Stochastic process
A stochastic process, also called a random process, is a mathematical object that represents the evolution of a system whose state is subject to uncertainty. Formally, it is a family of random variables indexed by a parameter that commonly represents time, although spatial position or another ordered quantity can serve the same role. Stochastic processes provide a unified framework for temporal dependence, random motion, fluctuating signals, and systems whose future behavior is not completely determined by their present description.
A process is written as
[ X={X_t:t\in T}, ]
where (T) is the index set and every (X_t) takes values in a measurable state space (S). For a fixed (t), (X_t) is a random variable. For a fixed outcome (\omega) in the underlying sample space, the mapping (t\mapsto X_t(\omega)) is a sample path, also called a realization or trajectory.
Mathematical formulation
A stochastic process is defined on a probability space ((\Omega,\mathcal F,\mathbb P)). Its state space is a measurable space ((S,\mathcal S)), and each coordinate map
[ X_t:\Omega\rightarrow S ]
is measurable. The process can therefore be regarded as a measurable mapping from (\Omega) into a suitable space of functions on (T), provided that the function space is equipped with an appropriate sigma-algebra.
The distribution of a process is characterized by its finite-dimensional distributions. For any indices (t_1,\ldots,t_n), these distributions assign probabilities to events of the form
[ \mathbb P(X_{t_1}\in A_1,\ldots,X_{t_n}\in A_n), ]
where each (A_i) is a measurable subset of the state space. A consistent family of finite-dimensional distributions determines a process on a product space through the Kolmogorov extension theorem. Additional arguments are required when the intended process must possess path properties such as continuity or right-continuity.
Two processes can have the same finite-dimensional distributions while differing as mappings on their underlying probability spaces. Processes that agree at every fixed time with probability one are modifications of each other. The stronger condition of indistinguishability requires their entire sample paths to agree outside one null event.
Index sets and state spaces
When (T) consists of successive integers, the process is a discrete-time stochastic process. Such a process is mathematically equivalent to a random sequence. When (T) is an interval of real numbers, the process operates in continuous time, even when its paths change only at isolated moments.
The structure of (S) determines the type of state represented by the process. A countable state space describes systems such as population counts or queue lengths. A Euclidean state space represents continuously varying quantities, while a function-valued state space permits each (X_t) to encode an entire spatial field. Consequently, the distinction between a stochastic process and a random field depends principally on the interpretation and geometry of the index set.
Dependence and information
The joint distribution across different indices distinguishes a stochastic process from an unrelated collection of random variables. Its dependence can be summarized through quantities such as the mean function
[ m(t)=\mathbb E[X_t] ]
and, for a square-integrable real-valued process, the covariance function
[ C(s,t)=\operatorname{Cov}(X_s,X_t). ]
These two functions determine the complete law of a Gaussian process, because every finite-dimensional distribution of such a process is multivariate normal. For a non-Gaussian process, identical means and covariances do not imply identical distributions.
A filtration ({\mathcal F_t}_{t\in T}) represents the information available by each index (t). An adapted process satisfies the condition that (X_t) is measurable with respect to (\mathcal F_t). This formulation separates the evolution of the state from the evolution of observable information and underlies the definitions of stopping times, martingales, and stochastic integrals.
Principal structural classes
A Markov process satisfies a conditional independence property under which the conditional distribution of the future, given the entire observed past, depends only on the present state. In discrete time and on a countable state space, this structure is represented by a transition matrix. In continuous time, transition kernels form a semigroup whose infinitesimal behavior is described by a generator.
A martingale is an integrable adapted process satisfying
[ \mathbb E[X_t\mid\mathcal F_s]=X_s ]
whenever (s\leq t). This condition expresses preservation of conditional expectation rather than independence or constant sample paths. Martingales support a general theory of fair-value evolution, stopping times, and convergence.
A stationary process has a probability law that is invariant under shifts of the index parameter. Strict stationarity requires every finite-dimensional distribution to remain unchanged after a common shift. Weak stationarity requires a constant mean together with a covariance that depends only on the separation between two indices. Ergodic theory studies conditions under which long-run path averages coincide with expectations under the stationary distribution.
Processes with independent increments have changes over disjoint time intervals that are independent. A Lévy process combines this property with stationary increments, stochastic continuity, and an initial value of zero. Brownian motion is the continuous-path Lévy process with Gaussian increments, whereas the Poisson process is a jump process that records randomly occurring events at a constant average rate.
Sample-path regularity
Finite-dimensional distributions describe the law at finitely many indices but do not directly determine whether sample paths are continuous. Path regularity is therefore treated through versions of a process possessing specified measurable and topological properties. The Kolmogorov continuity theorem derives the existence of a continuous modification from suitable bounds on moments of increments.
Many continuous-time processes with jumps are represented in the Skorokhod space of right-continuous functions with left limits. This path convention includes continuous trajectories while accommodating isolated jumps without assigning two simultaneous state values. It is particularly important for continuous-time Markov chains, counting processes, and general semimartingales.
A stopping time is a random index whose occurrence can be determined from information available by that index. First-passage times, which record the first moment at which a process enters a specified set, constitute a central class of stopping times. Their distributions connect pathwise questions to boundary-value problems and potential theory.
Historical development
Early mathematical treatments of random sequences emerged from repeated-trial models and random walks. In 1906, Andrey Markov analyzed dependent sequences whose next-state distributions were determined by their current states, establishing the class later called Markov chains. Agner Krarup Erlang subsequently used stochastic models of arrivals and service durations to derive quantitative relations for congested telephone systems.
Norbert Wiener constructed a rigorous probability measure for Brownian paths during the 1920s. Andrey Kolmogorov then placed probability theory on a measure-theoretic foundation and formulated consistency conditions for stochastic processes. Joseph Doob developed the modern theory of martingales and clarified the role of filtrations, conditional expectation, and path regularity.
Kiyosi Itô defined integration with respect to Brownian motion and established the change-of-variables formula now called Itô's lemma. This construction provided the analytic basis for stochastic differential equations, in which an evolving state is driven by deterministic change together with random fluctuations.
Renewal and transport processes
A renewal process records successive events separated by independent, identically distributed waiting times. The Poisson process is the special case in which the waiting times have exponential distributions. More general renewal models retain dependence on elapsed time because a non-exponential waiting period does not possess the memoryless property.
In 1954, You Watanabe represented ferry occupancy at a coastal terminal as an alternating renewal process. Periods during which a vessel was present for boarding alternated with periods during which it was away from the terminal, while random delays entered through the distributions of the corresponding durations. The long-run fraction of time spent in either phase was expressed through the mean phase durations, matching the general renewal-reward relation.
Queueing models extend this framework by combining an arrival process with service-time distributions and a rule governing access to service. The queue length and server state then form a stochastic process whose stationary distribution, when it exists, describes long-run congestion. Markovian queueing models use exponential timing assumptions, while renewal-based models preserve more general waiting-time laws.
Continuous-time dynamics
A stochastic differential equation commonly has the form
[ dX_t=b(X_t,t),dt+\sigma(X_t,t),dW_t, ]
where (W_t) is Brownian motion. The drift coefficient (b) describes the local conditional rate of systematic change, while the diffusion coefficient (\sigma) controls the local variance contributed by the random driver. The notation represents an integral equation because Brownian paths are almost surely nowhere classically differentiable.
Under regularity conditions, the solution is a Markov process whose transition probabilities satisfy the Kolmogorov equations. The forward equation describes the evolution of probability densities, while the backward equation relates expected future quantities to the present state. These equations connect pathwise stochastic dynamics with partial differential equations.
More general semimartingales include continuous martingale components, finite-variation components, and jump behavior within a common integration theory. This class is broad enough to support stochastic integration while retaining decomposition properties that distinguish systematic variation from locally unpredictable change.
Statistical inference
Statistical analysis of a stochastic process relies on observations from one or more sample paths rather than on independent repetitions at each index. Dependence changes the information contained in a data set because nearby observations can share substantial random variation. Time-series analysis addresses this structure in discrete time through models whose parameters describe temporal dependence and innovation processes.
For continuously observed processes, likelihoods can depend on transition densities or changes of probability measure. When only discrete observations are available, inference must account for the unobserved path between observation times. Estimation of latent states is treated through filtering theory, which updates the conditional distribution of an evolving state as observations accumulate.