Stochastic Process

A stochastic process is a collection of random variables indexed by time, location, or another parameter and defined on a common probability space. It describes both uncertainty at individual indices and dependence between different indices. In probability theory, a process can be viewed as a random trajectory rather than a single random value: an outcome determines one possible evolution of the system. Familiar examples include randomly fluctuating signals, particle trajectories, and counts of arriving events. (ocw.mit.edu)

Mathematical definition

Formally, a stochastic process is a family

on a probability space $(\Omega,\mathcal F,P)$, taking values in a measurable state space $S$. The set $T$ is the index set. For each fixed $t$, $X_t$ is a measurable map from $\Omega$ to $S$; measurability is specified using appropriate sigma-algebras. For each fixed outcome $\omega$, the function $t\mapsto X_t(\omega)$ is a sample path, also called a realization or trajectory. These two perspectives distinguish uncertainty across outcomes from variation along an individual path. (public.econ.duke.edu)

Time may be discrete, as with $T=\{0,1,2,\ldots\}$, or continuous, as with $T=[0,\infty)$. The state space may independently be discrete or continuous: an event count has integer-valued states even when time is continuous. Processes indexed by spatial coordinates, or by several parameters, are commonly called random fields. (ocw.mit.edu)

Distribution and dependence

For any finite selection $t_1,\ldots,t_n$, the joint distribution of

is a finite-dimensional distribution. The collection of these distributions specifies the process’s joint probabilistic behavior at finite sets of indices. Individual marginal distributions are insufficient: two processes can have identical distributions at each time but different temporal dependence. (public.econ.duke.edu)

The Kolmogorov extension theorem establishes that, under suitable state-space assumptions, a consistent family of finite-dimensional distributions can be realized by a stochastic process. Consistency requires agreement when coordinates are reordered or omitted. This construction does not by itself guarantee continuous sample paths; path regularity requires additional analysis. (public.econ.duke.edu)

For a real-valued process with finite second moments, useful descriptors are its mean and covariance functions:

The mean uses expectation, while covariance describes second-order dependence. In particular, $C(t,t)$ is the variance at time $t$. These functions generally do not determine the entire process distribution. (sia.mit.edu)

Stationarity and increments

A strictly stationary process has finite-dimensional distributions unchanged by a common shift of all time indices, whenever the shifted indices remain in its domain. Weak stationarity, or covariance stationarity, instead requires finite second moments, a constant mean, and covariance depending only on the time difference. Strict stationarity implies weak stationarity when the required moments exist; the converse generally fails. (sia.mit.edu)

An increment is a difference $X_t-X_s$. Stationary increments means that its distribution depends only on $t-s$, not on the starting time. Independent increments means increments over nonoverlapping time intervals have statistical independence. Neither property should be confused with stationarity of the process itself: standard Brownian motion has stationary increments, but its variance grows with time. (ocw.mit.edu)

Important examples

A random walk is formed by accumulating random steps:

For a simple symmetric walk, the steps independently equal $+1$ or $-1$, each with probability one-half. Despite independent steps, successive positions are dependent because they share accumulated increments. (ocw.mit.edu)

A Markov chain evolves between states according to transition probabilities. Its defining Markov property states that, conditional on the present state, the future does not additionally depend on the past. For a finite, time-homogeneous chain, these probabilities can be represented by a transition matrix. Markov processes may also evolve in continuous time. (ocw.mit.edu)

A homogeneous Poisson process $N_t$ counts events occurring at rate $\lambda>0$. It starts at zero, has independent stationary increments, and satisfies

The increment therefore follows a Poisson distribution, and paths increase through unit jumps. Such processes model random arrivals under specific assumptions about their timing. (ocw.mit.edu)

A standard Wiener process $W_t$, the mathematical model of Brownian motion, starts at zero, has continuous paths almost surely, and has independent increments satisfying

Its increments follow a normal distribution. Its mean is zero and its covariance is $\min(s,t)$. (public.econ.duke.edu)

A Gaussian process has a multivariate normal distribution at every finite selection of indices. Unlike a general process, its finite-dimensional distributions are completely determined by its mean and covariance functions. Gaussian processes also provide models for unknown functions in regression and spatial prediction. (stat.berkeley.edu)

Information and martingales

A filtration $(\mathcal F_t)$ is an increasing family of sigma-algebras representing information available over time. A process is adapted when its value at time $t$ is measurable with respect to $\mathcal F_t$. An integrable adapted process $M_t$ is a martingale if

This conditional expectation identity formalizes the absence of predictable changes in its conditional mean. Martingales need not have independent increments; their definition depends on the specified information structure. (ocw.mit.edu)