In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over timesuch as the growth of a bacterial population, an electrical current fluctuating due to thermal noiseor the movement of a gas molecule.

They have applications in many disciplines including sciences such as biology[6] chemistry[7] ecology[8] neuroscience[9] and physics [10] as well as technology and engineering fields such as image processingsignal processing[11] information theory[12] computer science[13] cryptography [14] and telecommunications. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse[21] and the Poisson processused by A.

Erlang to study the number of phone calls occurring in a certain period of time. The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space.

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. Historically, the index set was some subset of the real linesuch as the natural numbersgiving the index set the interpretation of time. A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables.

One common way of classification is by the cardinality of the index set and the state space. When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time.

The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space.

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. A classic example of a random walk is known as the simple random walkwhich is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one.

The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.

Almost surelya sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. The Poisson or the Poisson point process is a stochastic process that has different forms and definitions. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter.

This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. The homogeneous Poisson process can be defined and generalized in different ways.

It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. Defined on the real line, the Poisson process can be interpreted as a stochastic process, [50] [] among other random objects.

The autoregressive and moving average processes are used in modelling discrete-time empirical time series data, especially in economics. The autoregressive model treats a stochastic variable as depending on its own prior values and on a current independently and identically distributed iid stochastic term. The moving average model treats a stochastic variable as depending on the current and past values of an iid stochastic variable.

Generalizations include the vector autoregression model involving the modelling of more than one stochastic variable, and the ARIMA model involving both autoregressive and moving average components of a single modelled variable. There are others ways to consider a stochastic process, with the above definition being considered the traditional one. A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.

An increment of a stochastic process is the difference between two random variables of the same stochastic process.

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For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. Markov processes are stochastic processes, traditionally in discrete or continuous timethat have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process.

In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process. The Brownian motion process and the Poisson process in one dimension are both examples of Markov processes [] in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.

A Markov chain is a type of Markov process that has either discrete state space or discrete index set often representing timebut the precise definition of a Markov chain varies.

Markov processes form an important class of stochastic processes and have applications in many areas. A martingale is a discrete-time or continuous-time stochastic process with the property that the expectation of the next value of a martingale is equal to the current value given all the previous values of the process.

The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, [] [] [] but they can also be complex-valued [] or even more general. A symmetric random walk and a Wiener process with zero drift are both examples of martingales, respectively, in discrete and continuous time.

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process on the real line resulting in a martingale called the compensated Poisson process.

Martingales mathematically formalize the idea of a fair game, [] and they were originally developed to show that it is not possible to win a fair game. Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line.

Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago, [] but very little analysis on them was done in terms of probability.

Jakob Bernoulli [c] later wrote Ars Conjectandiwhich is considered a significant event in the history of probability theory. Bernoulli's book was published, also posthumously, in and inspired many mathematicians to study probability.

In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausiusmost of the work had little or no randomness.

In at the International Congress of Mathematicians in Paris David Hilbert presented a list of mathematical problemswhere he asked in his sixth problem for a mathematical treatment of physics and probability involving axioms. In s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergei BernsteinAleksandr Khinchin[e] and Andrei Kolmogorov.

In Andrei Kolmogorov published in German his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung[g] where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.

After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with signification contributions made in many areas of probability and mathematics as well as the icici bank matrix forex card of new areas. Also starting in the s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theorywith early ideas by Shizuo Kakutani and then later work by Joseph Doob.

In Doob published his book Stochastic processeswhich had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes. Other fields of probability were developed and used to study stochastic processes, with cboe options put call ratio main approach being the theory of large deviations.

Later in the s and s fundamental work was done by Alexander Wentzell in the Soviet Union and Monroe D. Donsker and Srinivasa Varadhan in the United States of America, [] which would later result in Varadhan winning the Abel Prize. The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.

Although Khinchin gave mathematical definitions of stochastic processes in the s, [] [] specific stochastic processes ludacris money maker video already been discovered in different settings, such as the Brownian motion process and the Poisson process.

The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. In Karl Pearson coined the term random walk while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields.

Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.

The French mathematician Louis Bachelier used a Wiener process in his thesis in order to model price changes on the Paris Boursea stock earn money in pakistan urdu[] without knowing the work of Thiele.

It is commonly thought that Bachelier's payoff of long call option gained little attention and was forgotten for decades until it was rediscovered in the s by the Leonard Savageand then become more popular after Bachelier's thesis was translated into English in But the work was never forgotten in the mathematical community, as Bachelier published a book in detailing his ideas, [] which was cited by mathematicians including Doob, Feller [] and Kolomogorov.

In Albert Einstein published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the kinetic theory of gases. Einstein derived a differential equationknown as a diffusion equationcall forward android 4.2.1 describing the probability of finding a particle in a certain region of space.

Shortly after Einstein's first metatrader 4 template forex on Brownian movement, Marian Smoluchowski published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method.

Einstein's work, as well as experimental results obtained by Jean Perrinlater inspired Norbert Wiener in the s [] to use a type of measure theory, developed by Percy Danielland Fourier analysis to prove the existence of the Wiener process as a mathematical object. Another discovery occurred in Denmark in when A. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval.

Erlang was not at currency converter free download windows xp time aware dj sava si andreea money maker download mp3 Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other.

He then stochastic analysis in stock market the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution. In Ernest Rutherford and Hans Geiger published experimental results on counting alpha particles.

Their experimental work had mathematical contributions from Harry Batemanwho derived Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process.

Markov processes and Markov chains are named after Andrey Markov who studied Markov chains in the early 20 th century. Markov was interested in studying an extension of independent random sequences. In his first paper on Markov chains, published inMarkov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, [] [] [] which had been commonly regarded as a requirement for such mathematical laws to hold.

Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest inand a branching process, introduced by Francis Galton and Henry William Watson inpreceding the work of Markov. Andrei Kolmogorov developed in a paper a large part of the early theory of continuous-time Stochastic analysis in stock market processes.

The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford Easter opening hours westfield hornsby Dictionary gives the year as its earliest occurrence. The term stochastic process first appeared in English in a paper by Joseph Doob.

Early occurrences of the word random in English with its current meaning, relating to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14 th century as a noun meaning "impetuosity, great speed, force, or violence in riding, running, striking, etc.

The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning to "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic processwhich the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in The definition of a stochastic process varies, [] but a stochastic process is traditionally defined as a collection of random variables indexed by some set.

The term random function is also used to refer to a stochastic or random process, [5] [] [] though sometimes it is only used when the stochastic process takes real values. The law of a stochastic process or a random variable is also called the probability lawprobability distributionor the distribution. The finite-dimensional distributions of stockton ca dog pound stochastic process satisfy two mathematical conditions known as consistency conditions.

Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line. This type of stochastic process can be used to describe a physical system that is in steady state, but iraqi dinar news usa today experiences random fluctuations.

A stochastic process with above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. A filtration saluhall market stockholm an increasing sequence of sigma-algebras defined in fx options clark to some probability space and an index set that has some total order relation, such in the case of the index set being some subset of the real numbers.

A modification of a stochastic havelock hunter stockbrokers tunbridge wells is another stochastic process, which is closely related to the original stochastic process.

Two stochastic processes that are modifications of each other have the same law [] and they are said to be stochastically equivalent or equivalent. Instead of modification, the term version is also used, [] [] [] [] however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse [] [].

If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version. Separability is a property of a stochastic process based on its index set in relation make money selling novelty items the probability measure.

The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space[h] which means that the index set has a dense countable subset. The concept of separability of a stochastic process was introduced by Joseph Doob[] where the underlying idea is to make a countable set of points of the index set determine the properties of the stochastic process.

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Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process on the real lineare also members of this space.

In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues. In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically.

One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.

Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem [j] to prove a corresponding stochastic process exists.

When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes. For example, both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions. Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined.

To overcome these two difficulties, different assumptions and approaches are possible. One approach for avoiding construction difficulties, proposed by Joseph Doobis to assume that the stochastic process is separable. Another approach is possible, originally developed by Anatoliy Skorokhod and Andrei Kolmogorov[] for a continuous-time stochastic process with any metric space as its state space.

For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits.

This approach is now more used than the separability assumption, [] [] but such a stochastic process based on this approach will be automatically separable. Although less used, the separability assumption is considered more general because every stochastic process has a separable version. From Wikipedia, the free encyclopedia. Autoregressive model and Moving average model. List of stochastic processes topics Covariance function Dynamics of Markovian particles Entropy rate for a stochastic process Ergodic process Gillespie algorithm Interacting particle system Law stochastic processes Markov chain Probabilistic cellular automaton Random field Randomness Stationary process Statistical model Stochastic calculus Stochastic control Deterministic system.

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Extremes in Random Fields: A Theory and Its Applications. Bernoulli process Branching process Chinese restaurant process Galton—Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy.

Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman—Yor process Point process Cox Poisson Random field Random graph. Autoregressive conditional heteroskedasticity ARCH model Autoregressive integrated moving average ARIMA model Autoregressive AR model Autoregressive—moving-average ARMA model Generalized autoregressive conditional heteroskedasticity GARCH model Moving-average MA model.

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