Applied Probability and Stochastic Processes is an edited work written in honor of Julien Keilson. This volume has attracted a host of scholars in applied probability, who have made major contributions to the field, and have written survey and state-of-the-art papers on a variety of applied probability topics, including, but not limited to: perturbation method, time reversible Markov chains, Poisson processes, Brownian techniques, Bayesian probability, optimal quality control, Markov decision processes, random matrices, queueing theory and a variety of applications of stochastic processes. The book has a mixture of theoretical, algorithmic, and application chapters providing examples of the cutting-edge work that Professor Keilson has done or influenced over the course of his highly-productive and energetic career in applied probability and stochastic processes. The book will be of interest to academic researchers, students, and industrial practitioners who seek to use the mathematics of applied probability in solving problems in modern society.

The first part of this book presents the essential topics for an introduction to deterministic optimal control theory. The second part introduces stochastic optimal control for Markov diffusion processes. It also inlcudes two other topics important for applications, namely, the solution to the stochastic linear regulator and the separation principle.

During the last fifty years, Gopinath Kallianpur has made extensive and significant contributions to diverse areas of probability and statistics, including stochastic finance, Fisher consistent estimation, non-linear prediction and filtering problems, zero-one laws for Gaussian processes and reproducing kernel Hilbert space theory, and stochastic differential equations in infinite dimensions. To honor Kallianpur's pioneering work and scholarly achievements, a number of leading experts have written research articles highlighting progress and new directions of research in these and related areas. This commemorative volume, dedicated to Kallianpur on the occasion of his seventy-fifth birthday, will pay tribute to his multi-faceted achievements and to the deep insight and inspiration he has so graciously offered his students and colleagues throughout his career. Contributors to the volume: S. Aida, N. Asai, K. B. Athreya, R. N. Bhattacharya, A. Budhiraja, P. S. Chakraborty, P. Del Moral, R. Elliott, L. Gawarecki, D. Goswami, Y. Hu, J. Jacod, G. W. Johnson, L. Johnson, T. Koski, N. V. Krylov, I. Kubo, H.-H. Kuo, T. G. Kurtz, H. J. Kushner, V. Mandrekar, B. Margolius, R. Mikulevicius, I. Mitoma, H. Nagai, Y. Ogura, K. R. Parthasarathy, V. Perez-Abreu, E. Platen, B. V. Rao, B. Rozovskii, I. Shigekawa, K. B. Sinha, P. Sundar, M. Tomisaki, M. Tsuchiya, C. Tudor, W. A. Woycynski, J. Xiong

A comprehensive overview of weak convergence of stochastic processes and its application to the study of financial markets. Split into three parts, the first recalls the mathematics of stochastic processes and stochastic calculus with special emphasis on contiguity properties and weak convergence of stochastic integrals. The second part is devoted to the analysis of financial theory from the convergence point of view. The main problems, which include portfolio optimization, option pricing and hedging are examined, especially when considering discrete-time approximations of continuous-time dynamics. The third part deals with lattice- and tree-based computational procedures for option pricing both on stocks and stochastic bonds. More general discrete approximations are also introduced and detailed. Includes detailed examples.

Pure and applied stochastic analysis and random fields form the subject of this book. The collection of articles on these topics represent the state of the art of the research in the field, with particular attention being devoted to stochastic models in finance. Some are review articles, others are original papers; taken together, they will apprise the reader of much of the current activity in the area.

This volume contains the Proceedings of the Fourth Seminar on Stochastic Analy sis, Random Fields and Applications, which took place at the Centro Stefano Fran scini (Monte Verita) in Ascona (Ticino), Switzerland, from May 20 to 24, 2002. The first three editions of this conference occured in 1993, 1996 and 1999. The Seminar covered several topics: fundamental aspects of stochastic analysis, such as stochastic partial differential equations and random fields, and applications to current active fields such as probabilistic methods in fluid dynamics, biomathe matics, and financial modeling. As in the previous editions, this last topic was the subject of the Fourth Minisymposium on Stochastic Methods in Financial Models. These proceedings aim to present key aspects of these topics to a larger audience. All papers in this volume have been refereed. A major topic within Stochastic Analysis is the area of random fields which includes as particular cases, Gaussian random fields, stochastic partial differential equations (s. p. d. e. 's) and stochastic differential equations with values in Banach spaces. In this framework, interesting new developments were presented in the theory of Gaussian random fields on manifolds with applications to astrophysics and neurosciences. Moreover, with the aim of modeling certain very irregular phe nomena, a theory of s. p. d. e. 's driven by noises concentrated on hyperplanes was presented."

This volume is a continuation of Unbiased Estimators and Their Applications, Vol. I: Univariate Case. It contains problems of parametric point estimation for multivariate probability distributions emphasizing problems of unbiased estimation. The volume consists of four chapters dealing, respectively, with some basic properties of multivariate continuous and discrete distributions, the general theory of point estimation in multivariate case, techniques for constructing unbiased estimators and applications of unbiased estimation theory in the multivariate case. These chapters contain numerous examples, many applications and are followed by a comprehensive Appendix which classifies and lists, in the form of tables, all known results relating to unbiased estimators of parameter functions for multivariate distributions. Audience: This volume will serve as a handbook on point unbiased estimation for researchers whose work involves statistics. It can also be recommended as a supplementary text for undergraduate and graduate students.

This volume deals with the analysis of nonlinear evolution problems described by partial differential equations having random or stochastic parameters. The emphasis throughout is on the actual determination of solutions, rather than on proving the existence of solutions, although mathematical proofs are given when this is necessary from an applications point of view. The content is divided into six chapters. Chapter 1 gives a general presentation of mathematical models in continuum mechanics and a description of the way in which problems are formulated. Chapter 2 deals with the problem of the evolution of an unconstrained system having random space-dependent initial conditions, but which is governed by a deterministic evolution equation. Chapter 3 deals with the initial-boundary value problem for equations with random initial and boundary conditions as well as with random parameters where the randomness is modelled by stochastic separable processes. Chapter 4 is devoted to the initial-boundary value problem for models with additional noise, which obey Ito-type partial differential equations. Chapter 5 is essential devoted to the qualitative and quantitative analysis of the chaotic behaviour of systems in continuum physics. Chapter 6 provides indications on the solution of ill-posed and inverse problems of stochastic type and suggests guidelines for future research. The volume concludes with an Appendix which gives a brief presentation of the theory of stochastic processes. Examples, applications and case studies are given throughout the book and range from those involving simple stochasticity to stochastic illposed problems. For applied mathematicians, engineers and physicists whosework involves solving stochastic problems.

Two-scale systems described by singularly perturbed SDEs have been the subject of ample literature. However, this new monograph develops subjects that were rarely addressed and could be given the collective description "Stochastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by stochastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and stochastic approximation.

This book contains 12 contributions on stochastic models in reliability and maintenance. Written by the leading researchers on each topic, each contribution surveys the current status on stochastic models emphasizing mathematical formulation and optimization applications. Each contribution is self-contained and has a thorough bibliography. The topics include renewal processes, semi-Markov processes, Markovian deterioration models, maintenance and replacement models, software reliability models and Monte-Carlo simulation. This book provides researchers, reliability engineers and graduate students with the current status of the field and future developments of the subject.

'Et moi *...* si j'avait su comment en revenir. One service mathematics has rendered the je n'y serais point aile: human race. It has put common sense back where it belongs. on the topmost shelf next Jules Verne (0 the dusty canister labelled 'discarded non- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

This book is for a general scientific and engineering audience as a guide to current ideas, methods, and models for stochastic modeling of microstructures.

It is a reference for professionals in material modeling, mechanical engineering, materials science, chemical, civil, environmental engineering and applied mathematics.

Initially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is only recently that the relation between the two theories has been thoroughly studied. The authors of this Grundlehren volume, two of the international leaders in the field, propose a systematic exposition of convergence in law for stochastic processes, from the point of view of semimartingale theory, with emphasis on results that are useful for mathematical theory and mathematical statistics. This leads them to develop in detail some particularly useful parts of the general theory of stochastic processes, such as martingale problems, and absolute continuity or contiguity results. The book contains an introduction to the theory of martingales and semimartingales, random measures stochastic integrales, Skorokhod topology, etc., as well as a large number of results which have never appeared in book form, and some entirely new results. The second edition contains some additions to the text and references. Some parts are completely rewritten.

We present here a selection of the seminars given at the Second International Workshop on Instabilities and Nonequilibrium Structures in Valparaiso, Chile, in December 1987. The Workshop was organized by Facultad de Ciencias Fisicas y Matematicas of Universidad de Chile and by Universidad Tecnica Federico Santa Maria where it took place. This periodic meeting takes place every two years in Chile and aims to contribute to the efforts of Latin America towards the development of scientific research. This development is certainly a necessary condition for progress in our countries and we thank our lecturers for their warm collaboration to fulfill this need. We are also very much indebted to the Chilean Academy of Sciences for sponsoring officially this Workshop. We thank also our sponsors and supporters for their valuable help, and most especially the Scientific Cooperation Program of France, UNESCO, Ministerio de Educaci6n of Chile and Fundaci6n Andes. We are grateful to Professor Michiel Hazewinkel for including this book in his series and to Dr. David Larner of Kluwer for his continuous interest and support to this project.

Nonlinearity and stochastic structural dynamics is of common interest to engineers and applied scientists belonging to many disciplines. Recent research in this area has been concentrated on the response and stability of nonlinear mechanical and structural systems subjected to random escitation. Simultaneously the focus of research has also been directed towards understanding intrinsic nonlinear phenomena like bifurcation and chaos in deterministic systems. These problems demand a high degree of sophistication in the analytical and numerical approaches. At the same time they arise from considerations of nonlinear system response to turbulence, earthquacke, wind, wave and guidancy excitations. The topic thus attracts votaries of both analytical rigour and practical applications. This books gives important and latest developments in the field presenting in a coherent fashion the research findings of leading international groups working in the area of nonlinear random vibration and chaos.

This book is devoted to an investigation of the basic problems of the the- ory of random fields which are characterized by certain singular properties (e. g., unboundedness, or vanishing) of their spectral densities. These ran- dom fields are called, the random fields with singular spectrum, long-memory fields, random fields with long-range dependence, fields with slowly decaying correlations or strongly dependent random fields by various authors. This phenomenon has been observed empirically by many scientists long before suitable mathematical models were known. The methods and results differ significantly from the theory of weakly dependent random fields. The first chapter presents basic concepts of the spectral theory of random fields, some examples of random processes and fields with singular spectrum, Tauberian and Abelian theorems for the covariance function of singular ran- dom fields. In the second chapter limit theorems for non-linear functionals of random fields with singular spectrum are proved. Chapter 3 summarizes some limit theorems for geometric functionals of random fields with long-range dependence. Limit distributions of the solutions of Burgers equation with random data via parabolic and hyperbolic rescaling are presented in chapter 4. And chapter 5 presents some problems of statistical analysis of random fields with singular spectrum. I would like to thank the editor, Michiel Hazewinkel, for his support. I am grateful to the following students and colleagues: 1. Deriev, A. Olenko, K. Rybasov, L. Sakhno, M. Sharapov, A. Sikorskii, M. Silac-BenSic. I would also like to thank V.Anh, O. Barndorff-Nielsen,Yu. Belyaev, P.

This book brings together the latest findings in the area of stochastic analysis and statistics. The individual chapters cover a wide range of topics from limit theorems, Markov processes, nonparametric methods, acturial science, population dynamics, and many others. The volume is dedicated to Valentin Konakov, head of the International Laboratory of Stochastic Analysis and its Applications on the occasion of his 70th birthday. Contributions were prepared by the participants of the international conference of the international conference "Modern problems of stochastic analysis and statistics", held at the Higher School of Economics in Moscow from May 29 - June 2, 2016. It offers a valuable reference resource for researchers and graduate students interested in modern stochastics.

One SCI\'ice mathematics bas rendered the 'Et moi, ...si j'avait su comment en revcnir. je n'y serais point aile: human race. It bas put common sc:nsc back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

In recent years, the study of the theory of Brownian motion has become a powerful tool in the solution of problems in mathematical physics. This self-contained and readable exposition by leading authors, provides a rigorous account of the subject, emphasizing the "explicit" rather than the "concise" where necessary, and addressed to readers interested in probability theory as applied to analysis and mathematical physics.
A distinctive feature of the methods used is the ubiquitous appearance of stopping time. The book contains much original research by the authors (some of which published here for the first time) as well as detailed and improved versions of relevant important results by other authors, not easily accessible in existing literature.

I became interested in Random Vibration during the preparation of my PhD dissertation, which was concerned with the seismic response of nuclear reactor cores. I was initiated into this field through the cla.ssical books by Y.K.Lin, S.H.Crandall and a few others. After the completion of my PhD, in 1981, my supervisor M.Gera.din encouraged me to prepare a course in Random Vibration for fourth and fifth year students in Aeronautics, at the University of Liege. There was at the time very little material available in French on that subject. A first draft was produced during 1983 and 1984 and revised in 1986. These notes were published by the Presses Poly techniques et Universitaires Romandes (Lausanne, Suisse) in 1990. When Kluwer decided to publish an English translation ofthe book in 1992, I had to choose between letting Kluwer translate the French text in-extenso or doing it myself, which would allow me to carry out a sustantial revision of the book. I took the second option and decided to rewrite or delete some of the original text and include new material, based on my personal experience, or reflecting recent technical advances. Chapter 6, devoted to the response of multi degree offreedom structures, has been completely rewritten, and Chapter 11 on random fatigue is entirely new. The computer programs which have been developed in parallel with these chapters have been incorporated in the general purpose finite element software SAMCEF, developed at the University of Liege.

'Et moi, ..~ si lavait su CO.llUlJalt en revc:nir, One acMcc matbcmatica bu JaIdcred the human rac:c. It bu put COIDIDOD _ beet je n'y serais point aBe.' Jules Verne wbac it bdoup, 0Jl !be~ IbcII _t to !be dusty cauialcr Iabc&d 'diMardod__ The series is divergent; thc:reforc we may be -'. I!.ticT. Bc:I1 able to do something with it. O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world when: both feedback and non- linearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statcmalts as: 'One service topology has rendered mathematical physics ...*; 'One service logic has rendered c0m- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series. This series, Mathematics and Its Applications. started in 19n. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However. the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branc:hes. It also happens, quite often in fact, that branches which were thought to be completely.

This is a collection of articles written by mathematicians and physicists, designed to describe the state of the art in climate models with stochastic input. Mathematicians will benefit from a survey of simple models, while physicists will encounter mathematically relevant techniques at work.

The main purpose of this handbook is to summarize and to put in order the ideas, methods, results and literature on the theory of random evolutions and their applications to the evolutionary stochastic systems in random media, and also to present some new trends in the theory of random evolutions and their applications. In physical language, a random evolution ( RE ) is a model for a dynamical sys tem whose state of evolution is subject to random variations. Such systems arise in all branches of science. For example, random Hamiltonian and Schrodinger equations with random potential in quantum mechanics, Maxwell's equation with a random refractive index in electrodynamics, transport equations associated with the trajec tory of a particle whose speed and direction change at random, etc. There are the examples of a single abstract situation in which an evolving system changes its "mode of evolution" or "law of motion" because of random changes of the "environment" or in a "medium." So, in mathematical language, a RE is a solution of stochastic operator integral equations in a Banach space. The operator coefficients of such equations depend on random parameters. Of course, in such generality, our equation includes any homogeneous linear evolving system. Particular examples of such equations were studied in physical applications many years ago. A general mathematical theory of such equations has been developed since 1969, the Theory of Random Evolutions."

The subject of this book is a new mathematical technique, the stochastic limit, developed for solving nonlinear problems in quantum theory involving systems with infinitely many degrees of freedom (typically quantum fields or gases in the thermodynamic limit). This technique is condensed into some easily applied rules (called "stochastic golden rules"), which allow us to single out the dominating contributions to the dynamical evolution of systems in regimes involving long times and small effects. In the stochastic limit the original Hamiltonian theory is approximated using a new Hamiltonian theory which is singular. These singular Hamiltonians still define a unitary evolution, and the new equations give much more insight into the relevant physical phenomena than the original Hamiltonian equations. Especially, one can explicitly compute multi-time correlations (e.g. photon statistics) or coherent vectors, which are beyond the reach of typical asymptotic techniques.

One of the most important methods in dealing with the optimization of large, complex systems is that of hierarchical decomposition. The idea is to reduce the overall complex problem into manageable approximate problems or subproblems, to solve these problems, and to construct a solution of the original problem from the solutions of these simpler prob lems. Development of such approaches for large complex systems has been identified as a particularly fruitful area by the Committee on the Next Decade in Operations Research (1988) [42] as well as by the Panel on Future Directions in Control Theory (1988) [65]. Most manufacturing firms are complex systems characterized by sev eral decision subsystems, such as finance, personnel, marketing, and op erations. They may have several plants and warehouses and a wide variety of machines and equipment devoted to producing a large number of different products. Moreover, they are subject to deterministic as well as stochastic discrete events, such as purchasing new equipment, hiring and layoff of personnel, and machine setups, failures, and repairs.