In probability theory and related fields, a stochastic (/stoʊˈkæstɪk/) 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 appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.[1][4][5] Stochastic processes have applications in many disciplines such as biology,[6] chemistry,[7] ecology,[8] neuroscience,[9] physics,[10] image processing, signal processing,[11] control theory,[12] information theory,[13] computer science,[14] cryptography[15] and telecommunications.[16] Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.[17][18][19]
Stochastic Processes And Models David Stirzaker.pdf
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,[22] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time.[23] These two stochastic processes are considered the most important and central in the theory of stochastic processes,[1][4][24] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.[22][25]
Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks,[32] martingales,[33] Markov processes,[34] Lévy processes,[35] Gaussian processes,[36] random fields,[37] renewal processes, and branching processes.[38] The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology[39][40][41] as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.[42][43][44] The theory of stochastic processes is considered to be an important contribution to mathematics[45] and it continues to be an active topic of research for both theoretical reasons and applications.[46][47][48]
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.[55][56] If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes.[49][57][58] Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.[59][60] If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence.[56]
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.[85][86][87][88][89] But some also use the term to refer to processes that change in continuous time,[90] particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism.[91] There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.[90][92]
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.[1][2][3][99][100][101][102] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.[103] But the process can be defined more generally so its state space can be n \displaystyle n -dimensional Euclidean space.[92][100][104] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ \displaystyle \mu , which is a real number, then the resulting stochastic process is said to have drift μ \displaystyle \mu .[105][106][107]
Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk.[50][106] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,[108][109] which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.[110][111][112]
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.[120][122] The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.[50]
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.[123][124] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t \displaystyle t , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.[125] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.[126][127]
all have the same probability distribution. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.[149][150] But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.[149][151][152]
Instead of modification, the term version is also used,[151][161][162][163] 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.[164][143]
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.[162][163][165] The theorem can also be generalized to random fields so the index set is n \displaystyle n -dimensional Euclidean space[166] as well as to stochastic processes with metric spaces as their state spaces.[167]
Two stochastic processes X \displaystyle X and Y \displaystyle Y defined on the same probability space ( Ω , F , P ) \displaystyle (\Omega ,\mathcal F,P) with the same index set T \displaystyle T and set space S \displaystyle S are said be indistinguishable if the following
Separability is a property of a stochastic process based on its index set in relation to 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,[b] which means that the index set has a dense countable subset.[151][169]
The concept of separability of a stochastic process was introduced by Joseph Doob,.[169] The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.[173] Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.[176] A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.[169][171][177] Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.[137]
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.[182][184] Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. 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 line), are also members of this space.[185][187]
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.[188][189] For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.[190][191]
Markov processes are stochastic processes, traditionally in discrete or continuous time, that 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.[192][193] 2ff7e9595c
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