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*  Ancestral reconstruction
Ornstein-Uhlenbeck process: in brief, an Ornstein-Uhlenbeck process is a continuous stochastic process that behaves like a ... Such models assume that the evolution of a trait through time may be modelled as a stochastic process. For discrete-valued ... that infer the stochastic process of evolution as it unfolds along each branch of a tree. Statistical justification. Without a ... the process is frequently taken to be a Brownian motion or an Ornstein-Uhlenbeck process. Using this model as the basis for ...
*  Weak Selection
Hence weak selection increases the impact of stochastic processes on the evolutionary dynamics of the trait being weakly ... In smaller populations, a weakly selected mutation could proliferate due to stochastic processes such as genetic drift even ... Weak selection can only be used to explain the maintenance of mutations in a Moran process [1]. A Moran process in one which ...
*  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 ... Let X : T × Ω → S be a stochastic process (so the map X t : Ω → S : ω ↦ X ( t , ω ) {\displaystyle X_{t}:\Omega \to S:\omega \ ... The law encodes a lot of information about the process; in the case of a random walk, for example, the law is the probability ... Indeed, many authors define Brownian motion to be a sample continuous process starting at the origin whose law is Wiener ...
*  Stochastic process
The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes ... then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the ... to denote the stochastic process. Markov processes are stochastic processes, traditionally in discrete or continuous time, that ... Lévy processes, Gaussian processes, and random fields, renewal processes and branching processes. The study of stochastic ...
*  Filtering problem (stochastic processes)
Stochastic Processes and Filtering Theory. New York: Academic Press. ISBN 0-12-381550-9. Øksendal, Bernt K. (2003). Stochastic ... In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems ... Maybeck, Peter S., Stochastic models, estimation, and control, Volume 141, Series Mathematics in Science and Engineering, 1979 ... Filtering (disambiguation) Not to be confused with Filter (signal processing) Kalman filter most famous filtering algorithm in ...
*  Infinitesimal generator (stochastic processes)
In mathematics - specifically, in stochastic analysis - the infinitesimal generator of a stochastic process is a partial ... The Ornstein-Uhlenbeck process on R, which satisfies the stochastic differential equation dXt = θ (μ − Xt) dt + σ dBt, has ... The two-dimensional process Y satisfying d Y t = ( d t d B t ) , {\displaystyle \mathrm {d} Y_{t}={\mathrm {d} t \choose \ ... which satisfies the stochastic differential equation dXt = rXt dt + αXt dBt, has generator A f ( x ) = r x f ′ ( x ) + 1 2 α 2 ...
*  Smoothing problem (stochastic processes)
Especially non-stochastic and non-Bayesian signal processing, without any hidden variables. 2. Estimation: The smoothing ... Filtering is causal but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on ... especially non-stochastic signal processing, often a name of various types of convolution). These names are used in the context ... whereas smoothing is batch processing of the given data. Filtering is the estimation of a (hidden) time-series process based on ...
*  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 ... this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, ... Let (Ω, Σ, P) be a probability space, let T be some interval of time, and let X : T × Ω → S be a stochastic process. For ... It is implicit here that the index of the stochastic process is a continuous variable. Some authors define a "continuous ( ...
*  List of stochastic processes topics
Markov chain Continuous-time Markov process Markov process Semi-Markov process Gauss-Markov processes: processes that are both ... bridge Brownian motion Chinese restaurant process CIR process Continuous stochastic process Cox process Dirichlet processes ... Stochastic control Stochastic differential equation Stochastic process Telegraph process Time series Wald's martingale Wiener ... See also Category:Stochastic processes Basic affine jump diffusion Bernoulli process: discrete-time processes with two possible ...
*  Stochastic Processes and their Applications
"Stochastic Processes and their Applications Abstracting and Indexing". Stochastic Processes and their Applications. Elsevier. ... Stochastic Processes and their Applications is a monthly peer-reviewed scientific journal published by Elsevier for the ... "Stochastic Processes and their Applications". 2012 Journal Citation Reports. Web of Science (Science ed.). Thomson Reuters. ... The principal focus of this journal is theory and applications of stochastic processes. It was established in 1973. The journal ...
*  Continuous-time stochastic process
... a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index ... Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are ... An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson process. An example ... A more restricted class of processes are the continuous stochastic processes: here the term often (but not always) implies both ...
*  Stochastic processes and boundary value problems
... the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic ... In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most ... Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ... X can be taken to be the solution to the stochastic differential equation d X t = b ( X t ) d t + σ ( X t ) d B t , {\ ...
*  Stochastic process rare event sampling
... (SPRES) is a Rare Event Sampling method in computer simulation, designed specifically ... The process of branching requires that identical paths can be made to diverge from each other, such as by changing the seed in ... for non-equilibrium calculations, including those for which the rare-event rates are time-dependent (non-stationary process). ...
*  Probability distribution of extreme points of a Wiener stochastic process
In the mathematical theory of probability, the Wiener process, named after Norbert Wiener, is a stochastic process used in ... If a Wiener stochastic process is chosen as a model for the objective function, it is possible to calculate the probability ... The stochastic process is taken as a model of the objective function, assuming that the probability distribution of its extrema ... Let X ( t ) {\displaystyle X(t)} be a Wiener stochastic process on an interval [ a , b ] {\displaystyle [a,b]} with initial ...
*  Doob-Meyer decomposition theorem
Doob decomposition theorem Doob 1953 Meyer 1952 Meyer 1963 Protter 2005 Protter (2005) Doob, J. L. (1953). Stochastic Processes ... Protter, Philip (2005). Stochastic Integration and Differential Equations. Springer-Verlag. pp. 107-113. ISBN 3-540-00313-4. ... Let Z {\displaystyle Z} be a cadlag submartingale of class D. Then there exists a unique, increasing, predictable process A {\ ... The Doob-Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may ...
*  Markov renewal process
Stochastic processes. New York: Wiley & Sons. ISBN 978-0-470-27000-4. Ross, Sheldon M. (1999). Stochastic processes (2nd ed.). ... Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP ( ... In probability and statistics a Markov renewal process is a random process that generalizes the notion of Markov jump processes ... The entire process is not Markovian, i.e., memoryless, as happens in a continuous time Markov chain/process (CTMC). Instead the ...
*  Joseph L. Doob
Books - (1953). Stochastic Processes. John Wiley & Sons. ISBN 0-471-52369-0. - (1984). Classical Potential Theory and Its ... 1975). "Stochastic process measurability conditions" (PDF). Annales de l'Institut Fourier. 25 (3-4): 163-176. doi:10.5802/aif. ... J.L. Doob Probability and statistics Doob J.L., Stochastic Processes Doob J.L., Classical Potential Theory and Its ... After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov ...
*  Markov chain
... and the Poisson process, which are considered the most important and central stochastic processes in the theory of stochastic ... The process described here is an approximation of a Poisson point process - Poisson processes are also Markov processes. ... Two important examples of Markov processes are the Wiener process, also known as the Brownian motion process, ... A Markov process is a stochastic process which satisfies the Markov property with respect to its natural filtration. In the ...
*  Tamer Başar
... stochastic control; estimation theory; stochastic processes; and mathematical economics. Tamer Başar was elected in 2000 as a ... For seminal contributions to dynamic games, stochastic and risk-sensitive control, control of networks, and hierarchical ...
*  Uniform integrability
ISBN 0-387-22833-0. Bass, Richard F. (2011). Stochastic Processes. Cambridge: Cambridge University Press. pp. 356-357. ISBN 978 ...
*  Stein's method
Barbour, A. D. & Brown, T. C. (1992). "Stein's method and point process approximation". Stochastic Processes and their ... such as Gaussian processes by Barbour (1990), the binomial distribution by Ehm (1991), Poisson processes by Barbour and Brown ( ... If A {\displaystyle {\mathcal {A}}} is the generator of a Markov process ( Z t ) t ≥ 0 {\displaystyle (Z_{t})_{t\geq 0}} (see ... ISBN 978-1-43983-574-6. Barbour, A. D. (1988). "Stein's method and Poisson process convergence". Journal of Applied Probability ...
*  Conditional expectation
doi:10.1090/s0002-9904-1953-09662-8. J. L. Doob (1953). Stochastic Processes. John Wiley & Sons. ISBN 0-471-52369-0. Olav ... Probability and Random Processes (3rd ed.). Oxford University Press. ISBN 0-19-857222-0. , pages 67-69 Ushakov, N.G. (2001) [ ... Replacing this limiting process by the Radon-Nikodym derivative yields an analogous definition that works more generally. ...
*  Chinese restaurant process
In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at ... The Generalized Chinese Restaurant Process is closely related to Pitman-Yor process. These processes have been used in many ... The Chinese restaurant process is closely connected to Dirichlet processes and Pólya's urn scheme, and therefore useful in ... At time n, the n customers have been partitioned among m ≤ n tables (or blocks of the partition). The results of this process ...
*  Regenerative process
A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process ... Else, the process is called a delayed regenerative process. Renewal processes are regenerative processes, with T1 being the ... a regenerative process is a class of stochastic process with the property that certain portions of the process can be treated ... These time point may themselves be determined by the evolution of the process. That is to say, the process {X(t), t ≥ 0} is a ...
*  Kolmogorov's zero-one law
ISBN 978-0-521-66349-6. . Brzezniak, Zdzislaw; Zastawniak, Thomasz (2000). Basic Stochastic Processes. Springer. ISBN 3-540- ...
*  Uncertainty quantification
Module 1: Gaussian process modeling for the computer model To address the issue from lack of simulation results, the computer ... Uncertainty quantification in stochastic systems using polynomial chaos expansion, International Journal of Applied Mechanics, ... Module 2: Gaussian process modeling for the discrepancy function Similarly with the first module, the discrepancy function is ... For example, the dimensions of a work piece in a process of manufacture may not be exactly as designed and instructed, which ...
*  Quantum probability
... was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes. ... "Quantum stochastic processes". Publ. Res. Inst. Math. Sci. 18 (1): 97-133. doi:10.2977/prims/1195184017. R.L. Hudson, K.R. ... Some recent advances are based on quantum filtering and feedback control theory as applications of quantum stochastic calculus ... for introduction or Belavkin, 1970s) gives the natural description of the measurement process. This new framework encapsulates ...