Stochastic process

thumb|A computer-simulated realization of a [[Wiener process|Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

Applications and real-world phenomena have repeatedly motivated mathematicians to propose new stochastic processes. Two classic examples are the Wiener process (also called the Brownian motion process) and the Poisson process. Louis Bachelier used the Wiener process to model price changes on the Paris Bourse, while A. K. Erlang used the Poisson process to model the number of phone calls occurring in a given period of time. These two processes are widely treated as central to the theory of stochastic processes, and they were invented repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.

The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

Introduction

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.

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. 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. 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. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.

According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th 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 run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888.

Terminology

The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. Both "collection", while instead of "index set", sometimes the terms "parameter set" though sometimes it is only used when the stochastic process takes real values. while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a manifold. <math>\{X(t)\}</math> or simply as <math>X</math>. Some authors mistakenly write <math>X(t)</math> even though it is an abuse of function notation. For example, <math>X(t)</math> or <math>X_t</math> are used to refer to the random variable with the index <math>t</math>, and not the entire stochastic process. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each idealised coin flip is an example of a Bernoulli trial.

Random walk

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. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in financial models, which has led to some confusion, resulting in its criticism. There are various other 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.

A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If <math>p=0.5</math>, this random walk is called a symmetric random walk.

Wiener process

The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. The Wiener process is named after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids.

thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift () and without drift ()

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. 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. But the process can be defined more generally so its state space can be <math>n</math>-dimensional Euclidean space. 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 <math> \mu</math>, which is a real number, then the resulting stochastic process is said to have drift <math> \mu</math>.

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. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.

The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes. It plays a central role in quantitative finance, where it is used, for example, in the Black–Scholes–Merton model. The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.

Poisson process

The Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. 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. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes. If the parameter constant of the Poisson process is replaced with some non-negative integrable function of <math>t</math>, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. 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.

Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.

Definitions

Stochastic process

A stochastic process is defined as a collection of random variables defined on a common probability space <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a sample space, <math>\mathcal{F}</math> is a <math>\sigma</math>-algebra, and <math>P</math> is a probability measure; and the random variables, indexed by some set <math>T</math>, all take values in the same mathematical space <math>S</math>, which must be measurable with respect to some <math>\sigma</math>-algebra <math>\Sigma</math>.

\{X(t):t\in T \}.

</math></div>

Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>. A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.

There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible functions from the set <math>T</math> into the space <math>S</math>.

Index set

The set <math>T</math> is called the index set of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set <math>T</math> the interpretation of time. such as the Cartesian plane <math>\mathbb{R}^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space. That said, many results and theorems are only possible for stochastic processes with a totally ordered index set.

State space

The mathematical space <math>S</math> of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, <math>n</math>-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.

Sample function

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. More precisely, if <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, the mapping

X(\cdot,\omega): T \rightarrow S,

</math></div>

is called a sample function, a realization, or, particularly when <math>T</math> is interpreted as time, a sample path of the stochastic process <math>\{X(t,\omega):t\in T \}</math>. This means that for a fixed <math>\omega\in\Omega</math>, there exists a sample function that maps the index set <math>T</math> to the state space <math>S</math>. or path.

Increment

An increment of a stochastic process is the difference between two random variables of the same stochastic process. 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. For example, if <math>\{X(t):t\in T \}</math> is a stochastic process with state space <math>S</math> and index set <math>T=[0,\infty)</math>, then for any two non-negative numbers <math>t_1\in [0,\infty)</math> and <math>t_2\in [0,\infty)</math> such that <math>t_1\leq t_2</math>, the difference <math>X_{t_2}-X_{t_1}</math> is a <math>S</math>-valued random variable known as an increment.

For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives

X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},

</math></div>

so the law of a <math>X</math> can be written as:

Finite-dimensional probability distributions

For a stochastic process <math>X</math> with law <math>\mu</math>, its finite-dimensional distribution for <math>t_1,\dots,t_n\in T</math> is defined as:

\mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1},

</math></div>

This measure <math>\mu_{t_1,..,t_n}</math> is the joint distribution of the random vector <math>

(X({t_1}),\dots, X({t_n}))

</math>; it can be viewed as a "projection" of the law <math>\mu</math> onto a finite subset of <math>T</math>.

For any measurable subset <math>C</math> of the <math>n</math>-fold Cartesian power <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.

When the index set <math>T</math> can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.

Filtration

A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math>\{\mathcal{F}_t\}_{t\in T} </math>, on a probability space <math>(\Omega, \mathcal{F}, P)</math> is a family of sigma-algebras such that <math> \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} </math> for all <math>s \leq t</math>, where <math>t, s\in T</math> and <math>\leq</math> denotes the total order of the index set <math>T</math>.

Modification

A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process <math>X</math> that has the same index set <math>T</math>, state space <math>S</math>, and probability space <math>(\Omega,{\cal F},P)</math> as another stochastic process <math>Y</math> is said to be a modification of <math>X</math> if for all <math>t\in T</math> the following

P(X_t=Y_t)=1 ,

</math></div>

holds. Two stochastic processes that are modifications of each other have the same finite-dimensional 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. The theorem can also be generalized to random fields so the index set is <math>n</math>-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces.

Indistinguishable

Two stochastic processes <math>X</math> and <math>Y</math> defined on the same probability space <math>(\Omega,\mathcal{F},P)</math> with the same index set <math>T</math> and set space <math>S</math> are said be indistinguishable if the following

P(X_t=Y_t \text{ for all } t\in T )=1 ,

</math></div>

holds.

Separability

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,(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right]</math> is zero for all times. Formally:

:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.

Independence implies uncorrelatedness

If two stochastic processes <math>X</math> and <math>Y</math> are independent, then they are also uncorrelated. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French phrase continue à droite, limite à gauche. A Skorokhod function space, introduced by Anatoliy Skorokhod, The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, <math>D[0,1]</math> denotes the space of càdlàg functions defined on the unit interval <math>[0,1]</math>.

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.

Regularity

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. 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.

Further examples

Markov processes and chains

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.

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 time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).

Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.

The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math>n</math>-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.

Martingale

A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. 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. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

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' where it is possible form reasonable expectations for payoffs, and they were originally developed to show that it is not possible to gain an 'unfair' advantage in such a game. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.

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. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments

X_{t_2}-X_{t_1}, \dots , X_{t_n}-X_{t_{n-1,

</math></div>

are all independent of each other, and the distribution of each increment only depends on the difference in time. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.

Point process

A point process is a collection of points randomly located on some mathematical space such as the real line, <math>n</math>-dimensional Euclidean space, or more abstract spaces. 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. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology. on which it is defined, such as the real line or <math>n</math>-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. but very little analysis on them was done in terms of probability. The year 1654 is often considered the birth of probability theory when French mathematicians Pierre Fermat and Blaise Pascal had a written correspondence on probability, motivated by a gambling problem. But there was earlier mathematical work done on the probability of gambling games such as Liber de Ludo Aleae by Gerolamo Cardano, written in the 16th century but posthumously published later in 1663.

After Cardano, Jakob Bernoulli wrote Ars Conjectandi, which is considered a significant event in the history of probability theory. Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability. But despite some renowned mathematicians contributing to probability theory, such as Pierre-Simon Laplace, Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev, most of the mathematical community did not consider probability theory to be part of mathematics until the 20th century.

Statistical mechanics

In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, are 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 Clausius, most of the work had little or no randomness.

This changed in 1859 when James Clerk Maxwell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he modelled the gas particles as moving in random directions at random velocities. The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement.

Measure theory and probability theory

At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth problem asked for a mathematical treatment of physics and probability involving axioms. and Andrei Kolmogorov. In the early 1930s, Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.

Birth of modern probability theory

In 1933, Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,