Category: Stochastic processes

Convergence of random variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered
Kinetic scheme
In physics, chemistry and related fields, a kinetic scheme is a network of states and connections between them representing the scheme of a dynamical process. Usually a kinetic scheme represents a Mar
Cheeger bound
In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special
Stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used
Girsanov theorem
In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how t
Moment closure
In probability theory, moment closure is an approximation method used to estimate moments of a stochastic process.
Long-tail traffic
A long-tailed or heavy-tailed probability distribution is one that assigns relatively high probabilities to regions far from the mean or median. A more formal mathematical definition is given below. I
Totally inaccessible stopping time
No description available.
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 (or process)
Filtration (mathematics)
In mathematics, a filtration is an indexed family of subobjects of a given algebraic structure , with the index running over some totally ordered index set , subject to the condition that if in , then
Feynman–Kac formula
The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman we
Hierarchical Dirichlet process
In statistics and machine learning, the hierarchical Dirichlet process (HDP) is a nonparametric Bayesian approach to clustering grouped data. It uses a Dirichlet process for each group of data, with t
Law of the iterated logarithm
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khin
Counting process
A counting process is a stochastic process {N(t), t ≥ 0} with values that are non-negative, integer, and non-decreasing: 1. * N(t) ≥ 0. 2. * N(t) is an integer. 3. * If s ≤ t then N(s) ≤ N(t). If s
Bessel process
In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process.
Stationary increments
In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many
Nuisance variable
In the theory of stochastic processes in probability theory and statistics, a nuisance variable is a random variable that is fundamental to the probabilistic model, but that is of no particular intere
Chinese restaurant process
In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant.Imagine a restaurant with an infinite number of ci
Predictable σ-algebra
No description available.
In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is th
Chernoff's distribution
In probability theory, Chernoff's distribution, named after Herman Chernoff, is the probability distribution of the random variable where W is a "two-sided" Wiener process (or two-sided "Brownian moti
Markov chain central limit theorem
In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theor
Stochastic control
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the s
Intensity of counting processes
The intensity of a counting process is a measure of the rate of change of its predictable part. If a stochastic process is a counting process, then it is a submartingale, and in particular its Doob-Me
Predictable process
In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smal
Frequency of exceedance
The frequency of exceedance, sometimes called the annual rate of exceedance, is the frequency with which a random process exceeds some critical value. Typically, the critical value is far from the mea
Local time (mathematics)
In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particl
Sample-continuous process
In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.
Two-state trajectory
A two-state trajectory (also termed two-state time trajectory or a trajectory with two states) is a dynamical signal that fluctuates between two distinct values: ON and OFF, open and closed, , etc. Ma
Information source (mathematics)
In mathematics, an information source is a sequence of random variables ranging over a finite alphabet Γ, having a stationary distribution. The uncertainty, or entropy rate, of an information source i
Ruin theory
In actuarial science and applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
Rough path
In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregul
Active Brownian particle
An active Brownian particle (ABP) is a model of self-propelled motion in a dissipative environment. It is a nonequilibrium generalization of a Brownian particle. The self-propulsion results from a for
Lorden's inequality
In probability theory, Lorden's inequality is a bound for the moments of overshoot for a stopped sum of random variables, first published by Gary Lorden in 1970. Overshoots play a central role in rene
Σ-Algebra of τ-past
The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory of stochastic processes, a branch of probabi
Oscillator linewidth
The concept of a linewidth is borrowed from laser spectroscopy. The linewidth of a laser is a measure of its phase noise. The spectrogram of a laser is produced by passing its light through a prism. T
List of stochastic processes topics
In the mathematics of probability, a stochastic process is a random function. In practical applications, the domain over which the function is defined is a time interval (time series) or a region of s
Hitting time
In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times
Risk of ruin
Risk of ruin is a concept in gambling, insurance, and finance relating to the likelihood of losing all one's investment capital or extinguishing one's bankroll below the minimum for further play. For
Numéraire
The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradabl
Random measure
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such
Stochastic simulation
A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities. Realizations of these random variables are generated and
Single-particle trajectory
Single-particle trajectories (SPTs) consist of a collection of successive discrete points causal in time. These trajectories are acquired from images in experimental data. In the context of cell biolo
Jump process
A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson
Regenerative process
In applied probability, a regenerative process is a class of stochastic process with the property that certain portions of the process can be treated as being statistically independent of each other.
Pitman–Yor process
In probability theory, a Pitman–Yor process denoted PY(d, θ, G0), is a stochastic process whose sample path is a probability distribution. A random sample from this process is an infinite discrete pro
Extinction probability
Extinction probability is the chance of an inherited trait becoming extinct as a function of time t. If t = ∞ this may be the complement of the chance of becoming a universal trait. * v * t * e
Recurrence period density entropy
Recurrence period density entropy (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal
Supersymmetric theory of stochastic dynamics
Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicabili
Random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random w
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the in
Continuous stochastic process
In 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
Progressively measurable process
In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the
Natural filtration
In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which r
Stopped process
In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time.
Dissociated press
Dissociated press is a parody generator (a computer program that generates nonsensical text). The generated text is based on another text using the Markov chain technique. The name is a play on "Assoc
Sazonov's theorem
In mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov (Вячесла́в Васи́льевич Сазо́нов), is a theorem in functional analysis. It states that a bounded linear operator between tw
Abstract Wiener space
The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emph
Increasing process
An increasing process is a stochastic process where the random variables which make up the process are increasing almost surely and adapted: A continuous increasing process is such a process where the
Spherical contact distribution function
In probability and statistics, a spherical contact distribution function, first contact distribution function, or empty space function is a mathematical function that is defined in relation to mathema
Stopping time
In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time) is a specific type of “random time
Ionescu-Tulcea theorem
In the mathematical theory of probability, the Ionescu-Tulcea theorem, sometimes called the Ionesco Tulcea extension theorem deals with the existence of probability measures for probabilistic events c
Feller-continuous process
In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend continuously
Borel right process
In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process. Let be a locally compact, separable, metric space.We
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimen
Jyotiprasad Medhi was a professor of statistics at Gauhati University and Institute of Advanced Study in Science and Technology.
Stochastic homogenization
In homogenization theory, a branch of mathematics, stochastic homogenization is a technique for understanding solutions to partial differential equations with oscillatory random coefficients.
Continuity in probability
In probability theory, a stochastic process is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge.
Random walk hypothesis
The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted.
Gibbs state
In probability theory and statistical mechanics, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system. For example, a stationary or ste
Stochastic process
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
Accessible stopping time
No description available.
Spitzer's formula
In probability theory, Spitzer's formula or Spitzer's identity gives the joint distribution of partial sums and maximal partial sums of a collection of random variables. The result was first published
Excursion probability
In probability theory, an excursion probability is the probability that a stochastic process surpasses a given value in a fixed time period. It is the probability Numerous approximation methods for th
Jump diffusion
Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, option pricing, and
Stochastic resonance
Stochastic resonance (SR) is a phenomenon in which a signal that is normally too weak to be detected by a sensor, can be boosted by adding white noise to the signal, which contains a wide spectrum of
Martingale (probability theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equ
Russo–Vallois integral
In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral for suitable functions and . The idea is to replace the derivat
Stochastic cellular automaton
Stochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting Markov chains are an important extension of cellular automaton. Cellular automa
System size expansion
The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of stochastic processes. Specifically, it allows on
Zero-order process (statistics)
In probability theory and statistics, a zero-order process is a stochastic process in which each observation is independent of all previous observations. For example, a zero-order process in marketing
Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the mos
Narrow escape problem
The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology. The mathematical formulation is the following: a Brownian particle (ion, molecule, or protein) is confine
Weakly dependent random variables
In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale. A (time) sequence of random variables is weakly dependent if di
Random Fibonacci sequence
In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation , where the signs + or − are chosen at random with equal probability
Moran process
A Moran process or Moran model is a simple stochastic process used in biology to describe finite populations. The process is named after Patrick Moran, who first proposed the model in 1958. It can be
Version (probability theory)
No description available.
Disorder problem
In the study of stochastic processes in mathematics, a disorder problem or quickest detection problem (formulated by Kolmogorov) is the problem of using ongoing observations of a stochastic process to
Basic affine jump diffusion
In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form where is a standard Brownian motion, and is an independent compound Poisson process w
Random dynamical system
In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are cha
Stationary process
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distri
Poisson boundary
In mathematics, the Poisson boundary is a measure space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when
Dependent Dirichlet process
In the mathematical theory of probability, the dependent Dirichlet process (DDP) provides a non-parametric prior over evolving mixture models. A construction of the DDP built on a Poisson point proces
Quasi-stationary distribution
In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for
Subordinator (mathematics)
In probability theory, a Subordinator is a stochastic process that is non-negative and whose increments are stationary and independent. Subordinators are a special class of Lévy process that play an i
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variable
Life-time of correlation
In probability theory and related fields, the life-time of correlation measures the timespan over which there is appreciable autocorrelation or cross-correlation in stochastic processes.
Gaussian free field
In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of
Càdlàg
In mathematics, a càdlàg (French: "continue à droite, limite à gauche"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function
Large deviations of Gaussian random functions
A random function – of either one variable (a random process), or two or more variables(a random field) – is called Gaussian if every finite-dimensional distribution is a multivariate normal distribut
Killed process
In probability theory — specifically, in stochastic analysis — a killed process is a stochastic process that is forced to assume an undefined or "killed" state at some (possibly random) time.
Kolmogorov's inequality
In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables ex
Stochastic geometry
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, h
Hawkes process
In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process. It has arrivals at times where the infinitesimal probability of an arrival
Stochastic quantization
In theoretical physics, stochastic quantization is a method for modelling quantum mechanics, introduced by Edward Nelson in 1966, and streamlined by Parisi and Wu.
Gaussian noise
Gaussian noise, named after Carl Friedrich Gauss, is a term from signal processing theory denoting a kind of signal noise that has a probability density function (pdf) equal to that of the normal dist
Onsager–Machlup function
The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to th
Dirichlet process
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions
Contact process (mathematics)
The contact process is a stochastic process used to model population growth on the set of sites of a graph in which occupied sites become vacant at a constant rate, while vacant sites become occupied
Law (stochastic processes)
In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space. The law encodes a lot of information abo
Statistical fluctuations
Statistical fluctuations are fluctuations in quantities derived from many identical random processes. They are fundamental and unavoidable. It can be proved that the relative fluctuations reduce as th
Bernoulli process
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes onl
Classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually n-dimensional
File dynamics
The term file dynamics is the motion of many particles in a narrow channel. In science: in chemistry, physics, mathematics and related fields, file dynamics (sometimes called, single file dynamics) is
Stochastic thermodynamics
Stochastic thermodynamics is an emergent field of research in statistical mechanics that uses stochastic variables to better understand the non-equilibrium dynamics present in many microscopic systems
Pregaussian class
In probability theory, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure, such that there exists a certain Gaussian
Schramm–Loewner evolution
In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling li
Stochastic
Stochastic (/stəˈkæstɪk/, from Greek στόχος (stókhos) 'aim, guess') refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are disti
Continuous-time stochastic process
In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a continuous set of