Non-linear stochastic systems are at the center of many engineering disciplines and progress in theoretical research had led to a better understanding of non-linear phenomena. This book provides information on new fundamental results and their applications which are beginning to appear across the entire spectrum of mechanics.
The outstanding points of these proceedings are Coherent compendium of the current state of modelling and analysis of non-linear stochastic systems from engineering, applied mathematics and physics point of view. Subject areas include: Multiscale phenomena, stability and bifurcations, control and estimation, computational methods and modelling.
For the Engineering and Physics communities, this book will provide first-hand information on recent mathematical developments. The applied mathematics community will benefit from the modelling and information on various possible applications.

tEL moi, .., ' si favait su comment en revenir. je One service mathematics has rendered the n 'y serais point alle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled' discarded nonsense'. The series is divergent; therefore we may be Eric T. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought A highly necessary tool in a world where both feedback and nonlineari ties abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser vice topology has rendered mathematical physics .. .'; 'One service logic has rendered computer 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 considers some models described by means of partial dif ferential equations and boundary conditions with chaotic stochastic disturbance. In a framework of stochastic Partial Differential Equa tions an approach is suggested to generalize solutions of stochastic Boundary Problems. The main topic concerns probabilistic aspects with applications to well-known Random Fields models which are representative for the corresponding stochastic Sobolev spaces. {The term "stochastic" in general indicates involvement of appropriate random elements. ) It assumes certain knowledge in general Analysis and Probability {Hilbert space methods, Schwartz distributions, Fourier transform) . I A very general description of the main problems considered can be given as follows. Suppose, we are considering a random field ~ in a region T ~ Rd which is associated with a chaotic (stochastic) source"' by means of the differential equation (*) in T. A typical chaotic source can be represented by an appropri ate random field"' with independent values, i. e. , generalized random function"' = ( cp, 'TJ), cp E C~(T), with independent random variables ( cp, 'fJ) for any test functions cp with disjoint supports. The property of having independent values implies a certain "roughness" of the ran dom field "' which can only be treated functionally as a very irregular Schwarz distribution. With the lack of a proper development of non linear analyses for generalized functions, let us limit ourselves to the 1 For related material see, for example, J. L. Lions, E.

This book is devoted to a domain of highest industrial and scienti?c interest, the complexity. The complexity understanding and management will be a main source of e?ciency and prosperity for the next decades. Complex systems areassembliesof multiple subsystemsand arecharact- ized by emergent behavior that results by nonlinear interactions among the subsystems at multiple levels of organization. Evolvability that is the ability to evolve is the method to confront and surpass the successive boundaries of complexity. Evolvability is not biological but should be considered here in the sense that the corresponding systems have, at di?erent levels, charact- istics that are naturally associated to the living systems. The signi?cance of the complexity and the phenomena of emergence are highlighted in the ?rst chapterofthe book.Theimplicationofconcepts aslevelofreality, circularity and closure for evolvable systems is evaluated. The second chapter of the book exposes the methodology to analyze and manage complex systems. The polystochastic models, PSMs, are the cons- ered mathematical tools. PSMs characterize systems emerging when several stochastic processes occurring at di?erent conditioning levels, are capable to interact with each other, resulting in qualitatively new processes and s- tems. Innovative are the higher categories approach and the introduction of apartialdi?erentialmodelfor multiple levelsmodeling.This imposes making use of appropriate notions of time, space, probabilities and entropy. Categorytheoryistheformalismcapabletooutlinethegeneralframework, shared by the functional organization of biological organisms, of cognitive systems, by the operational structure of evolvable technologies and devices and after all by the scienti?c and engineering methods

Stochastic Resonance: Theory and Applications deals with the theory of noise-added systems and in particular with Stochastic Resonance, a quite novel theory that was introduced in the 1980s to provide better understanding of some natural phenomena (e.g. ice age recurrence). Following the very first works, a number of different applications to both natural and human-produced phenomena were proposed. The book aims to improve the understanding of noise-based techniques and to focus on practical applications of this class of phenomena (an aspect that has been very poorly investigated up to now). Based on this objective, the book is roughly divided into two parts. The first part deals with the essential theory of noise-added systems and in particular a new approach to noise-added techniques that allows a number of strategies proposed in previous years to be unified. The proposed approach also allows real-time control of the noise characteristics, assuring optimal system performance. In the second part a large number of applications are described in detail in the field of electric and electronic devices, with the aim of allowing readers to build their own experimental set. The book comes with a diskette of educational software that the authors developed. Stochastic Resonance: Theory and Applications is an invaluable reference for students, researchers and engineering professionals working in the fields of electric and electronic measurements, electronics and signal theory.

The focus of the present volume is stochastic optimization of dynamical systems in discrete time where - by concentrating on the role of information regarding optimization problems - it discusses the related discretization issues. There is a growing need to tackle uncertainty in applications of optimization. For example the massive introduction of renewable energies in power systems challenges traditional ways to manage them. This book lays out basic and advanced tools to handle and numerically solve such problems and thereby is building a bridge between Stochastic Programming and Stochastic Control. It is intended for graduates readers and scholars in optimization or stochastic control, as well as engineers with a background in applied mathematics.

Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry random sets, point processes, random mosaics and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes."

This book shows how, by rather simple models, we can gain remarkable insights into the behavior of complex systems. It is devoted to the discussion of functional self-organization in large populations of interacting active elements. The possible forms of self-organization in such systems range from coherent collective motions in the physical coordinate space to the mutual synchronization of internal dynamics, the development of coherently operating groups, the rise of hierarchical structures, and the emergence of dynamical networks. Such processes play an important role in biological and social phenomena. The authors have chosen a series of models from physics, biochemistry, biology, sociology and economics, and will systematically discuss their general properties. The book addresses researchers and graduate students in a variety of disciplines, such as physics, chemistry, biology and the social sciences.

The focus of this monograph is on generalizing the notion of variation in a set of numbers to variation in a set of probability distributions. The authors collect some known ways of comparing stochastic matrices in the context of information theory, statistics, economics, and population sciences. They then generalize these comparisons, introduce new comparisons, and establish the relations of implication or equivalence among sixteen of these comparisons. Some of the possible implications among these comparisons remain open questions. The results in this book establish a new field of investigation for both mathematicians and scientific users interested in the variations among multiple probability distributions. The work is divided into two parts. The first deals with finite stochastic matrices, which may be interpreted as collections of discrete probability distributions. The first part is presented in a fairly elementary mathematical setting. The introduction provides sketches of applications of concepts and methods to discrete memory-less channels in information theory, to the design and comparison of experiments in statistics, to the measurement of inequality in economics, and to various analytical problems in population genetics, ecology, and demography. Part two is more general and entails more difficult analysis involving Markov kernels. Here, many results of the first part are placed in a more general setting, as required in more sophisticated applications. A great strength of this text is the resulting connections among ideas from diverse fields: mathematics, statistics, economics, and population biology. In providing this array of new tools and concepts, the work will appeal to the practitioner. At the same time, it will serve as an excellent resource for self-study of for a graduate seminar course, as well as a stimulus to further research.

This text is an Elementary Introduction to Stochastic Processes in discrete and continuous time with an initiation of the statistical inference. The material is standard and classical for a first course in Stochastic Processes at the senior/graduate level (lessons 1-12). To provide students with a view of statistics of stochastic processes, three lessons (13-15) were added. These lessons can be either optional or serve as an introduction to statistical inference with dependent observations. Several points of this text need to be elaborated, (1) The pedagogy is somewhat obvious. Since this text is designed for a one semester course, each lesson can be covered in one week or so. Having in mind a mixed audience of students from different departments (Math ematics, Statistics, Economics, Engineering, etc.) we have presented the material in each lesson in the most simple way, with emphasis on moti vation of concepts, aspects of applications and computational procedures. Basically, we try to explain to beginners questions such as "What is the topic in this lesson?" "Why this topic?," "How to study this topic math ematically?." The exercises at the end of each lesson will deepen the stu dents' understanding of the material, and test their ability to carry out basic computations. Exercises with an asterisk are optional (difficult) and might not be suitable for homework, but should provide food for thought."

Identifying the input-output relationship of a system or discovering the evolutionary law of a signal on the basis of observation data, and applying the constructed mathematical model to predicting, controlling or extracting other useful information constitute a problem that has been drawing a lot of attention from engineering and gaining more and more importance in econo metrics, biology, environmental science and other related areas. Over the last 30-odd years, research on this problem has rapidly developed in various areas under different terms, such as time series analysis, signal processing and system identification. Since the randomness almost always exists in real systems and in observation data, and since the random process is sometimes used to model the uncertainty in systems, it is reasonable to consider the object as a stochastic system. In some applications identification can be carried out off line, but in other cases this is impossible, for example, when the structure or the parameter of the system depends on the sample, or when the system is time-varying. In these cases we have to identify the system on line and to adjust the control in accordance with the model which is supposed to be approaching the true system during the process of identification. This is why there has been an increasing interest in identification and adaptive control for stochastic systems from both theorists and practitioners."

Since the pioneering work of Black, Scholes, and Merton in the field of financial mathematics, research has led to the rapid development of a substantial body of knowledge, with plenty of applications to the common functioning of the world 's financial institutions.

Mathematics, as the language of science, has always played a role in the development of knowledge and technology. Presently, the high-tech character of modern business has increased the need for advanced methods, which rely to a large extent on mathematical techniques. It has become essential for the financial analyst to possess a high degree of proficiency in these mathematical techniques.

This volume contains twenty-eight refereed research or review papers presented at the 5th Seminar on Stochastic Processes, Random Fields and Applications, which took place at the Centro Stefano Franscini (Monte Verita) in Ascona, Switzerland, from May 30 to June 3, 2005. The seminar focused mainly on stochastic partial differential equations, random dynamical systems, infinite-dimensional analysis, approximation problems, and financial engineering. The book will be a valuable resource for researchers in stochastic analysis and professionals interested in stochastic methods in finance.

Stochastic geometry is the branch of mathematics that studies geometric structures associated with random configurations, such as random graphs, tilings and mosaics. Due to its close ties with stereology and spatial statistics, the results in this area are relevant for a large number of important applications, e.g. to the mathematical modeling and statistical analysis of telecommunication networks, geostatistics and image analysis. In recent years - due mainly to the impetus of the authors and their collaborators - a powerful connection has been established between stochastic geometry and the Malliavin calculus of variations, which is a collection of probabilistic techniques based on the properties of infinite-dimensional differential operators. This has led in particular to the discovery of a large number of new quantitative limit theorems for high-dimensional geometric objects. This unique book presents an organic collection of authoritative surveys written by the principal actors in this rapidly evolving field, offering a rigorous yet lively presentation of its many facets.

This book highlights the latest advances in stochastic processes, probability theory, mathematical statistics, engineering mathematics and algebraic structures, focusing on mathematical models, structures, concepts, problems and computational methods and algorithms important in modern technology, engineering and natural sciences applications. It comprises selected, high-quality, refereed contributions from various large research communities in modern stochastic processes, algebraic structures and their interplay and applications. The chapters cover both theory and applications, illustrated by numerous figures, schemes, algorithms, tables and research results to help readers understand the material and develop new mathematical methods, concepts and computing applications in the future. Presenting new methods and results, reviews of cutting-edge research, and open problems and directions for future research, the book serves as a source of inspiration for a broad spectrum of researchers and research students in probability theory and mathematical statistics, applied algebraic structures, applied mathematics and other areas of mathematics and applications of mathematics. The book is based on selected contributions presented at the International Conference on "Stochastic Processes and Algebraic Structures - From Theory Towards Applications" (SPAS2017) to mark Professor Dmitrii Silvestrov's 70th birthday and his 50 years of fruitful service to mathematics, education and international cooperation, which was held at Malardalen University in Vasteras and Stockholm University, Sweden, in October 2017.

In view of Professor Wendell Fleming's many fundamental contributions, his profound influence on the mathematical and systems theory communi ties, his service to the profession, and his dedication to mathematics, we have invited a number of leading experts in the fields of control, optimiza tion, and stochastic systems to contribute to this volume in his honor on the occasion of his 70th birthday. These papers focus on various aspects of stochastic analysis, control theory and optimization, and applications. They include authoritative expositions and surveys as well as research papers on recent and important issues. The papers are grouped according to the following four major themes: (1) large deviations, risk sensitive and Hoc control, (2) partial differential equations and viscosity solutions, (3) stochastic control, filtering and parameter esti mation, and (4) mathematical finance and other applications. We express our deep gratitude to all of the authors for their invaluable contributions, and to the referees for their careful and timely reviews. We thank Harold Kushner for having graciously agreed to undertake the task of writing the foreword. Particular thanks go to H. Thomas Banks for his help, advice and suggestions during the entire preparation process, as well as for the generous support of the Center for Research in Scientific Computation. The assistance from the Birkhauser professional staff is also greatly appreciated."

The book provides a self-contained treatment of stochastic finite element methods. It helps the reader to establish a solid background on stochastic and reliability analysis of structural systems and enables practicing engineers to better manage the concepts of analysis and design in the presence of uncertainty. The book covers the basic topics of computational stochastic mechanics focusing on the stochastic analysis of structural systems in the framework of the finite element method. The target audience primarily comprises students in a postgraduate program specializing in structural engineering but the book may also be beneficial to practicing engineers and research experts alike.

The theory of stochastic processes provides a huge arsenal of methods suitable for analyzing the influence of noise on a wide range of systems. Noise-induced, noise-supported or noise-enhanced effects sometimes offer an explanation for as yet open problems (information transmission in the nervous system and information processing in the brain, processes at the cell level, enzymatic reactions, etc.), or pave the way to novel technological applications. Noise can play a prominent role in structure formation in physics, chemistry and biology, e.g. current filaments in semiconductors, catalytic reactions on surfaces, complex dynamics of the heart, brain, or of ecosystems. The book reviews those aspects of applied stochastics addressing researchers as well as students.

Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. This text offers easy access to this fundamental topic for many students of applied sciences at many levels. It includes examples, exercises, applications, and computational procedures. It is uniquely useful for beginners and non-beginners in the field. No knowledge of measure theory is presumed.

A comprehensive account of the statistical theory of exponential families of stochastic processes. The book reviews the progress in the field made over the last ten years or so by the authors - two of the leading experts in the field - and several other researchers. The theory is applied to a broad spectrum of examples, covering a large number of frequently applied stochastic process models with discrete as well as continuous time. To make the reading even easier for statisticians with only a basic background in the theory of stochastic process, the first part of the book is based on classical theory of stochastic processes only, while stochastic calculus is used later. Most of the concepts and tools from stochastic calculus needed when working with inference for stochastic processes are introduced and explained without proof in an appendix. This appendix can also be used independently as an introduction to stochastic calculus for statisticians. Numerous exercises are also included.

'Et moi, ..., si j'avait su comment en revenIT, One service mathematics has rendered the je n'y serais point allt\.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. 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 addresses stochastic optimization procedures in a broad manner. The first part offers an overview of relevant optimization philosophies; the second deals with benchmark problems in depth, by applying a selection of optimization procedures. Written primarily with scientists and students from the physical and engineering sciences in mind, this book addresses a larger community of all who wish to learn about stochastic optimization techniques and how to use them.

It is well known that the normal distribution is the most pleasant, one can even say, an exemplary object in the probability theory. It combines almost all conceivable nice properties that a distribution may ever have: symmetry, stability, indecomposability, a regular tail behavior, etc. Gaussian measures (the distributions of Gaussian random functions), as infinite-dimensional analogues of tht< classical normal distribution, go to work as such exemplary objects in the theory of Gaussian random functions. When one switches to the infinite dimension, some "one-dimensional" properties are extended almost literally, while some others should be profoundly justified, or even must be reconsidered. What is more, the infinite-dimensional situation reveals important links and structures, which either have looked trivial or have not played an independent role in the classical case. The complex of concepts and problems emerging here has become a subject of the theory of Gaussian random functions and their distributions, one of the most advanced fields of the probability science. Although the basic elements in this field were formed in the sixties-seventies, it has been still until recently when a substantial part of the corresponding material has either existed in the form of odd articles in various journals, or has served only as a background for considering some special issues in monographs.

In a stochastic network, such as those in computer/telecommunications and manufacturing, discrete units move among a network of stations where they are processed or served. Randomness may occur in the servicing and routing of units, and there may be queueing for services. This book describes several basic stochastic network processes, beginning with Jackson networks and ending with spatial queueing systems in which units, such as cellular phones, move in a space or region where they are served. The focus is on network processes that have tractable (closed-form) expressions for the equilibrium probability distribution of the numbers of units at the stations. These distributions yield network performance parameters such as expectations of throughputs, delays, costs, and travel times. The book is intended for graduate students and researchers in engineering, science and mathematics interested in the basics of stochastic networks that have been developed over the last twenty years. Assuming a graduate course in stochastic processes without measure theory, the emphasis is on multi-dimensional Markov processes. There is also some self-contained material on point processes involving real analysis. The book also contains rather complete introductions to reversible Markov processes, Palm probabilities for stationary systems, Little laws for queueing systems and space-time Poisson processes. This material is used in describing reversible networks, waiting times at stations, travel times and space-time flows in networks. Richard Serfozo received the Ph.D. degree in Industrial Engineering and Management Sciences at Northwestern University in 1969 and is currently Professor of Industrial and Systems Engineering at Georgia Institute of Technology. Prior to that he held positions in the Boeing Company, Syracuse University, and Bell Laboratories. He has held

Safety critical and high-integrity systems, such as industrial plants and economic systems can be subject to abrupt changes - for instance due to component or interconnection failure, and sudden environment changes etc.

Combining probability and operator theory, Discrete-Time Markov Jump Linear Systems provides a unified and rigorous treatment of recent results for the control theory of discrete jump linear systems, which are used in these areas of application.

The book is designed for experts in linear systems with Markov jump parameters, but is also of interest for specialists in stochastic control since it presents stochastic control problems for which an explicit solution is possible - making the book suitable for course use.

From the reviews:

"This text is very well written...it may prove valuable to those who work in the area, are at home with its mathematics, and are interested in stability of linear systems, optimal control, and filtering." Journal of the American Statistical Association, December 2005