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Stopping Rule
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”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a stopping rule, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time. Stopping times occur in decision theory, and the optional stopping theorem is an important result in this context. Stopping times are also frequently applied in mathematical proofs to “tame the continuum of time”, as Chung put it in his book (1982). Definition Discrete time Let \tau be a random variable, which is defined on the filtered probability space (\Omega, \mathcal F, (\mathcal F_n)_, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Martingale
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale. Local martingales are essential in stochastic analysis (see Itō calculus, semimartingale, and Girsanov theorem). Definition Let (\Omega,F,P) be a probability space; let F_*=\ be a filtration of F; let X: adapted stochastic process on the set S. Then X is called an F_*-local martingale if there exists a sequence of F_*-stopping rule">stopping times \tau_k : \Omega \to [0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas S
Thomas may refer to: People * List of people with given name Thomas * Thomas (name) * Thomas (surname) * Saint Thomas (other) * Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church * Thomas the Apostle * Thomas (bishop of the East Angles) (fl. 640s–650s), medieval Bishop of the East Angles * Thomas (Archdeacon of Barnstaple) (fl. 1203), Archdeacon of Barnstaple * Thomas, Count of Perche (1195–1217), Count of Perche * Thomas (bishop of Finland) (1248), first known Bishop of Finland * Thomas, Earl of Mar (1330–1377), 14th-century Earl, Aberdeen, Scotland Geography Places in the United States * Thomas, Illinois * Thomas, Indiana * Thomas, Oklahoma * Thomas, Oregon * Thomas, South Dakota * Thomas, Virginia * Thomas, Washington * Thomas, West Virginia * Thomas County (other) * Thomas Township (other) Elsewhere * Thomas Glacier (Greenland) Arts, entertainment, and media * ''Thomas'' (Burton novel) 1969 novel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 detect as soon as possible when the probabilistic properties of the process have changed. This is a type of change detection problem. An example case is to detect the change in the drift parameter of a Wiener process.Shiryaev (2007) page 208 See also *Compound Poisson process A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisso ... Notes References * * * * Kolmogorov, A. N., Prokhorov, Yu. V. and Shiryaev, A. N. (1990). Methods of detecting spontaneously occurring effects. Proc. Steklov Inst. Math. 1, 1–21. Stochastic processes Optimal decisions {{probability-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definition Let * (\Omega, \mathcal, \mathbb) be a probability space; * (\mathbb, \mathcal) be a measurable space; * X : ,_+_\infty)_\times_\Omega_\to_\mathbb_be_a_stochastic_process; *_\tau_:_\Omega_\to_[0,_+_\infty/math>_be_a_stopping_rule">stopping_time_with_respect_to_some__be_a_stopping_rule.html"__"title=",_+_\infty.html"_;"title=",_+_\infty)_\times_\Omega_\to_\mathbb_be_a_stochastic_process; *_\tau_:_\Omega_\to_[0,_+_\infty">,_+_\infty)_\times_\Omega_\to_\mathbb_be_a_stochastic_process; *_\tau_:_\Omega_\to_[0,_+_\infty/math>_be_a_stopping_rule">stopping_time_with_respect_to_some_filtration_(abstract_algebra)">filtration_\_of_\mathcal. Then_the_stopped_process_X^_is_defined_for_t_\geq_0_and_\omega_\in_\Omega_by :X_^_(\omega)_:=_X__(\omega). _Examples _Gambling Consider_a_Gambling.html" "title="filtration_(abstract_algebra).html" "tit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secretary Problem
The secretary problem demonstrates a scenario involving optimal stopping theory For French translation, secover storyin the July issue of ''Pour la Science'' (2009). that is studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem. The basic form of the problem is the following: imagine an administrator who wants to hire the best secretary out of n rankable applicants for a position. The applicants are interviewed one by one in random order. A decision about each particular applicant is to be made immediately after the interview. Once rejected, an applicant cannot be recalled. During the interview, the administrator gains information sufficient to rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants. The question is about the optimal strategy ( sto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odds Algorithm
The odds algorithm (or Bruss algorithm) is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the ''odds strategy'', and the importance of the odds strategy lies in its optimality, as explained below. The odds algorithm applies to a class of problems called ''last-success'' problems. Formally, the objective in these problems is to maximize the probability of identifying in a sequence of sequentially observed independent events the last event satisfying a specific criterion (a "specific event"). This identification must be done at the time of observation. No revisiting of preceding observations is permitted. Usually, a specific event is defined by the decision maker as an event that is of true interest in the view of "stopping" to take a well-defined action. Such problems are encountered in several situations. Examples Two different situations exemplify the interes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Stopping
In mathematics, the theory of optimal stopping or early stopping : (For French translation, secover storyin the July issue of ''Pour la Science'' (2009).) is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming. Definition Discrete time case Stopping rule problems are associated with two objects: # A sequence of random variables X_1, X_2, \ldots, whose joint distribution is something assumed to be known # A sequence of 'reward' functions (y_i)_ which depend on the observed values of the random variables in 1: #: y_i=y_i (x_1, \ldots ,x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequential Analysis
In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined stopping rule as soon as significant results are observed. Thus a conclusion may sometimes be reached at a much earlier stage than would be possible with more classical hypothesis testing or estimation, at consequently lower financial and/or human cost. History The method of sequential analysis is first attributed to Abraham Wald with Jacob Wolfowitz, W. Allen Wallis, and Milton Friedman while at Columbia University's Statistical Research Group as a tool for more efficient industrial quality control during World War II. Its value to the war effort was immediately recognised, and led to its receiving a "restricted" classification. At the same time, George Barnard led a group working on optimal stopping in Great Britain. Ano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996. biology,H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988. ecology,H. Thompson. Spatial point processes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adapted Process
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 that ''X'' is adapted if and only if, for every realisation and every ''n'', ''Xn'' is known at time ''n''. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process. Definition Let * (\Omega, \mathcal, \mathbb) be a probability space; * I be an index set with a total order \leq (often, I is \mathbb, \mathbb_0, , T/math> or filtration of the sigma algebra \mathcal; * (S,\Sigma) be a measurable space, the ''state space''; * X: I \times \Omega \to S be a stochastic process. The process X is said to be adapted to the filtration \left(\mathcal_i\right)_ if the random variable X_i: \Omega \to S is a (\mathcal_i, \Sigma)-measurable function for each i \in I. Examples Consider a stochastic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 are also examples of hitting times. Definitions Let ''T'' be an ordered index set such as the natural numbers, N, the non-negative real numbers, , +∞), or a subset of these; elements ''t'' ∈ ''T'' can be thought of as "times". Given a probability space (Ω, Σ, Pr) and a measurable space">measurable state space ''S'', let ''X'' : Ω × ''T'' → ''S'' be a stochastic process, and let ''A'' be a measurable set, measurable subset of the state space ''S''. Then the first hit time ''τ''''A'' : Ω → [0, +∞] is the random variable defined by :\tau_A (\omega) := \inf \. The first exit time (from ''A'') is defined to be the first hit time for ''S'' \ ''A'', the complement (set theory), c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
