Stochastic processes model systems that change randomly
A stochastic process is a family of random variables on a probability space, usually indexed by time. From bacterial growth to thermal noise to gas molecules, it models systems that seem to wander at random, and it underwrites modern finance from Bachelier onward.
In probability theory, a stochastic process is typically a collection of random variables on a shared probability space, each tied to an index often interpreted as time. Values need not be numbers—they can be vectors or other objects. When indexed by integers or real intervals the terms random process and stochastic process are interchangeable; when indexed by the plane or higher dimensions, the collection is usually called a random field.
Each random variable maps to a state space—the integers, the real line, or n-dimensional Euclidean space. An increment measures change between two index points. A single outcome trajectory is called a sample function or realization. Processes classify by index cardinality (discrete vs continuous time) and state type (integer-valued vs real-valued). Discrete-time processes are generally easier because continuous-time index sets are uncountable.
Two cornerstone examples recur throughout the field. The Wiener process—Brownian motion—has stationary, normally distributed independent increments; Norbert Wiener proved its existence, though Louis Bachelier applied it to Paris Bourse prices and it models physical Brownian movement. The Poisson process counts events in fixed intervals; A. K. Erlang modelled telephone traffic with it. Both were invented independently in multiple settings.
Beyond those classics lie many specialised families studied for their mathematical properties and real-world fit—path-dependent walks, memoryless Markov chains, drifted Lévy jumps, correlated Gaussian fields, and branching populations. The Bernoulli process—repeated independent coin flips taking values zero or one—is among the simplest. A symmetric simple random walk on integers moves up or down with equal probability. Wiener sample paths are continuous everywhere yet nowhere differentiable, arising as limits of scaled random walks via Donsker's invariance principle.
Source: Stochastic process
