The random process X (t) is a group ofrandom variable, t generally has the meaning of time.The set of all possible values of the random process {X (t), t ∈ T} is called the state space of the random process and is recorded as S.[12]The number of customers received by a store during the period from time t0 to time tK is a group of random variables that depend on time t, that is, a random process.
The theory of stochastic process came into being in the early 20th century[1], which is based on physics, biologymanagement scienceAnd other aspects.In automatic controlpublic utility, management science, etc.[2]
A family of infinitely many, interrelatedrandom variableIs a random process.[12]In the study of random processeschanceDescribing the inherent laws of necessity and describing them in the form of probability is the charm of this discipline.
random process[3]The theoretical basis of the whole discipline is that Kolmogorov andDubeLaid.This discipline originated from the study of physics, such as GibbsBoltzmannPang Jialai and othersstatistical mechanicsAnd later EinsteinWiener、LevyAnd othersBrownian motionThe pioneering work of.
Research on stochastic process
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research method
There are many methods to study stochastic processes, which can be mainly divided into two categories:
In practical research, two methods are often used together.In addition,Combination methodandalgebraic methodIt also plays a role in the study of some special stochastic processes.
The mathematical stochastic process is a mathematical structure caused by the concept of actual stochastic process.givenProbability space(Ω, F, P)，random variableX (ω) is defined insample space Ω, taken from RMeasurable function, random process X (t) is a group of random variables with parameter t as the index, which can be regarded asBivariate function{X(t, ω),(t, ω) ∈ R × Ω}。If ω is fixed, we will get aindependent variableThis is the "realization" of random process X (t) in an experiment, and this function is called one of random process X (t)Sample functionOr sample track.On the other hand, if t is fixed, then a random variable will be obtained. Let the distribution of the random variable be FX(t)(x) This distribution is called the one-dimensional distribution of random process X (t).[12]
People study this process because it is a real stochastic processmathematical model Or because of its intrinsic mathematical meaning and itsprobability theoryApplications outside the field.
1923 NWienerThe mathematical definition of Brownian motion is given, and this process is still an important research object.Nevertheless, the general theory of stochastic processes is generally believed to have started in the 1930s.
In 1931, A. H. Kolmogorov published Analytic Methods of Probability Theory;Three years later, A. ∨. Xinqin published《Stationary processRelated theories.These two important papers aremarkov process And stationary process.Later, PLevyHe has published two books on Brownian motion and additive process, which contain richProbability idea。
1953, J.LDubeThe famous work "Theory of Stochastic Process" was published, which systematically and strictly describes thefundamental theory。
In the 1960s,french school Based on Markov process andGeopotentialSome ideas and results in the theory have developed the general theory of stochastic processes to a considerable extent, includingSectionTheorem andProjection of processTheory, etc. Chinese scholars have studied the stability process, Markov processmartingale 、limit theorem , stochastic differential equation, etc.
Statistical characteristics of stochastic processes
