Stochastic Processes: Processes that incorporate some element of randomness, used particularly to refer to a time series of random variables.Computer Simulation: Computer-based representation of physical systems and phenomena such as chemical processes.Models, Biological: Theoretical representations that simulate the behavior or activity of biological processes or diseases. For disease models in living animals, DISEASE MODELS, ANIMAL is available. Biological models include the use of mathematical equations, computers, and other electronic equipment.Models, Statistical: Statistical formulations or analyses which, when applied to data and found to fit the data, are then used to verify the assumptions and parameters used in the analysis. Examples of statistical models are the linear model, binomial model, polynomial model, two-parameter model, etc.Markov Chains: A stochastic process such that the conditional probability distribution for a state at any future instant, given the present state, is unaffected by any additional knowledge of the past history of the system.Models, Genetic: Theoretical representations that simulate the behavior or activity of genetic processes or phenomena. They include the use of mathematical equations, computers, and other electronic equipment.Biological Evolution: The process of cumulative change over successive generations through which organisms acquire their distinguishing morphological and physiological characteristics.Population Dynamics: The pattern of any process, or the interrelationship of phenomena, which affects growth or change within a population.Probability: The study of chance processes or the relative frequency characterizing a chance process.Ecosystem: A functional system which includes the organisms of a natural community together with their environment. (McGraw Hill Dictionary of Scientific and Technical Terms, 4th ed)Models, Theoretical: Theoretical representations that simulate the behavior or activity of systems, processes, or phenomena. They include the use of mathematical equations, computers, and other electronic equipment.Monte Carlo Method: In statistics, a technique for numerically approximating the solution of a mathematical problem by studying the distribution of some random variable, often generated by a computer. The name alludes to the randomness characteristic of the games of chance played at the gambling casinos in Monte Carlo. (From Random House Unabridged Dictionary, 2d ed, 1993)Evolution, Molecular: The process of cumulative change at the level of DNA; RNA; and PROTEINS, over successive generations.Algorithms: A procedure consisting of a sequence of algebraic formulas and/or logical steps to calculate or determine a given task.Phylogeny: The relationships of groups of organisms as reflected by their genetic makeup.Selection, Genetic: Differential and non-random reproduction of different genotypes, operating to alter the gene frequencies within a population.Time Factors: Elements of limited time intervals, contributing to particular results or situations.Genetic Variation: Genotypic differences observed among individuals in a population.Mutation: Any detectable and heritable change in the genetic material that causes a change in the GENOTYPE and which is transmitted to daughter cells and to succeeding generations.Kinetics: The rate dynamics in chemical or physical systems.

*  Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control | Optimization | Discrete Mathematics |...

Appendix E. Markov Processes.. Answers to Selected Exercises.. References.. Frequently Used Notation. ... Stochastic search and optimization techniques are used in a vast number of areas, including aerospace, medicine, transportation ... Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control is a graduate-level introduction to the ... Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. James C. Spall ...

*  Stochastic Processes : Poisson Process and Markov Chains

Suppose that shocks occur according to a Poisson process with rate A, 0. Also suppose that each shock independently causes the ... Stochastic Processes : Poisson Process and Markov Chains. Add. Remove. 1. Suppose that shocks occur according to a Poisson ... PS If you are not well versed in stochastic processes then please do not sign this problem out. ... 2. Let {N(t)},0 be a nonhomogeneous Poisson process with intensity function A(t) , 0. Then the mean value function 4u(t) is ...

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*  Law (stochastic processes) - Wikipedia

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 - Wikipedia

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) - Wikipedia

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) - Wikipedia

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) - Wikipedia

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 - Wikipedia

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 - Wikipedia

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 - Wikipedia

"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 - Wikipedia

... 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 - Wikipedia

... 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 - Wikipedia

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 - Wikipedia

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 ...

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MATH 4740 - Stochastic Processes. spring 2018. Lawler, Gregory F., Introduction to Stochastic Processes, CRC Press LLC, 2006 ( ... MATH 7720 - Topics in Stochastic Processes: Random Walks on Amenable Groups. spring 2018. (optional) Woess, Wolfgang, Random ...

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Doob decomposition theorem: In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero. The theorem was proved by and is named for Joseph L.Interval boundary element method: Interval boundary element method is classical boundary element method with the interval parameters.
Matrix model: == Mathematics and physics ==Inverse probability weighting: Inverse probability weighting is a statistical technique for calculating statistics standardized to a population different from that in which the data was collected. Study designs with a disparate sampling population and population of target inference (target population) are common in application.Vladimir Andreevich Markov: Vladimir Andreevich Markov (; May 8, 1871 – January 18, 1897) was a Russian mathematician, known for proving the Markov brothers' inequality with his older brother Andrey Markov. He died of tuberculosis at the age of 25.Matrix population models: Population models are used in population ecology to model the dynamics of wildlife or human populations. Matrix population models are a specific type of population model that uses matrix algebra.Negative probability: The probability of the outcome of an experiment is never negative, but quasiprobability distributions can be defined that allow a negative probability for some events. These distributions may apply to unobservable events or conditional probabilities.EcosystemVon Neumann regular ring: In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R such that . To avoid the possible confusion with the regular rings and regular local rings of commutative algebra (which are unrelated notions), von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left module is flat.Monte Carlo methods for option pricing: In mathematical finance, a Monte Carlo option model uses Monte Carlo methods Although the term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he asked about the results of dropping a needle randomly on a striped floor or table. See Buffon's needle.Molecular evolution: Molecular evolution is a change in the sequence composition of cellular molecules such as DNA, RNA, and proteins across generations. The field of molecular evolution uses principles of evolutionary biology and population genetics to explain patterns in these changes.Clonal Selection Algorithm: In artificial immune systems, Clonal selection algorithms are a class of algorithms inspired by the clonal selection theory of acquired immunity that explains how B and T lymphocytes improve their response to antigens over time called affinity maturation. These algorithms focus on the Darwinian attributes of the theory where selection is inspired by the affinity of antigen-antibody interactions, reproduction is inspired by cell division, and variation is inspired by somatic hypermutation.Branching order of bacterial phyla (Gupta, 2001): There are several models of the Branching order of bacterial phyla, one of these was proposed in 2001 by Gupta based on conserved indels or protein, termed "protein signatures", an alternative approach to molecular phylogeny. Some problematic exceptions and conflicts are present to these conserved indels, however, they are in agreement with several groupings of classes and phyla.Selection (relational algebra): In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written asTemporal analysis of products: Temporal Analysis of Products (TAP), (TAP-2), (TAP-3) is an experimental technique for studyingGenetic variation: right|thumbSilent mutation: Silent mutations are mutations in DNA that do not significantly alter the phenotype of the organism in which they occur. Silent mutations can occur in non-coding regions (outside of genes or within introns), or they may occur within exons.Burst kinetics: Burst kinetics is a form of enzyme kinetics that refers to an initial high velocity of enzymatic turnover when adding enzyme to substrate. This initial period of high velocity product formation is referred to as the "Burst Phase".

(1/2792) A processive single-headed motor: kinesin superfamily protein KIF1A.

A single kinesin molecule can move "processively" along a microtubule for more than 1 micrometer before detaching from it. The prevailing explanation for this processive movement is the "walking model," which envisions that each of two motor domains (heads) of the kinesin molecule binds coordinately to the microtubule. This implies that each kinesin molecule must have two heads to "walk" and that a single-headed kinesin could not move processively. Here, a motor-domain construct of KIF1A, a single-headed kinesin superfamily protein, was shown to move processively along the microtubule for more than 1 micrometer. The movement along the microtubules was stochastic and fitted a biased Brownian-movement model.  (+info)

(2/2792) Influence of sampling on estimates of clustering and recent transmission of Mycobacterium tuberculosis derived from DNA fingerprinting techniques.

The availability of DNA fingerprinting techniques for Mycobacterium tuberculosis has led to attempts to estimate the extent of recent transmission in populations, using the assumption that groups of tuberculosis patients with identical isolates ("clusters") are likely to reflect recently acquired infections. It is never possible to include all cases of tuberculosis in a given population in a study, and the proportion of isolates found to be clustered will depend on the completeness of the sampling. Using stochastic simulation models based on real and hypothetical populations, the authors demonstrate the influence of incomplete sampling on the estimates of clustering obtained. The results show that as the sampling fraction increases, the proportion of isolates identified as clustered also increases and the variance of the estimated proportion clustered decreases. Cluster size is also important: the underestimation of clustering for any given sampling fraction is greater, and the variability in the results obtained is larger, for populations with small clusters than for those with the same number of individuals arranged in large clusters. A considerable amount of caution should be used in interpreting the results of studies on clustering of M. tuberculosis isolates, particularly when sampling fractions are small.  (+info)

(3/2792) A fast, stochastic threading algorithm for proteins.

MOTIVATION: Sequences for new proteins are being determined at a rapid rate, as a result of the Human Genome Project, and related genome research. The ability to predict the three-dimensional structure of proteins from sequence alone would be useful in discovering and understanding their function. Threading, or fold recognition, aims to predict the tertiary structure of a protein by aligning its amino acid sequence with a large number of structures, and finding the best fit. This approach depends on obtaining good performance from both the scoring function, which simulates the free energy for given trial alignments, and the threading algorithm, which searches for the lowest-score alignment. It appears that current scoring functions and threading algorithms need improvement. RESULTS: This paper presents a new threading algorithm. Numerical tests demonstrate that it is more powerful than two popular approximate algorithms, and much faster than exact methods.  (+info)

(4/2792) Declining survival probability threatens the North Atlantic right whale.

The North Atlantic northern right whale (Eubalaena glacialis) is considered the most endangered large whale species. Its population has recovered only slowly since the cessation of commercial whaling and numbers about 300 individuals. We applied mark-recapture statistics to a catalog of photographically identified individuals to obtain the first statistically rigorous estimates of survival probability for this population. Crude survival decreased from about 0.99 per year in 1980 to about 0.94 in 1994. We combined this survival trend with a reported decrease in reproductive rate into a branching process model to compute population growth rate and extinction probability. Population growth rate declined from about 1. 053 in 1980 to about 0.976 in 1994. Under current conditions the population is doomed to extinction; an upper bound on the expected time to extinction is 191 years. The most effective way to improve the prospects of the population is to reduce mortality. The right whale is at risk from entanglement in fishing gear and from collisions with ships. Reducing this human-caused mortality is essential to the viability of this population.  (+info)

(5/2792) FORESST: fold recognition from secondary structure predictions of proteins.

MOTIVATION: A method for recognizing the three-dimensional fold from the protein amino acid sequence based on a combination of hidden Markov models (HMMs) and secondary structure prediction was recently developed for proteins in the Mainly-Alpha structural class. Here, this methodology is extended to Mainly-Beta and Alpha-Beta class proteins. Compared to other fold recognition methods based on HMMs, this approach is novel in that only secondary structure information is used. Each HMM is trained from known secondary structure sequences of proteins having a similar fold. Secondary structure prediction is performed for the amino acid sequence of a query protein. The predicted fold of a query protein is the fold described by the model fitting the predicted sequence the best. RESULTS: After model cross-validation, the success rate on 44 test proteins covering the three structural classes was found to be 59%. On seven fold predictions performed prior to the publication of experimental structure, the success rate was 71%. In conclusion, this approach manages to capture important information about the fold of a protein embedded in the length and arrangement of the predicted helices, strands and coils along the polypeptide chain. When a more extensive library of HMMs representing the universe of known structural families is available (work in progress), the program will allow rapid screening of genomic databases and sequence annotation when fold similarity is not detectable from the amino acid sequence. AVAILABILITY: FORESST web server at http://absalpha.dcrt.nih.gov:8008/ for the library of HMMs of structural families used in this paper. FORESST web server at http://www.tigr.org/ for a more extensive library of HMMs (work in progress). CONTACT: valedf@tigr.org; munson@helix.nih.gov; garnier@helix.nih.gov  (+info)

(6/2792) A quantitative method for the detection of edges in noisy time-series.

A modification of the edge detector of Chung & Kennedy is proposed in which the output provides confidence limits for the presence or absence of sharp edges (steps) in the input waveform. Their switching method with forward and backward averaging windows is retained, but the output approximates an ideal output function equal to the difference in these averages divided by the standard deviation of the noise. Steps are associated with peak output above a pre-set threshold. Formulae for the efficiency and reliability of this ideal detector are derived for input waveforms with Gaussian white noise and sharp edges, and serve as benchmarks for the switching edge detector. Efficiency is kept high if the threshold is a fixed fraction of the step size of interest relative to noise, and reliability is improved by increasing the window width W to reduce false output. For different steps sizes D, the window width for fixed efficiency and reliability scales as 1/D2. Versions with weighted averaging (flat, ramp, triangular) or median averaging but the same window width perform similarly. Binned above-threshold output is used to predict the locations and signs of detected steps, and simulations show that efficiency and reliability are close to ideal. Location times are accurate to order square root of W. Short pulses generate reduced output if the number of data points in the pulse is less than W. They are optimally detected by choosing W as above and collecting data at a rate such that the pulse contains approximately W data points. A Fortran program is supplied.  (+info)

(7/2792) Strength of a weak bond connecting flexible polymer chains.

Bond dissociation under steadily rising force occurs most frequently at a time governed by the rate of loading (Evans and Ritchie, 1997 Biophys. J. 72:1541-1555). Multiplied by the loading rate, the breakage time specifies the force for most frequent failure (called bond strength) that obeys the same dependence on loading rate. The spectrum of bond strength versus log(loading rate) provides an image of the energy landscape traversed in the course of unbonding. However, when a weak bond is connected to very compliant elements like long polymers, the load applied to the bond does not rise steadily under constant pulling speed. Because of nonsteady loading, the most frequent breakage force can differ significantly from that of a bond loaded at constant rate through stiff linkages. Using generic models for wormlike and freely jointed chains, we have analyzed the kinetic process of failure for a bond loaded by pulling the polymer linkages at constant speed. We find that when linked by either type of polymer chain, a bond is likely to fail at lower force under steady separation than through stiff linkages. Quite unexpectedly, a discontinuous jump can occur in bond strength at slow separation speed in the case of long polymer linkages. We demonstrate that the predictions of strength versus log(loading rate) can rationalize conflicting results obtained recently for unfolding Ig domains along muscle titin with different force techniques.  (+info)

(8/2792) Visual form created solely from temporal structure.

In several experiments, it was found that global perception of spatial form can arise exclusively from unpredictable but synchronized changes among local features. Within an array of nonoverlapping apertures, contours move in one of two directions, with direction reversing randomly over time. When contours within a region of the array reverse directions in synchrony, they stand out conspicuously from the rest of the array where direction reversals are unsynchronized. Clarity of spatial structure from synchronized change depends on the rate of motion reversal and on the proportion of elements reversing direction in synchrony. Evidently, human vision is sensitive to the rich temporal structure in these stochastic events.  (+info)



Brownian

  • The law of the process X is then defined to be the pushforward measure L X := ( Φ X ) ∗ ( P ) = P ∘ Φ X − 1 {\displaystyle {\mathcal {L}}_{X}:=\left(\Phi _{X}\right)_{*}(\mathbf {P} )=\mathbf {P} \circ \Phi _{X}^{-1}} on ST. The law of standard Brownian motion is classical Wiener measure. (wikipedia.org)
  • Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. (wikipedia.org)
  • Standard Brownian motion on Rn, which satisfies the stochastic differential equation dXt = dBt, has generator ½Δ, where Δ denotes the Laplace operator. (wikipedia.org)
  • A geometric Brownian motion on R, which satisfies the stochastic differential equation dXt = rXt dt + αXt dBt, has generator A f ( x ) = r x f ′ ( x ) + 1 2 α 2 x 2 f ″ ( x ) . {\displaystyle Af(x)=rxf'(x)+{\frac {1}{2}}\alpha ^{2}x^{2}f''(x). (wikipedia.org)
  • Birth-death process Branching process Branching random walk Brownian bridge Brownian motion Chinese restaurant process CIR process Continuous stochastic process Cox process Dirichlet processes Finite-dimensional distribution First Passage Time Galton-Watson process Gamma process Gaussian process - a process where all linear combinations of coordinates are normally distributed random variables. (wikipedia.org)
  • In the mathematical theory of probability, the Wiener process, named after Norbert Wiener, is a stochastic process used in modeling various phenomena, including Brownian motion and fluctuations in financial markets. (wikipedia.org)
  • Stochastic integration with respect to continuous local martingales, Ito's formula, Levy's Characterization of Brownian motion, Girsanov transformation, Stochastic Differential Equations with Lipschitz Coefficients. (uconn.edu)
  • Looking for advisable mathematical properties (for instance, the stationarity of the increments), the corresponding self-similar stochastic processes are represented in terms of fractional Brownian motions with stochastic variance, whose profile is modelled by using the M -Wright density or the Lévy stable density. (royalsocietypublishing.org)
  • Moreover, normal diffusion is also associated with the Gaussian PDF for particle displacement, and, by using the stochastic process terminology, normal diffusion is also called Brownian motion (BM). (royalsocietypublishing.org)
  • Two important examples of Markov processes are the Wiener process, also known as the Brownian motion process, and the Poisson process, which are considered the most important and central stochastic processes in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after 1906, in various settings. (wikipedia.org)

differential

  • In mathematics - specifically, in stochastic analysis - the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process. (wikipedia.org)
  • The Ornstein-Uhlenbeck process on R, which satisfies the stochastic differential equation dXt = θ (μ − Xt) dt + σ dBt, has generator A f ( x ) = θ ( μ − x ) f ′ ( x ) + σ 2 2 f ″ ( x ) . {\displaystyle Af(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x). (wikipedia.org)
  • Stochastic Differential Equations: An Introduction with Applications (Sixth ed. (wikipedia.org)
  • Numerical solution of stochastic differential equations. (wikipedia.org)
  • However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation. (wikipedia.org)
  • Stochastic Integration and Differential Equations. (wikipedia.org)

mathematics

  • 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. (wikipedia.org)
  • 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. (wikipedia.org)
  • In the mathematics of probability, a stochastic process is a random function. (wikipedia.org)
  • Karhunen-Loève theorem Lévy process Local time (mathematics) Loop-erased random walk Markov processes are those in which the future is conditionally independent of the past given the present. (wikipedia.org)
  • In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. (wikipedia.org)
  • In 1987 he obtained his Ph.D. degree in Physics and Mathematics from the Institute of Applied Physics, Academy of Science, Gorky, for a dissertation on the stochastic dynamics of structure in nonequilibrium media. (oup.com)

Itō

  • Sample continuity is the appropriate notion of continuity for processes such as Itō diffusions. (wikipedia.org)

mathematical

  • In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. (wikipedia.org)
  • Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. (wikipedia.org)
  • The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. (wikipedia.org)
  • The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. (wikipedia.org)
  • 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. (wikipedia.org)
  • 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. (wikipedia.org)
  • 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. (wikipedia.org)
  • In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems in signal processing and related fields. (wikipedia.org)
  • Stochastic Processes and their Applications is a monthly peer-reviewed scientific journal published by Elsevier for the Bernoulli Society for Mathematical Statistics and Probability. (wikipedia.org)
  • The book should be of interest to students and researchers in probability, stochastic modelling, and mathematical statistics. (worldcat.org)
  • like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. (wikipedia.org)

martingales

  • After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov processes, and stationary processes, Doob realized that there was a real need for a book showing what is known about the various types of stochastic processes, so he wrote the book Stochastic Processes. (wikipedia.org)

Wiener

  • This is used in the context of World War 2 defined by people like Norbert Wiener, in (stochastic) control theory, radar, signal detection, tracking, etc. (wikipedia.org)
  • A formula for the conditional probability distribution of the extremum of the Wiener process and a sketch of its proof appears in work of H. J. Kusher published in 1964. (wikipedia.org)
  • If a Wiener stochastic process is chosen as a model for the objective function, it is possible to calculate the probability distribution of the model extreme points inside each interval, conditioned by the known values at the interval boundaries. (wikipedia.org)

Poisson

  • b) FindP{NrrnT=t}. (c) Describe how the results from part (b) could have alternatively been determined by considering an appropriate decomposition of the original Poisson process. (brainmass.com)
  • Poisson Processes and Markov Chains are investigated. (brainmass.com)
  • An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson process. (wikipedia.org)
  • Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP (Markov renewal process). (wikipedia.org)
  • citation needed] In the 1990s the method was adapted to a variety of distributions, such as Gaussian processes by Barbour (1990), the binomial distribution by Ehm (1991), Poisson processes by Barbour and Brown (1992), the Gamma distribution by Luk (1994), and many others. (wikipedia.org)

Gaussian

  • Recently, Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein-Uhlenbeck process, driven by a positive Levy process without Gaussian component. (repec.org)
  • Inference with non-Gaussian Ornstein-Uhlenbeck processes for stochastic volatility ," Journal of Econometrics , Elsevier, vol. 134(2), pages 605-644, October. (repec.org)
  • This can lead, for instance, to a Gaussian process such as the fractional BM (fBM) that generalizes the standard BM. (royalsocietypublishing.org)
  • A physical idea useful to model anomalous diffusion is related to time subordination of the Gaussian process. (royalsocietypublishing.org)

calculus

  • This paper presents a graphical representation for the stochastic pi-calculus, which builds on previous formal and informal notations. (microsoft.com)
  • The graphical representation can also be used as a front end to a simulator for the stochastic pi-calculus. (microsoft.com)
  • The Doob-Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. (wikipedia.org)

1962

  • Continuous signal Parzen, E. (1962) Stochastic Processes, Holden-Day. (wikipedia.org)

stationary

  • every stationary process in N outcomes is a Bernoulli scheme, and vice versa. (wikipedia.org)
  • Special cases include stationary processes, also called time-homogeneous. (wikipedia.org)
  • Stochastic Process Rare Event Sampling (SPRES) is a Rare Event Sampling method in computer simulation, designed specifically for non-equilibrium calculations, including those for which the rare-event rates are time-dependent (non-stationary process). (wikipedia.org)

thermodynamics

  • ED: Taschenbuch / Paperback], [PU: Springer, Berlin], AUSFÜHRLICHERE BESCHREIBUNG: This book brings theories in nonlinear dynamics, stochastic processes, irreversible thermodynamics, physical chemistry and biochemistry together in an introductory but formal and comprehensive manner. (eurobuch.de)

displaystyle

  • Similarly, the graph of the Ornstein-Uhlenbeck process has generator A f ( t , x ) = ∂ f ∂ t ( t , x ) + θ ( μ − x ) ∂ f ∂ x ( t , x ) + σ 2 2 ∂ 2 f ∂ x 2 ( t , x ) . {\displaystyle Af(t,x)={\frac {\partial f}{\partial t}}(t,x)+\theta (\mu -x){\frac {\partial f}{\partial x}}(t,x)+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x). (wikipedia.org)
  • displaystyle {\lim _{y\to x}u(y)}=g(x),&x\in \partial D.\end{cases}}\quad {\mbox{(P1)}}} The idea of the stochastic method for solving this problem is as follows. (wikipedia.org)
  • Let Z {\displaystyle Z} be a cadlag submartingale of class D. Then there exists a unique, increasing, predictable process A {\displaystyle A} with A 0 = 0 {\displaystyle A_{0}=0} such that M t = Z t − A t {\displaystyle M_{t}=Z_{t}-A_{t}} is a uniformly integrable martingale. (wikipedia.org)

numerical

  • Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. (wikipedia.org)

probability density functions

  • its L2 Hermitian adjoint is used in evolution equations such as the Fokker-Planck equation (which describes the evolution of the probability density functions of the process). (wikipedia.org)

convergence

  • The relationships between the various types of continuity of stochastic processes are akin to the relationships between the various types of convergence of random variables. (wikipedia.org)
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discrete-time processes

  • See also Category:Stochastic processes Basic affine jump diffusion Bernoulli process: discrete-time processes with two possible states. (wikipedia.org)
  • Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are called continuous-time random walks. (wikipedia.org)

Markov Chains

  • Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, exchange rates of currencies, storage systems such as dams, and population growths of certain animal species. (wikipedia.org)

estimation

  • The Smoothing problem (not to be confused with smoothing in signal processing and other contexts) refers to Recursive Bayesian estimation also known as Bayes filter is the problem of estimating an unknown probability density function recursively over time using incremental incoming measurements. (wikipedia.org)
  • Both the smoothing problem (in sense of estimation) and the filtering problem (in sense of estimation) are often confused with smoothing and filtering in other contexts (especially non-stochastic signal processing, often a name of various types of convolution). (wikipedia.org)
  • Filtering is causal but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on serial incremental observations. (wikipedia.org)
  • Filtering is the estimation of a (hidden) time-series process based on serial incremental observations. (wikipedia.org)

ISBN

  • ISBN 0-19-920613-9 (Entry for "continuous process") Kloeden, Peter E. (wikipedia.org)

representation

  • The algorithm is based on Markov chain Monte Carlo methods and we use a series representation of Levy processes. (repec.org)
  • Inference for such models is complicated by the fact that parameter changes will often induce a change of dimension in the representation of the process and the associated problem of overconditioning. (repec.org)

continuity

  • Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to analyze. (wikipedia.org)
  • Continuous time random processes, Kolmogorov's continuity theorem. (uconn.edu)

telegraph

  • Even worse, even if A is an event, P(A) can be strictly positive even if P(At) = 0 for every t ∈ T. This is the case, for example, with the telegraph process. (wikipedia.org)

Filtering

  • For example, moving average, low-pass filtering, convolution with a kernel, or blurring using Laplace filters in image processing. (wikipedia.org)
  • Filtering is causal, whereas smoothing is batch processing of the given data. (wikipedia.org)

define

  • 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 is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). (wikipedia.org)

repeatedly

  • These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. (wikipedia.org)

outcomes

  • A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. (wikipedia.org)

simulation

  • Furthermore, Markov processes are the basis for general stochastic simulation methods known as Gibbs sampling and Markov Chain Monte Carlo, are used for simulating random objects with specific probability distributions, and have found extensive application in Bayesian statistics. (wikipedia.org)

Examples

  • Random walks on integers and the gambler's ruin problem are examples of Markov processes. (wikipedia.org)
  • These two processes are 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. (wikipedia.org)

applications

  • They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. (wikipedia.org)
  • Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. (wikipedia.org)
  • The principal focus of this journal is theory and applications of stochastic processes. (wikipedia.org)

Furthermore

  • Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. (wikipedia.org)

continuous time

  • The entire process is not Markovian, i.e., memoryless, as happens in a continuous time Markov chain/process (CTMC). (wikipedia.org)
  • Usually the term "Markov chain" is reserved for a process with a discrete set of times, i.e. a discrete-time Markov chain (DTMC), but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention. (wikipedia.org)

random

  • 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. (wikipedia.org)
  • A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. (wikipedia.org)
  • The process of branching requires that identical paths can be made to diverge from each other, such as by changing the seed in the computer's random number generator. (wikipedia.org)
  • This is a book on coupling, the method of establishing properties of random variables and processes (or any random things) through a joint construction on a common probability space. (worldcat.org)
  • In probability and statistics a Markov renewal process is a random process that generalizes the notion of Markov jump processes. (wikipedia.org)
  • Note the main difference between an MRP and a semi-Markov process is that the former is defined as a two-tuple of states and times, whereas the latter is the actual random process that evolves over time and any realisation of the process has a defined state for any given time. (wikipedia.org)

types

  • The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. (wikipedia.org)

analysis

  • Bayesian Analysis of Stochastic Volatility Models ," Journal of Business & Economic Statistics , American Statistical Association, vol. 20(1), pages 69-87, January. (repec.org)
  • Bayesian Analysis of Stochastic Volatility Models ," Journal of Business & Economic Statistics , American Statistical Association, vol. 12(4), pages 371-389, October. (repec.org)

CTMC

  • A semi-Markov process (defined in the above bullet point) where all the holding times are exponentially distributed is called a CTMC. (wikipedia.org)

time

  • An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. (wikipedia.org)
  • 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. (wikipedia.org)
  • 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. (wikipedia.org)
  • 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. (wikipedia.org)
  • 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 values, as contrasted with a discrete-time process for which the index variable takes only distinct values. (wikipedia.org)
  • Continuous-time stochastic volatility models are becoming a more and more popular way to describe moderate and high-frequency financial data. (repec.org)
  • Applied Materials' Sym3 etch chamber features hardware that provides pulsed energy at dual frequencies along with low residence time of reactant byproducts to allow for precise tuning of process parameters no matter what chemistry is needed. (semimd.com)
  • Two different modelling approaches related to time subordination are considered and unified in the framework of self-similar stochastic processes. (royalsocietypublishing.org)
  • 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. (wikipedia.org)
  • The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. continuous time: Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. (wikipedia.org)
  • While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. (wikipedia.org)

renewal process

  • So, if the states are ignored and we have a chain of iid times, then we have a renewal process. (wikipedia.org)

methods

  • Emphasis is on a broad description of the general methods and processes for the synthesis, modification and characterization of macromolecules. (springer.com)

index

  • It is implicit here that the index of the stochastic process is a continuous variable. (wikipedia.org)
  • A more restricted class of processes are the continuous stochastic processes: here the term often (but not always) implies both that the index variable is continuous and that sample paths of the process are continuous. (wikipedia.org)

expectation

  • Here x denotes the initial state of the process X, and Ex denotes expectation conditional upon the event that X starts at x. (wikipedia.org)

problem

  • PS If you are not well versed in stochastic processes then please do not sign this problem out. (brainmass.com)

Models

  • An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes is used. (repec.org)

distributions

  • Gauss-Markov process (cf. below) Girsanov's theorem Homogeneous processes: processes where the domain has some symmetry and the finite-dimensional probability distributions also have that symmetry. (wikipedia.org)
  • The comparison of the obtained distributions provides a criterion for selecting the interval in which the process should be iterated. (wikipedia.org)

model

  • The stochastic process is taken as a model of the objective function, assuming that the probability distribution of its extrema gives the best indication about extrema of the objective function. (wikipedia.org)
  • A Markov chain is "a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. (wikipedia.org)
  • In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see Markov model). (wikipedia.org)

jump

  • Instead the process is Markovian only at the specified jump instants. (wikipedia.org)