In the

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

, a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

is a random function. In practical applications, the domain over which the function is defined is a time interval (''

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

'') or a region of space (''

random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other d ...

''). Familiar examples of time series include

stock market A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on businesses; these may include ''securities'' listed on a public stock exchange, as ...

and exchange rate fluctuations, signals such as speech, audio and video; medical data such as a patient's

EKG Electrocardiography is the process of producing an electrocardiogram (ECG or EKG), a recording of the heart's electrical activity. It is an electrogram of the heart which is a graph of voltage versus time of the electrical activity of the hear ...

EEG Electroencephalography (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in the neocortex ...

, blood pressure or temperature; and random movement such as Brownian motion or

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

s. Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material.

Stochastic processes topics

:''This list is currently incomplete.'' See also :Stochastic processes *

Basic affine jump diffusion In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form : dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt\,dB_t+dJ_t,\qquad t\geq 0, Z_\geq 0, where B is a standard Brownian motion, and ...

Bernoulli process In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. Th ...

: discrete-time processes with two possible states. **

Bernoulli scheme In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical sys ...

s: discrete-time processes with ''N'' possible states; every stationary process in ''N'' outcomes is a Bernoulli scheme, and vice versa. *

Bessel process In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. Formal definition The Bessel process of order ''n'' is the real-valued process ''X'' given (when ''n'' ≥ 2) by :X_t = \, W_t \, , whe ...

Birth–death process The birth–death process (or birth-and-death process) is a special case of continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state ...

* Branching process *

Branching random walk In probability theory, a branching random walk is a stochastic process that generalizes both the concept of a random walk and of a branching process. At every generation (a point of discrete time), a branching random walk's value is a set of ele ...

Brownian bridge A Brownian bridge is a continuous-time stochastic process ''B''(''t'') whose probability distribution is the conditional probability distribution of a standard Wiener process ''W''(''t'') (a mathematical model of Brownian motion) subject to the co ...

* Brownian motion *

Chinese restaurant process In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant. Imagine a restaurant with an infinite number of circular tables, each with infinite capacity. C ...

CIR process CIR or Cir may refer to: Locations * Cairo Regional Airport, FAA/IATA code, CIR * CIR, station code for the Caledonian Road & Barnsbury railway station in the UK * Christmas Island Resort, a casino/resort in the northeastern Indian Oceans Organ ...

Continuous stochastic process In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have, ...

* Cox process *

Dirichlet process In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a pro ...

es *

Finite-dimensional distribution In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or fin ...

* First passage time *

Galton–Watson process The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal (passed from father to son), while offspr ...

Gamma process In mathematics and probability theory, a gamma process, also known as (Moran-)Gamma subordinator, is a random process with independent gamma distributed increments. Often written as \Gamma(t;\gamma,\lambda), it is a pure-jump increasing Lévy ...

Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...

– a process where all linear combinations of coordinates are normally distributed random variables. ** Gauss–Markov process (cf. below) * GenI process *

Girsanov's theorem In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which des ...

* Hawkes process * Homogeneous processes: processes where the domain has some

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

and the finite-dimensional probability distributions also have that symmetry. Special cases include stationary processes, also called time-homogeneous. * Karhunen–Loève theorem * Lévy process * Local time (mathematics) *

Loop-erased random walk In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See als ...

Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

es are those in which the future is conditionally independent of the past given the present. **

Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

Markov chain central limit theorem In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the vari ...

Continuous-time Markov process A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of ...

Semi-Markov process In probability and statistics, a Markov renewal process (MRP) is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chains, Poisson processes and renewal processes can be derived as special ...

** Gauss–Markov processes: processes that are both Gaussian and Markov * Martingales – processes with constraints on the expectation * Onsager–Machlup function *

Ornstein–Uhlenbeck process In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...

* Percolation theory *

Point process In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th edition. ...

es: random arrangements of points in a space

S

. They can be modelled as stochastic processes where the domain is a sufficiently large family of subsets of ''S'', ordered by inclusion; the range is the set of natural numbers; and, if ''A'' is a subset of ''B'', ''ƒ''(''A'') ≤ ''ƒ''(''B'') with probability 1. * Poisson process **

Compound Poisson process A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisso ...

* Population process * Probabilistic cellular automaton * Queueing theory **

Queue __NOTOC__ Queue () may refer to: * Queue area, or queue, a line or area where people wait for goods or services Arts, entertainment, and media *''ACM Queue'', a computer magazine * The Queue (Sorokin novel), ''The Queue'' (Sorokin novel), a 198 ...

Random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other d ...

Gaussian random field A Gaussian random field (GRF) within statistics, is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process. An important special case of a GRF is the Gaussian fre ...

** Markov random field * Sample-continuous process * Stationary process * Stochastic calculus ** Itô calculus ** Malliavin calculus ** Semimartingale ** Stratonovich integral *

Stochastic control Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. The system designer assumes, in a Bayes ...

* Stochastic differential equation *

Stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

Telegraph process In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal). If the two possible values that a rando ...

Time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

Wald's martingale In probability theory Wald's martingale, named after Abraham Wald and more commonly known as the geometric Brownian motion, is a stochastic process of the form :: \left\ for any real value ''λ'' where ''W't'' is a Wiener process In ma ...

Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...

{{DEFAULTSORT:Stochastic processes topics Mathematics-related lists *List Statistics-related lists