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- Fluid Limits of Optimally Controlled Queueing Networks
- Diffusions, Markov Processes and Martingales. Volume 1, Foundations, 2nd Edition
- Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
- MA946 - Introduction to Graduate Probability
Many stochastic processes can be represented by time series. However, a stochastic process is by nature continuous while a time series is a set of observations indexed by integers. A stochastic process may involve several related random variables.
Fluid Limits of Optimally Controlled Queueing Networks
Many stochastic processes can be represented by time series. However, a stochastic process is by nature continuous while a time series is a set of observations indexed by integers.
A stochastic process may involve several related random variables. Common examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise , or the movement of a gas molecule. They have applications in many disciplines such as biology ,  chemistry ,  ecology ,  neuroscience ,  physics ,  image processing , signal processing ,  control theory ,  information theory ,  computer science ,  cryptography  and telecommunications.
Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse ,  and the Poisson process , used by A.
Erlang to study the number of phone calls occurring in a certain period of time. 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.
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. Historically, the index set was some subset of the real line , such as the natural numbers , giving the index set the interpretation of time. 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.
One common way of classification is by the cardinality of the index set and the state space. 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. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes.
If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process.
If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year as its earliest occurrence. The term stochastic process first appeared in English in a paper by Joseph Doob.
According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence in riding, running, striking, etc.
The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process , which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in The definition of a stochastic process varies,  but a stochastic process is traditionally defined as a collection of random variables indexed by some set.
The term random function is also used to refer to a stochastic or random process,    though sometimes it is only used when the stochastic process takes real values. Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.
A classic example of a random walk is known as the simple random walk , which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.
Almost surely , a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The Poisson process is a stochastic process that has different forms and definitions.
The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set.
This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. The homogeneous Poisson process can be defined and generalized in different ways.
It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.
Defined on the real line, the Poisson process can be interpreted as a stochastic process,   among other random objects. There are other ways to consider a stochastic process, with the above definition being considered the traditional one.
The state space is defined using elements that reflect the different values that the stochastic process can take. A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period.
The law of a stochastic process or a random variable is also called the probability law , probability distribution , or the distribution. The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.
Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations.
A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers.
A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process.
Two stochastic processes that are modifications of each other have the same finite-dimensional law  and they are said to be stochastically equivalent or equivalent. Instead of modification, the term version is also used,     however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.
Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space , [b] which means that the index set has a dense countable subset.
The concept of separability of a stochastic process was introduced by Joseph Doob ,. Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.
But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process on the real line , are also members of this space. In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.
Markov processes are stochastic processes, traditionally in discrete or continuous time , that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process. The Brownian motion process and the Poisson process in one dimension are both examples of Markov processes  in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.
A Markov chain is a type of Markov process that has either discrete state space or discrete index set often representing time , but the precise definition of a Markov chain varies.
Markov processes form an important class of stochastic processes and have applications in many areas. A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value.
In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes.
Martingales are usually defined to be real-valued,    but they can also be complex-valued  or even more general. A symmetric random walk and a Wiener process with zero drift are both examples of martingales, respectively, in discrete and continuous time.
Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process on the real line resulting in a martingale called the compensated Poisson process.
Martingales mathematically formalize the idea of a fair game,  and they were originally developed to show that it is not possible to win a fair game. Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.
In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field.
Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,   but very little analysis on them was done in terms of probability. After Cardano, Jakob Bernoulli [e] wrote Ars Conjectandi , which is considered a significant event in the history of probability theory. In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics , where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles.
Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius , most of the work had little or no randomness. At the International Congress of Mathematicians in Paris in , David Hilbert presented a list of mathematical problems , where his sixth problem asked for a mathematical treatment of physics and probability involving axioms.
In s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergei Bernstein , Aleksandr Khinchin , [g] and Andrei Kolmogorov. In Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung , [i] where Kolmogorov used measure theory to develop an axiomatic framework for probability theory.
The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics. After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas.
Also starting in the s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potential theory , with early ideas by Shizuo Kakutani and then later work by Joseph Doob. In Doob published his book Stochastic processes , which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Techniques and theory were developed to study Markov processes and then applied to martingales.
Conversely, methods from the theory of martingales were established to treat Markov processes. Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations. The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.
Although Khinchin gave mathematical definitions of stochastic processes in the s,   specific stochastic processes had already been discovered in different settings, such as the Brownian motion process and the Poisson process. The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. In Karl Pearson coined the term random walk while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields.
Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks. The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.
The French mathematician Louis Bachelier used a Wiener process in his thesis   in order to model price changes on the Paris Bourse , a stock exchange ,  without knowing the work of Thiele. It is commonly thought that Bachelier's work gained little attention and was forgotten for decades until it was rediscovered in the s by the Leonard Savage , and then become more popular after Bachelier's thesis was translated into English in
Diffusions, Markov Processes and Martingales. Volume 1, Foundations, 2nd Edition
Guodong Pang, Martin V. We consider a class of queueing processes represented by a Skorokhod problem coupled with a controlled point process. Posing a discounted control problem for such processes, we show that the optimal value functions converge, in the fluid limit, to the value of an analogous deterministic control problem for fluid processes. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Special Issues. Guodong Pang 1 and Martin V.
Now available in paperback, this celebrated book has been prepared with readers' needs in mind, giving a systematic treatment of the subject whilst retaining its vitality. The authors' aim is not o present the subject of Brownian motion as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of the theory of stochastic processes. Chapter III is a lively and readable treatment of the theory of Markov processes. Cambridge University Press has a long and honourable history of publishing in mathematics and counts many classics of the mathematical literature within its list.
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Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
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Lecturer: Stefan Adams. Term 1: Three lectures per week are scheduled for f2f teaching in MS. If the situation changes lectures will be online.
Энсей Танкадо - это Северная Дакота… Сьюзан попыталась расставить все фрагменты имеющейся у нее информации по своим местам. Если Танкадо - Северная Дакота, выходит, он посылал электронную почту самому себе… а это значит, что никакой Северной Дакоты не существует.
MA946 - Introduction to Graduate Probability
Немец схватил ее и нетерпеливо стянул с нее рубашку. Его толстые пальцы принялись методично, сантиметр за сантиметром, ощупывать ее тело. Росио упала на него сверху и начала стонать и извиваться в поддельном экстазе. Когда он перевернул ее на спину и взгромоздился сверху, она подумала, что сейчас он ее раздавит. Его массивная шея зажала ей рот, и Росио чуть не задохнулась.
- Вы что-то нашли. - Вроде. - У Соши был голос провинившегося ребенка. - Помните, я сказала, что на Нагасаки сбросили плутониевую бомбу. - Да, - ответил дружный хор голосов. - Так вот… - Соши шумно вздохнула.
Это нацарапал мой дружок… ужасно глупо, правда. Беккер не мог выдавить ни слова. Проваливай и умри. Он не верил своим глазам. Немец не хотел его оскорбить, он пытался помочь. Беккер посмотрел на ее лицо. В свете дневных ламп он увидел красноватые и синеватые следы в ее светлых волосах.
Diffusions, Markov processes, and martingales, Volume One: Foundations, Second. Edition, by L. C. G. Rogers and D. Williams, John Wiley & Sons, Chichester.
Чрезвычайная. В шифровалке. Она не могла себе этого представить. - С-слушаюсь, сэр.
- Глаза коммандера, сузившись, пристально смотрели на Чатрукьяна. - Ну, что еще - до того как вы отправитесь домой. В одно мгновение Сьюзан все стало ясно. Когда Стратмор загрузил взятый из Интернета алгоритм закодированной Цифровой крепости и попытался прогнать его через ТРАНСТЕКСТ, цепная мутация наткнулась на фильтры системы Сквозь строй.
Но ведь у нас есть ТРАНСТЕКСТ, почему бы его не расшифровать? - Но, увидев выражение лица Стратмора, она поняла, что правила игры изменились. - О Боже, - проговорила Сьюзан, сообразив, в чем дело, - Цифровая крепость зашифровала самое. Стратмор невесело улыбнулся: - Наконец ты поняла. Формула Цифровой крепости зашифрована с помощью Цифровой крепости.
Мне кажется, мистер Беккер опаздывает на свидание. Проследите, чтобы он вылетел домой немедленно. Смит кивнул: - Наш самолет в Малаге.
Не имеет понятия. Рассказ канадца показался ему полным абсурдом, и он подумал, что старик еще не отошел от шока или страдает слабоумием. Тогда он посадил его на заднее сиденье своего мотоцикла, чтобы отвезти в гостиницу, где тот остановился.
Он жестом предложил старику перешагнуть через него, но тот пришел в негодование и еле сдержался. Подавшись назад, он указал на целую очередь людей, выстроившихся в проходе. Беккер посмотрел в другую сторону и увидел, что женщина, сидевшая рядом, уже ушла и весь ряд вплоть до центрального прохода пуст. Не может быть, что служба уже закончилась.
Фонд электронных границ усилил свое влияние, доверие к Фонтейну в конгрессе резко упало, и, что еще хуже, агентство перестало быть анонимным. Внезапно домохозяйки штата Миннесота начали жаловаться компаниям Америка онлайн и Вундеркинд, что АНБ, возможно, читает их электронную почту, - хотя агентству, конечно, не было дела до рецептов приготовления сладкого картофеля. Провал Стратмора дорого стоил агентству, и Мидж чувствовала свою вину - не потому, что могла бы предвидеть неудачу коммандера, а потому, что эти действия были предприняты за спиной директора Фонтейна, а Мидж платили именно за то, чтобы она эту спину прикрывала.
Да! - Соши ткнула пальцем в свой монитор. - Смотрите. Все прочитали: - …в этих бомбах использовались разные виды взрывчатого вещества… обладающие идентичными химическими характеристиками.