In the mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 of probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, a stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

can be thought of as a random function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

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

Time series

In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

) or a region of space (random field

Random field

A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....

).

Familiar examples of time series include stock market

Stock market

A stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...

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

Electroencephalography

Electroencephalography is the recording of electrical activity along the scalp. EEG measures voltage fluctuations resulting from ionic current flows within the neurons of the brain...

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

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

 or random walk

Random walk

A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...

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 :Category:Stochastic processes

• Basic affine jump diffusion 
• Bernoulli process
  Bernoulli process
  In probability and statistics, a Bernoulli process 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. The component Bernoulli variables Xi are identical and independent...
  
   Bernoulli process: discrete-time processes with two possible states.
  • Bernoulli scheme
    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 are important in the study of dynamical systems, as most such systems exhibit a repellor that is the product of the Cantor set and a smooth...
    
    s: discrete-time processes with N possible states; every stationary process in N outcomes is a Bernoulli scheme, and vice-versa.
• Birth-death process
  Birth-death process
  The birth–death process is a special case of continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths...
  
   
• Branching process
  Branching process
  In probability theory, a branching process is a Markov process that models a population in which each individual in generation n produces some random number of individuals in generation n + 1, according to a fixed probability distribution that does not vary from individual to...
  
   
• Branching random walk
  Branching random walk
  In probability theory, a branching random walk is a stochastic process that generalizes both the concept of random walk and of branching process. At every generation , a branching random walk's value is a set of elements that are located in some linear space, such as the real line...
  
   
• Brownian bridge
  Brownian bridge
  A Brownian bridge is a continuous-time stochastic process B whose probability distribution is the conditional probability distribution of a Wiener process W given the condition that B = B = 0.The expected value of the bridge is zero, with variance t, implying that the most...
  
   
• Brownian motion
  Brownian motion
  Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
  
   
• Chinese restaurant process 
• CIR process
  CIR process
  The CIR process is a Markov process with continuous paths defined by the following stochastic differential equation :dr_t = \theta \,dt + \sigma\, \sqrt r_t dW_t\,...
  
   
• Continuous stochastic process
  Continuous stochastic process
  In the 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 a process to have, since it implies that they are well-behaved in some sense, and,...
  
   
• Cox process
  Cox process
  A Cox process , also known as a doubly stochastic Poisson process or mixed Poisson process, is a stochastic process which is a generalization of a Poisson process...
  
   
• Dirichlet process
  Dirichlet process
  In probability theory, a Dirichlet process is a stochastic process that can be thought of as a probability distribution whose domain is itself a random distribution...
  
  es
• Finite-dimensional distribution
  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 onto a finite-dimensional vector space .-Finite-dimensional distributions of a measure:Let be a measure space...
  
   
• Galton–Watson process 
• Gamma process 
• Gaussian process
  Gaussian process
  In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...
  
     – a process where all linear combinations of coordinates are normally distributed random variables.
  • Gauss–Markov process   (cf. below)
• Girsanov's theorem 
• Homogeneous processes: processes where the domain has some symmetry
  Symmetry
  Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
  
   and the finite-dimensional probability distributions also have that symmetry. Special cases include stationary process
  Stationary process
  In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
  
  es, also called time-homogeneous.
• Karhunen–Loève theorem
• Lévy process
  Lévy process
  In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...
  
   
• Local time (mathematics)
  Local time (mathematics)
  In the mathematical theory of stochastic processes, local time is a stochastic process associated with diffusion processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level...
  
   
• Loop-erased random walk
  Loop-erased random walk
  In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics and, in physics, quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree...
  
   
• Markov process
  Markov process
  In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
  
  es are those in which the future is conditionally independent of the past given the present.
  • Markov chain
    Markov chain
    A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
    
     
  • Continuous-time Markov process 
  • Markov process
    Markov process
    In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
    
     
  • Semi-Markov process
    Semi-Markov process
    A continuous-time stochastic process is called a semi-Markov process or 'Markov renewal process' if the embedded jump chain is a Markov chain, and where the holding times are random variables with any distribution, whose distribution function may depend on the two states between which the move is...
    
     
  • Gauss–Markov processes: processes that are both Gaussian and Markov
• Martingale
  Martingale (probability theory)
  In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
  
  s – processes with constraints on the expectation
• Ornstein–Uhlenbeck process 
• Point process
  Point process
  In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces...
  
  es: random arrangements of points in a space . 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
  Poisson process
  A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
  
   
  • Compound Poisson process
    Compound Poisson process
    A compound Poisson process is a continuous-time 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...
    
     
• Population process
  Population process
  In applied probability, a population process is a Markov chain in which the state of the chain is analogous to the number of individuals in a population , and changes to the state are analogous to the addition or removal of individuals from the population.Although named by analogy to biological...
  
   
• Queueing theory
  Queueing theory
  Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
  
   
  • Queue 
• Random field
  Random field
  A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....
  
   
  • Gaussian random field
    Gaussian random field
    A Gaussian random field is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process....
    
     
  • Markov random field 
• Sample-continuous process 
• Stationary process
  Stationary process
  In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
  
   
• Stochastic calculus
  Stochastic calculus
  Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes...
  
   
  • Itō calculus
    Ito calculus
    Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion . It has important applications in mathematical finance and stochastic differential equations....
    
     
  • Malliavin calculus
    Malliavin calculus
    The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....
    
     
  • Semimartingale
    Semimartingale
    In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....
    
     
  • Stratonovich integral
    Stratonovich integral
    In stochastic processes, the Stratonovich integral is a stochastic integral, the most common alternative to the Itō integral...
    
     
• Stochastic differential equation
  Stochastic differential equation
  A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
  
   
• Stochastic process
  Stochastic process
  In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
  
   
• Telegraph process
  Telegraph process
  In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values.If these are called a and b, the process can be described by the following master equations:...
  
   
• Time series
  Time series
  In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
  
   
• Wiener process
  Wiener process
  In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...