Probability and Stochastic Modeling

Elements of Reliability Theory. Counting and Queuing Processes. Birth and Death Processes: Comparison of Random Variables. Rotar has authored four books and more than scientific papers on probability theory and its applications in leading mathematical journals. So, is there something making this book different or better than many others available in the market? The answer is yes! The author presents the material systematically and rigorously. Of primary importance, however, is the emphasis on the modeling aspects.

The large number of well-chosen and carefully described practical cases is a tremendous advantage of the book.

New concepts are introduced with illuminating discussion and illustrations. Pakes, Mathematical Reviews , September This includes upper division students in science and engineering including statistics and mathematics, as well as students in fields such as economics and finance.

In addition, it will be a wonderful book for self study for many others. Important and well-chosen examples illustrate the theory throughout, and a large body of exercises supplements the text. It gives a lucid presentation of basic probability theory, including Markov chains and martingales.

1st Edition

A special feature of this book is a marvelous exposition of many interesting aspects of financial mathematics that are generally considered rather intricate and inaccessible at this level. This book carries the imprint of a distinguished mathematician and teacher with expertise in probability theory and many of its special applications to mathematical economics and finance. It is an outstanding addition to the field requiring only a modest background in mathematics.

The author, having extensive teaching experience and an undoubted literary talent, has managed to create an original introduction to modern probability theory. The successful combination of a variety of examples, exercises and applications with deep and nontrivial ideas makes the book interesting not only for beginning students, but also for professionals working with probabilistic problems.

I believe that the book can serve as an ideal textbook for anyone interested in probability theory and its applications. The book will take a worthy place in the literature on probabilistic issues. The text is suitable for students with a standard background in calculus and linear algebra. The approach is rigorous without being pedantic, and the text is liberally sprinkled with examples.

Throughout, there is a welcome emphasis on stochastic modeling. Of note is the fairly early introduction and use of conditional expectations. The main text is complemented by a large collection of exercises with a wide range of difficulty. This book is a welcome and attractive addition to the list of textbooks available for an upper division probability course and would even be suitable for a graduate-level introduction to non-measure-theoretic probability and stochastic processes. You will be prompted to fill out a registration form which will be verified by one of our sales reps.

We provide complimentary e-inspection copies of primary textbooks to instructors considering our books for course adoption. Learn More about VitalSource Bookshelf. CPD consists of any educational activity which helps to maintain and develop knowledge, problem-solving, and technical skills with the aim to provide better health care through higher standards. It could be through conference attendance, group discussion or directed reading to name just a few examples.

We provide a free online form to document your learning and a certificate for your records. The Wiener process or Brownian motion process has its origins in different fields including statistics, finance and physics.

Stochastic process - Wikipedia

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 But the work was never forgotten in the mathematical community, as Bachelier published a book in detailing his ideas, [] which was cited by mathematicians including Doob, Feller [] and Kolomogorov. In Albert Einstein published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the kinetic theory of gases.

Einstein derived a differential equation , known as a diffusion equation , for describing the probability of finding a particle in a certain region of space. Shortly after Einstein's first paper on Brownian movement, Marian Smoluchowski published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method. Einstein's work, as well as experimental results obtained by Jean Perrin , later inspired Norbert Wiener in the s [] to use a type of measure theory, developed by Percy Daniell , and Fourier analysis to prove the existence of the Wiener process as a mathematical object.

Another discovery occurred in Denmark in when A. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.

In Ernest Rutherford and Hans Geiger published experimental results on counting alpha particles. Motivated by their work, Harry Bateman studied the counting problem and derived Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process.

Markov processes and Markov chains are named after Andrey Markov who studied Markov chains in the early 20th century. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in , and a branching process, introduced by Francis Galton and Henry William Watson in , preceding the work of Markov. Andrei Kolmogorov developed in a paper a large part of the early theory of continuous-time Markov processes. In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically.

One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.

Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem [j] to prove a corresponding stochastic process exists. When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes.

For example, both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions. Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined. To overcome these two difficulties, different assumptions and approaches are possible.

One approach for avoiding mathematical construction issues of stochastic processes, proposed by Joseph Doob , is to assume that the stochastic process is separable. Another approach is possible, originally developed by Anatoliy Skorokhod and Andrei Kolmogorov , [] for a continuous-time stochastic process with any metric space as its state space.

For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption, [70] [] but such a stochastic process based on this approach will be automatically separable. Although less used, the separability assumption is considered more general because every stochastic process has a separable version.

From Wikipedia, the free encyclopedia. List of stochastic processes topics Covariance function Dynamics of Markovian particles Entropy rate for a stochastic process Ergodic process Gillespie algorithm Interacting particle system Law stochastic processes Markov chain Probabilistic cellular automaton Random field Randomness Stationary process Statistical model Stochastic calculus Stochastic control Deterministic system.

For a stochastic process to be separable in a probabilistic sense , its index set must be a separable space in a topological or analytic sense , in addition to other conditions. Rogers; David Williams 13 April Diffusions, Markov Processes, and Martingales: Michael Steele 6 December Stochastic Calculus and Financial Applications. Introduction to the Theory of Random Processes. From Theory to Implementation and Experimentation. Bressloff 22 August Stochastic Processes in Cell Biology.

Van Kampen 30 August Stochastic Processes in Physics and Chemistry. Stochastic Population Dynamics in Ecology and Conservation. Stochastic Methods in Neuroscience. From Physics to Finance. Random processes for image and signal processing. Thomas 28 November Elements of Information Theory. Introduction to Modern Cryptography: Stochastic Geometry and Wireless Networks. Martingale Methods in Financial Modelling.

Navigation menu

Shreve 3 June Stochastic Calculus for Finance II: Miller 6 December Random Point Processes in Time and Space. Some History of Stochastic Point Processes". Theory of Stochastic Processes: Basic Principles and Applications of Probability Theory. Foundations of Modern Probability. Taylor 29 January Random Fields and Geometry. Lawler; Vlada Limic 24 June Lectures on Gaussian Processes. Adler 28 January The Geometry of Random Fields. Taylor 2 December A First Course in Stochastic Processes. Random Processes for Engineers.

Ramaswami 1 January Daley; David Vere-Jones 12 November An Introduction to the Theory of Point Processes: General Theory and Structure. Fourier Analysis and Stochastic Processes. Functional Analysis for Probability and Stochastic Processes: From probability to finance and quantum groups". Notices of the AMS. Surveys in Stochastic Processes. Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems.

Probability and Stochastic Processes. Borovkov 22 June World Scientific Publishing Co Inc. Kloeden; Eckhard Platen 17 April Numerical Solution of Stochastic Differential Equations. An Introduction to Random Fields. Oxford English Dictionary 3rd ed. Subscription or UK public library membership required. Theory of probability and statistics as exemplified in short dictums.

Gray 21 November A Modern Approach to Probability Theory. Applied Probability and Queues. Stochastic Processes and Models. Gubner 1 June Essentials of Stochastic Processes. Bass 6 October Introduction to Stochastic Calculus with Applications. Ibe 29 August Elements of Random Walk and Diffusion Processes. Probability Theory and Statistical Inference: Econometric Modeling with Observational Data. Probability and Random Processes. Archive for History of Exact Sciences. Brownian Motion and Stochastic Calculus.

Stochastic process

Continuous Martingales and Brownian Motion. Wiersema 6 August Tijms 6 May A First Course in Stochastic Models. Vere-Jones 10 April Elementary Theory and Methods. Pinsky; Samuel Karlin An Introduction to Stochastic Modeling. Kingman 17 December Stochastic Geometry for Wireless Networks. Kendall; Joseph Mecke 27 June Stochastic Geometry and Its Applications.

Streit 15 September Imaging, Tracking, and Sensing. Basics of Applied Stochastic Processes. Rozanov 6 December Kroese 20 September Simulation and the Monte Carlo Method. Lopes 10 May Markov Chain Monte Carlo: Martingales in Banach Spaces.

• ?
• .
• Retail Strategy: The View from the Bridge.

Heyde 10 July Martingale Limit Theory and Its Application. Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences. Sinai 10 August Theory of Probability and Random Processes. Cox; Valerie Isham 17 July Stationary Stochastic Processes for Scientists and Engineers. An Introduction with Applications. Victoir 4 February Multidimensional Stochastic Processes as Rough Paths: