The 1988 Seminar on Stochastic Processes was held at the University of Florida, Gainesville, March 3 through March 5, 1988. It was the eighth seminar in a continuing series of meetings which provide opportunities for researchers to discuss current work in stochastic processes in an informal and enjoyable atmosphere. Previous seminars were held at Princeton University, Northwestern University, the University of Florida and the University of Virginia. The participants' enthusiasm and interest have created stimulating and successful seminars. We thank those participants who have permitted us to publish their research in this volume. This year's invited participants included B. Atkinson, J. Azema, D. Bakry, P. Baxendale, J. Brooks, G. Brosamler, K. Burdzy, E. Cinlar, R. Darling, N. Dinculeanu, E. Dynkin, S. Evans, N. Falkner, P. Fitzsimmons, R. Getoor, J. Glover, V. Goodman, P. Hsu, J.-F. Le Gall, M. Liao, P. March, P. McGill, J. Mitro, T. Mountford, C. Mueller, A. Mukherjea, V. Papanicolaou, E. Perkins, M. Pinsky, L. Pitt, A. O. Pittenger, Z. Pop-Stojanovic, M. Rao, J. Rosen, T. Salisbury, C. Shih, M. Taksar, J. Taylor, S. J. Taylor, E. Toby, R. Williams, Wu Rong, and Z. Zhao. The seminar was made possible through the generous support of the Department of Mathematics, the Center for Applied Mathematics, the Division of Sponsored Research and the College of Liberal Arts and Sciences of the University of Florida. We extend our thanks for local arrangements to our host, Zoran Pop-Stojanovic. 1. G.

Labor omnia vincit improbus. VIRGIL, Georgica I, 144-145. In the first part of his Theoria combinationis observationum erroribus min- imis obnoxiae, published in 1821, Carl Friedrich Gauss [Gau80, p.10] deduces a Chebyshev-type inequality for a probability density function, when it only has the property that its value always decreases, or at least does l not increase, if the absolute value of x increases . One may therefore conjecture that Gauss is one of the first scientists to use the property of 'single-humpedness' of a probability density function in a meaningful probabilistic context. More than seventy years later, zoologist W.F.R. Weldon was faced with 'double- humpedness'. Indeed, discussing peculiarities of a population of Naples crabs, possi- bly connected to natural selection, he writes to Karl Pearson (E.S. Pearson [Pea78, p.328]): Out of the mouths of babes and sucklings hath He perfected praise! In the last few evenings I have wrestled with a double humped curve, and have overthrown it. Enclosed is the diagram...If you scoff at this, I shall never forgive you. Not only did Pearson not scoff at this bimodal probability density function, he examined it and succeeded in decomposing it into two 'single-humped curves' in his first statistical memoir (Pearson [Pea94]).

  It has been thirteen years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus even after thirteen years and many intervening texts, it seems worthwhile nevertheless to publish a second edition. We will no longer call it "a new approach" however. The second edition has several significant changes. The most obvious is the addition of exercises for solution. These exercises are intended to supplement the text, and in no cases have lemmas needed in a proof been relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue University and Cornell University. Chapter three has been nearly completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter four treats sigma martingales which have become important in finance theory, as well as a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space $\mathcal{H}^1$ can be identified with BMO martingales. Last, there are of course small changes throughout the book.

This volume contains the papers presented at the meeting "Distributions with given marginals and statistical modelling", held in Barcelona (Spain), July 17- 20, 2000. This is the fourth meeting on given marginals, showing that this topic has aremarkable interest. BRIEF HISTORY The construction of distributions with given marginals started with the seminal papers by Hoeffding (1940) and Fn!chet (1951). Since then, many others have contributed on this topic: Dall' Aglio, Farlie, Gumbel, Johnson, Kellerer, Kotz, Morgenstern, Marshali, Olkin, Strassen, Vitale, Whitt, etc., as weIl as Arnold, Cambanis, Deheuvels, Genest, Frank, Joe, Kirneldorf, Nelsen, Ruschendorf, Sampson, Scarsini, Tiit, etc. In 1957 Sklar and Schweizer introduced probabilistic metric spaces. In 1975 Kirneldorf and Sampson studied the uniform representation of a bivariate dis- tribution and proposed the desirable conditions that should be satisfied by any bivariate family. In 1991 Darsow, Nguyen and Olsen defined a natural operation between cop- ulas, with applications in stochastic processes. In 1993, AIsina, Nelsen and Schweizer introduced the notion of quasi-copula.

The second conference on Fractal Geometry and Stochastics was held at Greifs wald/Koserow, Germany from August 28 to September 2, 1998. Four years had passed after the first conference with this theme and during this period the interest in the subject had rapidly increased. More than one hundred mathematicians from twenty-two countries attended the second conference and most of them presented their newest results. Since it is impossible to collect all these contributions in a book of moderate size we decided to ask the 13 main speakers to write an account of their subject of interest. The corresponding articles are gathered in this volume. Many of them combine a sketch of the historical development with a thorough discussion of the most recent results of the fields considered. We believe that these surveys are of benefit to the readers who want to be introduced to the subject as well as to the specialists. We also think that this book reflects the main directions of research in this thriving area of mathematics. We express our gratitude to the Deutsche Forschungsgemeinschaft whose financial support enabled us to organize the conference. The Editors Introduction Fractal geometry deals with geometric objects that show a high degree of irregu larity on all levels of magnitude and, therefore, cannot be investigated by methods of classical geometry but, nevertheless, are interesting models for phenomena in physics, chemistry, biology, astronomy and other sciences."

Collecting together selected pioneering works of the celebrated mathematician Anatolii V. Skorokhod, this volume serves as a guide to the theory of stochastic processes from its beginning to its current state. It offers both an excellent bibliographic resource and a unique opportunity for readers to gain a better understanding of Skorokhod's original and beautiful ideas, which had a deep impact on the development of the subject. The modern theory of stochastic processes is a fast-growing branch of probability theory which is now an independent science in its own right, with its own methods and philosophy. It has many applications in various fields, including financial mathematics, quantum physics and engineering. A clear understanding of this theory is impossible without knowledge of the ideas which form its base, many of which are due to Skorokhod. The book is intended for a broad audience of researchers and students with an interest in probability theory, stochastic processes and their applications.

Stochastic modeling is a set of quantitative techniques for analyzing practical systems with random factors. This area is highly technical and mainly developed by mathematicians. Most existing books are for those with extensive mathematical training; this book minimizes that need and makes the topics easily understandable. Fundamentals of Stochastic Models offers many practical examples and applications and bridges the gap between elementary stochastics process theory and advanced process theory. It addresses both performance evaluation and optimization of stochastic systems and covers different modern analysis techniques such as matrix analytical methods and diffusion and fluid limit methods. It goes on to explore the linkage between stochastic models, machine learning, and artificial intelligence, and discusses how to make use of intuitive approaches instead of traditional theoretical approaches. The goal is to minimize the mathematical background of readers that is required to understand the topics covered in this book. Thus, the book is appropriate for professionals and students in industrial engineering, business and economics, computer science, and applied mathematics.

Stochastic Differential Equations for Science and Engineering is aimed at students at the M.Sc. and PhD level. The book describes the mathematical construction of stochastic differential equations with a level of detail suitable to the audience, while also discussing applications to estimation, stability analysis, and control. The book includes numerous examples and challenging exercises. Computational aspects are central to the approach taken in the book, so the text is accompanied by a repository on GitHub containing a toolbox in R which implements algorithms described in the book, code that regenerates all figures, and solutions to exercises. Features: Contains numerous exercises, examples, and applications Suitable for science and engineering students at Master's or PhD level Thorough treatment of the mathematical theory combined with an accessible treatment of motivating examples GitHub repository available at: https://github.com/Uffe-H-Thygesen/SDEbook and https://github.com/Uffe-H-Thygesen/SDEtools

This book reviews problems associated with rare events arising in a wide range of circumstances, treating such topics as how to evaluate the probability an insurance company will be bankrupted, the lifetime of a redundant system, and the waiting time in a queue. Well-grounded, unique mathematical evaluation methods of basic probability characteristics concerned with rare events are presented, which can be employed in real applications, as the volume also contains relevant numerical and Monte Carlo methods. The various examples, tables, figures and algorithms will also be appreciated. Audience: This work will be useful to graduate students, researchers and specialists interested in applied probability, simulation and operations research.

This book contains an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms. Moreover this analytic theory is unified with the probabilistic potential theory based on symmetric Markov processes and developed further in conjunction with the stochastic analysis based on additive functional. Since the publication of the first edition in 1994, this book has attracted constant interests from readers and is by now regarded as a standard reference for the theory of Dirichlet forms. For the present second edition, the authors not only revised the existing text, but also added sections on capacities and Sobolev type inequalities, irreducible recurrence and ergodicity, recurrence and Poincare type inequalities, the Donsker-Varadhan type large deviation principle, as well as several new exercises with solutions. The book addresses to researchers and graduate students who wish to comprehend the area of Dirichlet forms and symmetric Markov processes.

Stemming from the IHP trimester "Stochastic Dynamics Out of Equilibrium", this collection of contributions focuses on aspects of nonequilibrium dynamics and its ongoing developments. It is common practice in statistical mechanics to use models of large interacting assemblies governed by stochastic dynamics. In this context "equilibrium" is understood as stochastically (time) reversible dynamics with respect to a prescribed Gibbs measure. Nonequilibrium dynamics correspond on the other hand to irreversible evolutions, where fluxes appear in physical systems, and steady-state measures are unknown. The trimester, held at the Institut Henri Poincare (IHP) in Paris from April to July 2017, comprised various events relating to three domains (i) transport in non-equilibrium statistical mechanics; (ii) the design of more efficient simulation methods; (iii) life sciences. It brought together physicists, mathematicians from many domains, computer scientists, as well as researchers working at the interface between biology, physics and mathematics. The present volume is indispensable reading for researchers and Ph.D. students working in such areas.

This open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet is a beautiful area of mathematics with links to many other subjects. While describing the general theory of Hardy, Rellich, Caffarelli-Kohn-Nirenberg, Sobolev, and other inequalities in the setting of general homogeneous groups, the authors pay particular attention to the special class of stratified groups. In this environment, the theory of Hardy inequalities becomes intricately intertwined with the properties of sub-Laplacians and subelliptic partial differential equations. These topics constitute the core of this book and they are complemented by additional, closely related topics such as uncertainty principles, function spaces on homogeneous groups, the potential theory for stratified groups, and the potential theory for general Hoermander's sums of squares and their fundamental solutions. This monograph is the winner of the 2018 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. As can be attested as the winner of such an award, it is a vital contribution to literature of analysis not only because it presents a detailed account of the recent developments in the field, but also because the book is accessible to anyone with a basic level of understanding of analysis. Undergraduate and graduate students as well as researchers from any field of mathematical and physical sciences related to analysis involving functional inequalities or analysis of homogeneous groups will find the text beneficial to deepen their understanding.

Quantitative Methods in Transportation provides the most useful, simple, and advanced quantitative techniques for solving real-life transportation engineering problems. It aims to help transportation engineers and analysts to predict travel and freight demand, plan new transportation networks, and develop various traffic control strategies that are safer, more cost effective, and greener. Transportation networks can be exceptionally large, and this makes many transportation problems combinatorial, and the challenges are compounded by the stochastic and independent nature of trip-planners decision making. Methods outlined in this book range from linear programming, multi-attribute decision making, data envelopment analysis, probability theory, and simulation to computer techniques such as genetic algorithms, simulated annealing, tabu search, ant colony optimization, and bee colony optimization. The book is supported with problems and has a solutions manual to aid course instructors.

It is well known that contemporary mathematics includes many disci plines. Among them the most important are: set theory, algebra, topology, geometry, functional analysis, probability theory, the theory of differential equations and some others. Furthermore, every mathematical discipline consists of several large sections in which specific problems are investigated and the corresponding technique is developed. For example, in general topology we have the following extensive chap ters: the theory of compact extensions of topological spaces, the theory of continuous mappings, cardinal-valued characteristics of topological spaces, the theory of set-valued (multi-valued) mappings, etc. Modern algebra is featured by the following domains: linear algebra, group theory, the theory of rings, universal algebras, lattice theory, category theory, and so on. Concerning modern probability theory, we can easily see that the clas sification of its domains is much more extensive: measure theory on ab stract spaces, Borel and cylindrical measures in infinite-dimensional vector spaces, classical limit theorems, ergodic theory, general stochastic processes, Markov processes, stochastical equations, mathematical statistics, informa tion theory and many others."

This book illustrates the application of the economic concept of stochastic dominance to option markets and presents an alternative option pricing paradigm to the prevailing no arbitrage simultaneous equilibrium in the frictionless underlying and option markets. This new methodology was developed primarily by the author, working independently or jointly with other co-authors, over the course of more than thirty years. Among others, it yields the fundamental Black-Scholes-Merton option value when markets are complete, presents a new approach to the pricing of rare event risk, and uncovers option mispricing that leads to tradeable strategies in the presence of transaction costs. In the latter case it shows how a utility-maximizing investor trading in the market and a riskless bond, subject to proportional transaction costs, can increase his/her expected utility by overlaying a zero-net-cost portfolio of options bought at their ask price and written at their bid price, irrespective of the specific form of the utility function. The book contains a unified presentation of these methods and results, making it a highly readable supplement for educators and sophisticated professionals working in the popular field of option pricing. It also features a foreword by George Constantinides, the Leo Melamed Professor of Finance at the Booth School of Business, University of Chicago, USA, who was a co-author in several parts of the book.

This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems - as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis - will find this text to be a valuable addition to the mathematical literature.

This book provides an extensive, systematic overview of the modern theory of telegraph processes and their multidimensional counterparts, together with numerous fruitful applications in financial modelling. Focusing on stochastic processes of bounded variation instead of classical diffusion, or more generally, Levy processes, has two obvious benefits. First, the mathematical technique is much simpler, which helps to concentrate on the key problems of stochastic analysis and applications, including financial market modelling. Second, this approach overcomes some shortcomings of the (parabolic) nature of classical diffusions that contradict physical intuition, such as infinite propagation velocity and infinite total variation of paths. In this second edition, some sections of the previous text are included without any changes, while most others have been expanded and significantly revised. These are supplemented by predominantly new results concerning piecewise linear processes with arbitrary sequences of velocities, jump amplitudes, and switching intensities. The chapter on functionals of the telegraph process has been significantly expanded by adding sections on exponential functionals, telegraph meanders and running extrema, the times of the first passages of telegraph processes with alternating random jumps, and distribution of the Euclidean distance between two independent telegraph processes. A new chapter on the multidimensional counterparts of the telegraph processes is also included. The book is intended for graduate students in mathematics, probability, statistics and quantitative finance, and for researchers working at academic institutions, in industry and engineering. It can also be used by university lecturers and professionals in various applied areas.

This book is devoted to the study and optimization of spatiotemporal stochastic processes, that is, processes which develop simultaneously in space and time under random influences. These processes are seen to occur almost everywhere when studying the global behavior of complex systems, including:

- Physical and technical systems

- Population dynamics

- Neural networks

- Computer and telecommunication networks

- Complex production networks

- Flexible manufacturing systems

- Logistic networks and transportation systems

-Environmental engineering

Climate modelling and prediction

Earth surface models

Classical stochastic dynamic optimization forms the framework of the book. Taken as a whole, the project undertaken in the book is to establish optimality or near-optimality for Markovian policies in the control of spatiotemporal Markovian processes. The authors apply this general principle to different frameworks of Markovian systems and processes. Depending on the structure of the systems and the surroundings of the model classes the authors arrive at different levels of simplicity for the policy classes which encompass optimal or nearly optimal policies. A set of examples accompanies the theoretical findings, and these examples should demonstrate some important application areas for the theorems discussed.

Over the course of his distinguished career, Vladimir Maz'ya has made a number of groundbreaking contributions to numerous areas of mathematics, including partial differential equations, function theory, and harmonic analysis. The chapters in this volume - compiled on the occasion of his 80th birthday - are written by distinguished mathematicians and pay tribute to his many significant and lasting achievements.

Stochastic differential equations have many applications in the natural sciences. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce solution of multi-dimensional problems for partial differential equations to integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. The authors propose many new special schemes, some published here for the first time. In the second part of the book they construct numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.

This volume highlights the main results of the research performed within the network "Harmonic and Complex Analysis and its Applications" (HCAA), which was a five-year (2007-2012) European Science Foundation Programme intended to explore and to strengthen the bridge between two scientific communities: analysts with broad backgrounds in complex and harmonic analysis and mathematical physics, and specialists in physics and applied sciences. It coordinated actions for advancing harmonic and complex analysis and for expanding its application to challenging scientific problems. Particular topics considered by this Programme included conformal and quasiconformal mappings, potential theory, Banach spaces of analytic functions and their applications to the problems of fluid mechanics, conformal field theory, Hamiltonian and Lagrangian mechanics, and signal processing. This book is a collection of surveys written as a result of activities of the Programme and will be interesting and useful for professionals and novices in analysis and mathematical physics, as well as for graduate students. Browsing the volume, the reader will undoubtedly notice that, as the scope of the Programme is rather broad, there are many interrelations between the various contributions, which can be regarded as different facets of a common theme.

This book covers a wide range of problems involving the applications of stochastic processes, stochastic calculus, large deviation theory, group representation theory and quantum statistics to diverse fields in dynamical systems, electromagnetics, statistical signal processing, quantum information theory, quantum neural network theory, quantum filtering theory, quantum electrodynamics, quantum general relativity, string theory, problems in biology and classical and quantum fluid dynamics. The selection of the problems has been based on courses taught by the author to undergraduates and postgraduates in Electronics and Communications Engineering. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan).

The stochastic calculus of variations of Paul Malliavin (1925 - 2010), known today as the Malliavin Calculus, has found many applications, within and beyond the core mathematical discipline. Stochastic analysis provides a fruitful interpretation of this calculus, particularly as described by David Nualart and the scores of mathematicians he influences and with whom he collaborates. Many of these, including leading stochastic analysts and junior researchers, presented their cutting-edge research at an international conference in honor of David Nualart's career, on March 19-21, 2011, at the University of Kansas, USA. These scholars and other top-level mathematicians have kindly contributed research articles for this refereed volume.

This volume is devoted to theoretical results which formalize the concept of state lumping: the transformation of evolutions of systems having a complex (large) phase space to those having a simpler (small) phase space. The theory of phase lumping has aspects in common with averaging methods, projection formalism, stiff systems of differential equations, and other asymptotic theorems. Numerous examples are presented in this book from the theory and applications of random processes, and statistical and quantum mechanics which illustrate the potential capabilities of the theory developed. The volume contains seven chapters. Chapter 1 presents an exposition of the basic notions of the theory of linear operators. Chapter 2 discusses aspects of the theory of semigroups of operators and Markov processes which have relevance to what follows. In Chapters 3--5, invertibly reducible operators perturbed on the spectrum are investigated, and the theory of singularly perturbed semigroups of operators is developed assuming that the perturbation is subordinated to the perturbed operator. The case of arbitrary perturbation is also considered, and the results are presented in the form of limit theorems and asymptotic expansions. Chapters 6 and 7 describe various applications of the method of phase lumping to Markov and semi-Markov processes, dynamical systems, quantum mechanics, etc. The applications discussed are by no means exhaustive and this book points the way to many more fruitful applications in various other areas. For researchers whose work involves functional analysis, semigroup theory, Markov processes and probability theory.

This volume in the Mastering Mathematical Finance series strikes just the right balance between mathematical rigour and practical application. Existing books on the challenging subject of stochastic interest rate models are often too advanced for Master's students or fail to include practical examples. Stochastic Interest Rates covers practical topics such as calibration, numerical implementation and model limitations in detail. The authors provide numerous exercises and carefully chosen examples to help students acquire the necessary skills to deal with interest rate modelling in a real-world setting. In addition, the book's webpage at www.cambridge.org/9781107002579 provides solutions to all of the exercises as well as the computer code (and associated spreadsheets) for all numerical work, which allows students to verify the results.