This book presents the current status and research trends in Stochastic Analysis. Several new and emerging research areas are described in detail, highlighting the present outlook in Stochastic Analysis and its impact on abstract analysis. The book focuses on treating problems in areas that serve as a launching pad for continual research.

"Covers the areas of modern analysis and probability theory. Presents a collection of papers given at the Festschrift held in honor of the 65 birthday of M. M. Rao, whose prolific published research includes the well-received Marcel Dekker, Inc. books Theory of Orlicz Spaces and Conditional Measures and Applications. Features previously unpublished research articles by a host of internationally recognized scholars."

Based on the proceedings of the first International Conference on Matrix-Analytic Methods (MAM) in Stochastic Models, held in Flint, Michigan, this book presents a general working knowledge of MAM through tutorial articles and application papers. It furnishes information on MAM studies carried out in the former Soviet Union.

The main focus of this book is on the development of the theory of Graph Directed Markov Systems. This far-reaching generalization of the theory of conformal iterated systems can be applied in many situations, including the theory of dynamical systems. Dan Mauldin and Mariusz Urbanski include much of the necessary background material to increase the appeal of this book to graduate students as well as researchers. They also include an extensive list of references for further reading.

This volume contains refereed research or review articles presented at the 7th Seminar on Stochastic Analysis, Random Fields and Applications which took place at the Centro Stefano Franscini (Monte Verita) in Ascona, Switzerland, in May 2011. The seminar focused mainly on: - stochastic (partial) differential equations, especially with jump processes, construction of solutions and approximations - Malliavin calculus and Stein methods, and other techniques in stochastic analysis, especially chaos representations and convergence, and applications to models of interacting particle systems - stochastic methods in financial models, especially models for power markets or for risk analysis, empirical estimation and approximation, stochastic control and optimal pricing. The book will be a valuable resource for researchers in stochastic analysis and for professionals interested in stochastic methods in finance. "

This compact yet thorough text zeros in on the parts of the theory that are particularly relevant to applications . It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential equations. It solves stochastic differential equations by a variety of methods and studies in detail the one-dimensional case. The book concludes with a treatment of semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions.

The presentation is unparalleled in its clarity and simplicity. Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together the material you need to effectively and efficiently impart the practical background they need.

This book provides an introduction to the asymptotic theory of random summation, combining a strict exposition of the foundations of this theory and recent results. It also includes a description of its applications to solving practical problems in hardware and software reliability, insurance, finance, and more. The authors show how practice interacts with theory, and how new mathematical formulations of problems appear and develop.
Attention is mainly focused on transfer theorems, description of the classes of limit laws, and criteria for convergence of distributions of sums for a random number of random variables. Theoretical background is given for the choice of approximations for the distribution of stock prices or surplus processes. General mathematical theory of reliability growth of modified systems, including software, is presented. Special sections deal with doubling with repair, rarefaction of renewal processes, limit theorems for supercritical Galton-Watson processes, information properties of probability distributions, and asymptotic behavior of doubly stochastic Poisson processes.
Random Summation: Limit Theorems and Applications will be of use to specialists and students in probability theory, mathematical statistics, and stochastic processes, as well as to financial mathematicians, actuaries, and to engineers desiring to improve probability models for solving practical problems and for finding new approaches to the construction of mathematical models.

This book collects significant contributions from the fifth conference on Fractal Geometry and Stochastics held in Tabarz, Germany, in March 2014. The book is divided into five topical sections: geometric measure theory, self-similar fractals and recurrent structures, analysis and algebra on fractals, multifractal theory, and random constructions. Each part starts with a state-of-the-art survey followed by papers covering a specific aspect of the topic. The authors are leading world experts and present their topics comprehensibly and attractively. Both newcomers and specialists in the field will benefit from this book.

The Wiley Classics Library consists of selected books originally published by John Wiley & Sons that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: Emil Artin Geometric Algebra Norman T. J. Bailey The Elements of Stochastic Processes With Applications to the Natural Sciences R. W. Carter Simple Groups of Lie Type Richard Courant Differential and Integral Calculus. Volume I Richard Courant Differential and Integral Calculus, Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume II Harold S.M. Coxeter Introduction to Modern Geometry, Second Edition Charles W. Curtis & Irving Reiner Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis & Irving Reiner Methods of Representation Theory With Applications to Finite Groups and Orders, Volume 1 W. Edwards Darning Sample Design in Business Research Amos deShalit & Herman Feshbach Theoretical Nuclear Physics, Volume I-Nuclear Structure J. L. Doob Stochastic Processes Nelson Dunford, Jacob T. Schwartz Linear Operators, Part One, General Theory Nelson Dunford, Jacob T. Schwartz Linear Operators, Part Two, Spectral Theory - Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob T. Schwartz Linear Operators, Part Three, Spectral Operators Peter Henrici Applied and Computational Complex Analysis, Volume I - Power Series-Integration-Conformal Mapping-Location of Zeros Peter Hilton, Yel-Chiang Wu A Course in Modern Algebra Harry Hochstadt Integral Equations Erwin Kreyszig Introductory Functional Analysis with Applications William H. Louisell Quantum Statistical Properties of Radiation P. M. Prenter Splines and Variational Methods Walter Rudin Fourier Analysis on Groups C. L. Siegel Topics in Complex Function Theory Volume I - Elliptic Functions and Uniformization Theory C. L. Siegel Topics in Complex Function Theory Volume II - Automorphic and Abelian Integrals C. L. Siegel Topics in Complex Function Theory, Volume III - Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry

The book treats two topics in the theory of stochastic partial differential equations: space-regularity of solutions and existence of stochastic flows. The equations considered in the book are linear parabolic with multiplicative noise, like those arising in non-linear filtering or diffusion models in randomly moving media. Regularity theory in Sobolev spaces is extensively investigated, for homogeneous and non-homogeneous boundary value problems, with a detailed analysis of the new geometrical conditions on coefficients arising as a consequence of the stochaticity. The book provides an account of regularity results that may represent a useful reference for the researcher in stochastic partial differential equations. Regularity theory is then applied to prove the existence of stochastic flows. In spite of the variety of results on stochastic flows obtained by this method, several open problems are pointed out, with the hope of stimulating further research on this subject.

Stochastic Modeling of Scientific Data combines stochastic modeling and statistical inference in a variety of standard and less common models, such as point processes, Markov random fields and hidden Markov models in a clear, thoughtful and succinct manner. The distinguishing feature of this work is that, in addition to probability theory, it contains statistical aspects of model fitting and a variety of data sets that are either analyzed in the text or used as exercises. Markov chain Monte Carlo methods are introduced for evaluating likelihoods in complicated models and the forward backward algorithm for analyzing hidden Markov models is presented. The strength of this text lies in the use of informal language that makes the topic more accessible to non-mathematicians. The combinations of hard science topics with stochastic processes and their statistical inference puts it in a new category of probability textbooks. The numerous examples and exercises are drawn from astronomy, geology, genetics, hydrology, neurophysiology and physics.

This book explores the remarkable connections between two domains that, a priori, seem unrelated: Random matrices (together with associated random processes) and integrable systems. The relations between random matrix models and the theory of classical integrable systems have long been studied. These appear mainly in the deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based upon the Riemann-Hilbert problem of analytic function theory and by related approaches to the study of nonlinear asymptotics in the large N limit. Associated with studies of matrix models are certain stochastic processes, the "Dyson processes", and their continuum diffusion limits, which govern the spectrum in random matrix ensembles, and may also be studied by related methods. Random Matrices, Random Processes and Integrable Systems provides an in-depth examination of random matrices with applications over a vast variety of domains, including multivariate statistics, random growth models, and many others. Leaders in the field apply the theory of integrable systems to the solution of fundamental problems in random systems and processes using an interdisciplinary approach that sheds new light on a dynamic topic of current research.

Highlighting modern computational methods, Applied Stochastic Modelling, Second Edition provides students with the practical experience of scientific computing in applied statistics through a range of interesting real-world applications. It also successfully revises standard probability and statistical theory. Along with an updated bibliography and improved figures, this edition offers numerous updates throughout. New to the Second Edition An extended discussion on Bayesian methods A large number of new exercises A new appendix on computational methods The book covers both contemporary and classical aspects of statistics, including survival analysis, Kernel density estimation, Markov chain Monte Carlo, hypothesis testing, regression, bootstrap, and generalised linear models. Although the book can be used without reference to computational programs, the author provides the option of using powerful computational tools for stochastic modelling. All of the data sets and MATLAB and R programs found in the text as well as lecture slides and other ancillary material are available for download at www.crcpress.com Continuing in the bestselling tradition of its predecessor, this textbook remains an excellent resource for teaching students how to fit stochastic models to data.

The first book to examine weakly stationary random fields and their connections with invariant subspaces (an area associated with functional analysis). It reviews current literature, presents central issues and most important results within the area. For advanced Ph.D. students, researchers, especially those conducting research on Gaussian theory.

Biomathematical Problems in Optimization of Cancer Radiotherapy provides insight into the role of cell population heterogeneity in the optimal control of fractionated irradiation of tumors. The book emphasizes the mathematical modeling aspect of the problem and presents the state of the art in the stochastic description of irradiated cell survival. Some of the results are of general theoretical interest and can be applied to other areas of optimal control methodology. Detailed explanations of all mathematical statements are provided throughout the text. The book is excellent for biomathematicians, radiotherapists, oncologists, health physicists, and other researchers and students interested in the topic.

Presents new computer methods in approximation, simulation, and visualization for a host of alpha-stable stochastic processes.

An extensive update to a classic text

Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences. They offer successful models for the description of random two- and three-dimensional micro and macro structures and statistical methods for their analysis.

The previous edition of this book has served as the key reference in its field for over 18 years and is regarded as the best treatment of the subject of stochastic geometry, both as a subject with vital applications to spatial statistics and as a very interesting field of mathematics in its own right.

"This editi""on" Presents a wealth of models for spatial patterns and related statistical methods.Provides a great survey of the modern theory of random tessellations, including many new models that became tractable only in the last few years.Includes new sections on random networks and random graphs to review the recent ever growing interest in these areas.Provides an excellent introduction to theory and modelling of point processes, which covers some very latest developments.Illustrate the forefront theory of random sets, with many applications.Adds new results to the discussion of fibre and surface processes.Offers an updated collection of useful stereological methods.Includes 700 new references.Is written in an accessible style enabling non-mathematicians to benefit from this book.Provides a companion website hosting information on recent developments in the field www.wiley.com/go/cskm

"Stochastic Geometry and its Applications" is ideally suited for researchers in physics, materials science, biology and ecological sciences as well as mathematicians and statisticians. It should also serve as a valuable introduction to the subject for students of mathematics and statistics.

The main mathematical ideas are presented in a context with which economists will be familiar. Using a binomial approximation to Brownian motion, the mathematics is reduced to simple algebra, progressing to some equally simple limits. The starting point of the calculus of Brownian motion -- "Ito's Lemma" -- emerges by analogy with the economics of risk-aversion. Conditions for the optimal regulation of Brownian motion, including the important, but often mysterious "smooth pasting" condition, are derived in a similar way. Each theoretical derivation is illustrated by developing a significant economic application, drawn mainly from recent research in macro-economics and international economics.
This book aims to widen the understanding and use of stochastic dynamic choice and equilibrium models. It offers a simplified and heuristic exposition of the theory of Brownian motion and its control or regulation, rendering such methods more accessible to economists who do not require a de

Semimartingale Theory and Stochastic Calculus presents a systematic and detailed account of the general theory of stochastic processes, the semimartingale theory, and related stochastic calculus. The book emphasizes stochastic integration for semimartingales, characteristics of semimartingales, predictable representation properties and weak convergence of semimartingales. It also includes a concise treatment of absolute continuity and singularity, contiguity, and entire separation of measures by semimartingale approach. Two basic types of processes frequently encountered in applied probability and statistics are highlighted: processes with independent increments and marked point processes encountered frequently in applied probability and statistics.

Semimartingale Theory and Stochastic Calculus is a self-contained and comprehensive book that will be valuable for research mathematicians, statisticians, engineers, and students.

Sojourns and Extremes of Stochastic Processes is a research monograph in the area of probability theory. During the past thirty years Berman has made many contributions to the theory of the extreme values and sojourn times of the sample functions of broad classes of stochastic processes. These processes arise in theoretical and applied models, and are presented here in a unified exposition.

This book covers the development of methods for detection and estimation of changes in complex systems. These systems are generally described by nonstationary stochastic models, which comprise both static and dynamic regimes, linear and nonlinear dynamics, and constant and time-variant structures of such systems. It covers both retrospective and sequential problems, particularly theoretical methods of optimal detection. Such methods are constructed and their characteristics are analyzed both theoretically and experimentally. Suitable for researchers working in change-point analysis and stochastic modelling, the book includes theoretical details combined with computer simulations and practical applications. Its rigorous approach will be appreciated by those looking to delve into the details of the methods, as well as those looking to apply them.

This book contains all of Wolfgang Doeblin's publications. In addition, it includes a reproduction of the pli cachete on l'equation de Kolmogoroff and previously unpublished material that Doeblin wrote in 1940. The articles are accompanied by commentaries written by specialists in Doeblin's various areas of interest. The modern theory of probability developed between the two World Wars thanks to the very remarkable work of Kolmogorov, Khinchin, S.N. Bernstein, Romanovsky, von Mises, Hostinsky, Onicescu, Frechet, Levy and others, among whom one name shines particularly brightly, that of Wolfgang Doeblin (1915-1940). The work of this young mathematician, whose life was tragically cut short by the war, remains even now, and indeed will remain into the future, an exemplar of originality and of mathematical power. This book was conceived and in essence brought to fruition by Marc Yor before his death in 2014. It is dedicated to him.

The fourth edition of this widely used textbook on pricing and hedging of financial derivatives now also includes dynamic equilibrium theory and continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, Arbitrage Theory in Continuous Time is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Bjoerk has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. This edition includes an entirely new section presenting dynamic equilibrium theory, covering unit net supply endowments models and the Cox-Ingersoll-Ross equilibrium factor model. Providing two full treatments of arbitrage theory-the classical delta hedging approach and the modern martingale approach-this book is written so that these approaches can be studied independently of each other, thus providing the less mathematically-oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action. This textbook is a natural choice for graduate students and advanced undergraduates studying finance and an invaluable introduction to mathematical finance for mathematicians and professionals in the market.

Most branches of science involving random fluctuations can be approached by Stochastic Calculus. These include, but are not limited to, signal processing, noise filtering, stochastic control, optimal stopping, electrical circuits, financial markets, molecular chemistry, population dynamics, etc. All these applications assume a strong mathematical background, which in general takes a long time to develop. Stochastic Calculus is not an easy to grasp theory, and in general, requires acquaintance with the probability, analysis and measure theory.The goal of this book is to present Stochastic Calculus at an introductory level and not at its maximum mathematical detail. The author's goal was to capture as much as possible the spirit of elementary deterministic Calculus, at which students have been already exposed. This assumes a presentation that mimics similar properties of deterministic Calculus, which facilitates understanding of more complicated topics of Stochastic Calculus.The second edition contains several new features that improved the first edition both qualitatively and quantitatively. First, two more chapters have been added, Chapter 12 and Chapter 13, dealing with applications of stochastic processes in Electrochemistry and global optimization methods.This edition contains also a final chapter material containing fully solved review problems and provides solutions, or at least valuable hints, to all proposed problems. The present edition contains a total of about 250 exercises.This edition has also improved presentation from the first edition in several chapters, including new material.

Ruin Probabilities: Smoothness, Bounds, Supermartingale Approach deals with continuous-time risk models and covers several aspects of risk theory. The first of them is the smoothness of the survival probabilities. In particular, the book provides a detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizon survival probabilities for different risk models. Next, it gives some possible applications of the results concerning the smoothness of the survival probabilities. Additionally, the book introduces the supermartingale approach, which generalizes the martingale one introduced by Gerber, to get upper exponential bounds for the infinite-horizon ruin probabilities in some generalizations of the classical risk model with risky investments.