\operatorname{Ber}(p)$ for some $p\in[0,1]$, so each $X_n$ takes values in $\{0,1\}$ with \begin{align*} \mathbb P(X_n=1)=p \quad \text{and} \quad \mathbb P(X_n=0)=1-p. \end{align*} The process $(X_n)_{n\in\mathbb N}$ records the outcome of each trial separately: $X_n=1$ means success at time $n$, and $X_n=0$ means failure at time $n$. Define the accumulated process by \begin{align*} S_n=\sum_{i=1}^n X_i. \end{align*} For example, \begin{align*} S_1=X_1. \end{align*} Also, \begin{align*} S_2=X_1+X_2. \end{align*} Since each summand is either $0$ or $1$, the value of $S_2$ counts how many of the first two trials were successes. Explicitly, if $(X_1,X_2)=(0,0)$ then \begin{align*} S_2=0+0=0. \end{align*} If $(X_1,X_2)=(1,0)$ then \begin{align*} S_2=1+0=1. \end{align*} If $(X_1,X_2)=(0,1)$ then \begin{align*} S_2=0+1=1. \end{align*} If $(X_1,X_2)=(1,1)$ then \begin{align*} S_2=1+1=2. \end{align*} Thus $S_n$ counts the number of successes among the first $n$ observations. The original Bernoulli process keeps the individual success-failure record, while the partial-sum process keeps the accumulated count up to each time. [/example]

### Continuous Time

Continuous time introduces a new tension: there are uncountably many random variables $X_t$. Pointwise measurability of each $X_t$ does not automatically guarantee good measurability of the map $(t,\omega)\mapsto X_t(\omega)$, and path regularity becomes a theorem or an assumption rather than an afterthought. To integrate a process over time or treat it as a random function of two variables, we need a measurability condition on the joint map.

[definition: Jointly Measurable Process] Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $T$ be a measurable space with sigma-algebra $\mathcal T$, and let $(E,\mathcal E)$ be a measurable space. A stochastic process $(X_t)_{t\in T}$ on $(\Omega,\mathcal F,\mathbb P)$ is jointly measurable if the map $X:T\times\Omega\to E$ defined by \begin{align*} X(t,\omega)=X_t(\omega) \end{align*} is $(\mathcal T\otimes\mathcal F,\mathcal E)$-measurable. [/definition]

Joint measurability is the entry ticket for applying Fubini-type arguments to random evolutions. It is stronger than asking each $X_t$ to be measurable as a random variable, and the distinction matters in continuous-time theory.

When $E$ has topology, one may ask for paths that are continuous, right-continuous, or have left limits. These conditions let us treat sample paths with analytic tools, so they must be named separately from the underlying probabilistic definition. There are two levels of path regularity: a property may hold for every $\omega$, or it may hold outside a single null set. Most continuous-time probability uses the almost-sure version, while the stricter version is useful when a process has already been modified on a null set.

[definition: Continuous Sample Paths] Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $T\subset\mathbb R$, let $(E,d)$ be a [metric space](/page/Metric%20Space) with Borel sigma-algebra $\mathcal B(E)$, and let $(X_t)_{t\in T}$ be an $E$-valued stochastic process on $(\Omega,\mathcal F,\mathbb P)$. The process has continuous sample paths if for every $\omega\in\Omega$, the function $t\mapsto X_t(\omega)$ from $T$ to $E$ is continuous. [/definition]

This condition is often too strong if demanded for every $\omega$, because stochastic processes are usually identified up to null sets. In standard process theory, the phrase "has continuous sample paths" often means the following almost-sure condition; whenever the distinction matters, the page will say which version is intended.