Das Buch liefert die Werkzeuge, um den Gesetzmassigkeiten der Stochastik auf die Spur zu kommen. Dafur wird, ausgehend von der elementaren beschreibenden Statistik, die Wahrscheinlichkeitstheorie bis hin zum Zentralen Grenzwertsatz entwickelt. Ein weiterer Schwerpunkt liegt in der Einfuhrung in aktuelle stochastische Fragestellungen - von der Informationstheorie bis zur Finanzmathematik

By reducing mathematical detail and focusing on real-world applications, this book provides engineers with an easy-to-understand overview of stochastic modeling. An entire chapter is included on how to set up the problem, and then another complete chapter presents examples of applications before doing any math. A previously unpublished computational method for solving equations related to Markov processes is added. The book shows how to add costs or revenues to the basic probability structures without much additional effort. In addition, numerous examples are included that show how the theory can be used. Engineers will also find explanations on how to formulate word problems into the models that the math worked on.

Provides a more accessible introduction than other books on Markov processes by emphasizing the structure of the subject and avoiding sophisticated measure theory

Leads the reader to a rigorous understanding of basic theory

From the Reviews: "Gihman and Skorohod have done an excellent job of presenting the theory in its present state of rich imperfection." --D.W. Stroock, Bulletin of the American Mathematical Society, 1980

Stochastic Processes and Random Vibrations Theory and Practice JAlA-us SA3lnes University of Iceland, ReykjavA-k, Iceland This book covers the fundamental theory of stochastic processes for analysing mechanical and structural systems subject to random excitation, and also for treating random signals of a general nature with special emphasis on earthquakes and turbulent winds. Starting with basic probability calculus and the fundamental theory of stochastic processes, the author progresses onto engineering applications: systems analysis and treatment of random signals. The random excitation and response of simple mechanical systems and complex structural systems is discussed in some detail. Extreme conditions such as distribution of large vibration peaks, random excursions above certain limits and mechanical failure due to fatigue are then addressed. The text also offers a discussion of some well-known stochastic models and an introduction to signal processing and digital filters. Numerous worked examples are included: distribution of extreme wind speeds, analysis of structural reliability, earthquake response of a tall multi-storey structure, wind loading of tall towers, generation of random earthquake signals and earthquake risk analysis.

This volume includes the five lecture courses given at the CIME-EMS School on "Stochastic Methods in Finance" held in Bressanone/Brixen, Italy 2003. It deals with innovative methods, mainly from stochastic analysis, that play a fundamental role in the mathematical modelling of finance and insurance: the theory of stochastic processes, optimal and stochastic control, stochastic differential equations, convex analysis and duality theory. Five topics are treated in detail: Utility maximization in incomplete markets; the theory of nonlinear expectations and its relationship with the theory of risk measures in a dynamic setting; credit risk modelling; the interplay between finance and insurance; incomplete information in the context of economic equilibrium and insider trading.

This thesis summarizes most of my recent research in the field of portfolio optimization. The main topics which I have addressed are portfolio problems with stochastic interest rates and portfolio problems with defaultable assets. The starting point for my research was the paper "A stochastic control ap proach to portfolio problems with stochastic interest rates" (jointly with Ralf Korn), in which we solved portfolio problems given a Vasicek term structure of the short rate. Having considered the Vasicek model, it was obvious that I should analyze portfolio problems where the interest rate dynamics are gov erned by other common short rate models. The relevant results are presented in Chapter 2. The second main issue concerns portfolio problems with default able assets modeled in a firm value framework. Since the assets of a firm then correspond to contingent claims on firm value, I searched for a way to easily deal with such claims in portfolio problems. For this reason, I developed the elasticity approach to portfolio optimization which is presented in Chapter 3. However, this way of tackling portfolio problems is not restricted to portfolio problems with default able assets only, but it provides a general framework allowing for a compact formulation of portfolio problems even if interest rates are stochastic."

This volume provides a systematic mathematical exposition of the conceptual problems of nonequilibrium statistical physics, such as entropy production, irreversibility, and ordered phenomena. Markov chains, diffusion processes, and hyperbolic dynamical systems are used as mathematical models of physical systems. A measure-theoretic definition of entropy production rate and its formulae in various cases are given. It vanishes if and only if the stationary system is reversible and in equilibrium. Moreover, in the cases of Markov chains and diffusion processes on manifolds, it can be expressed in terms of circulations on directed cycles. Regarding entropy production fluctuations, the Gallavotti-Cohen fluctuation theorem is rigorously proved.

The subject of time series is of considerable interest, especially among researchers in econometrics, engineering, and the natural sciences. As part of the prestigious Wiley Series in Probability and Statistics, this book provides a lucid introduction to the field and, in this new Second Edition, covers the important advances of recent years, including nonstationary models, nonlinear estimation, multivariate models, state space representations, and empirical model identification. New sections have also been added on the Wold decomposition, partial autocorrelation, long memory processes, and the Kalman filter.

Major topics include:

• Moving average and autoregressive processes
• Introduction to Fourier analysis
• Spectral theory and filtering
• Large sample theory
• Estimation of the mean and autocorrelations
• Estimation of the spectrum
• Parameter estimation
• Regression, trend, and seasonality
• Unit root and explosive time series

To accommodate a wide variety of readers, review material, especially on elementary results in Fourier analysis, large sample statistics, and difference equations, has been included.

Quantum probability and the theory of operator algebras are both concerned with the study of noncommutative dynamics. Focusing on stationary processes with discrete-time parameter, this book presents (without many prerequisites) some basic problems of interest to both fields, on topics including extensions and dilations of completely positive maps, Markov property and adaptedness, endomorphisms of operator algebras and the applications arising from the interplay of these themes. Much of the material is new, but many interesting questions are accessible even to the reader equipped only with basic knowledge of quantum probability and operator algebras.

This book is the first comprehensive treatment of rational matrix equations in stochastic systems, including various aspects of the field, previously unpublished results and explicit examples. Topics include modelling with stochastic differential equations, stochastic stability, reformulation of stochastic control problems, analysis of the rational matrix equation and numerical solutions. Primarily a survey in character, this monograph is intended for researchers, graduate students and engineers in control theory and applied linear algebra.

The second Symposium on Stochastic Algorithms, Foundations and Applications (SAGA 2003), took place on September 22 23,2003, in Hat?eld, England.The present volume comprises 12 contributed papers and 3 invited talks. The contributed papers included in the proceedings present results in the following areas: ant colony optimization; randomized algorithmsfor the intersection problem; - cal search for constraint satisfaction problems; randomized local search methods for combinatorial optimization, in particular, simulated annealing techniques; probabilistic global search algorithms; network communication complexity; open shop scheduling; aircraft routing; traf?c control; randomized straight-line programs; and stochastic - tomata and probabilistic transformations. TheinvitedtalkbyRolandKirschnerprovidesabriefintroductiontoquantuminf- matics. The requirements and the prospects of the physical implementation of a qu- tum computer are addressed. Lucila Ohno-Machado and Winston P. Kuo describe the factors that make the an- ysis of high-throughput gene expression data especially challenging, and indicate why properly evaluated stochastic algorithms can play a particularly important role in this process. John Vaccaro et al. review a fundamental element of quantum information theory, source coding, which entails the compression of quantum data. A recent experiment that demonstrates this fundamental principle is presented and discussed. Our special thanks go to all who supported SAGA 2003, to all authors who subm- ted papers, to the members of the program committee, to the invited speakers, and to the members of the organizing committee. Andreas Albrecht Kathleen Steinhofel ] Organization SAGA2003wasorganizedbytheUniversityofHertfordshire, DepartmentofComputer Science, Hat?eld, Hertfordshire AL10 9AB, United Kingdom."

Sequential Stochastic Optimization provides mathematicians and applied researchers with a well-developed framework in which stochastic optimization problems can be formulated and solved. Offering much material that is either new or has never before appeared in book form, it lucidly presents a unified theory of optimal stopping and optimal sequential control of stochastic processes. This book has been carefully organized so that little prior knowledge of the subject is assumed; its only prerequisites are a standard graduate course in probability theory and some familiarity with discrete-parameter martingales.

Major topics covered in Sequential Stochastic Optimization include:

• Fundamental notions, such as essential supremum, stopping points, accessibility, martingales and supermartingales indexed by INd
• Conditions which ensure the integrability of certain suprema of partial sums of arrays of independent random variables
• The general theory of optimal stopping for processes indexed by Ind
• Structural properties of information flows
• Sequential sampling and the theory of optimal sequential control
• Multi-armed bandits, Markov chains and optimal switching between random walks

The theory of stochastic processes indexed by a partially ordered set has been the subject of much research over the past twenty years. The objective of this CIME International Summer School was to bring to a large audience of young probabilists the general theory of spatial processes, including the theory of set-indexed martingales and to present the different branches of applications of this theory, including stochastic geometry, spatial statistics, empirical processes, spatial estimators and survival analysis. This theory has a broad variety of applications in environmental sciences, social sciences, structure of material and image analysis. In this volume, the reader will find different approaches which foster the development of tools to modelling the spatial aspects of stochastic problems.

This book contains the proceedings of the conference "Fractals in Graz 2001 - Analysis, Dynamics, Geometry, Stochastics" that was held in the second week of June 2001 at Graz University of Technology, in the capital of Styria, southeastern province of Austria. The scientific committee of the meeting consisted of M. Barlow (Vancouver), R. Strichartz (Ithaca), P. Grabner and W. Woess (both Graz), the latter two being the local organizers and editors of this volume. We made an effort to unite in the conference as well as in the present pro ceedings a multitude of different directions of active current work, and to bring together researchers from various countries as well as research fields that all are linked in some way with the modern theory of fractal structures. Although (or because) in Graz there is only a very small group working on fractal structures, consisting of "non-insiders," we hope to have been successful with this program of wide horizons. All papers were written upon explicit invitation by the editors, and we are happy to be able to present this representative panorama of recent work on poten tial theory, random walks, spectral theory, fractal groups, dynamic systems, fractal geometry, and more. The papers presented here underwent a refereeing process."

The book provides an easily accessible computationally oriented introduction into the numerical solution of stochastic differential equations using computer experiments. It develops in the reader an ability to apply numerical methods solving stochastic differential equations in their own fields. Furthermore, it creates an intuitive understanding of the necessary theoretical background from stochastic and numeric analysis. The book is related to the more theoretical monograph P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 1992, but can be independently used. It provides solutions to over 100 exercises used in this monograph to illustrate the theory. Corresponding Turbo Pascal programs are given on a floppy disk; furthermore commentaries on the programs and their use are carefully worked out in the book.

This book presents a new method for the diagnosis and observation of dynamical systems. This approach is presented with a strong theoretical background. The given methods are developed for engineering applications and are illustrated with numerous graphic and practical examples. In the first part of the book, new results in the area of automata theory, such as the solution to supervision problems for stochastic automata, are presented as well as an elaborated study on automata networks. The second part presents a new approach to qualitative modelling of dynamical systems based on quantized systems. This methodology opens the path towards application and is described and illustrated in detail. In conclusion practical applications of the developed methods are demonstrated.

Markov Chains are widely used as stochastic models to study a broad spectrum of system performance and dependability characteristics. This monograph is devoted to compositional specification and analysis of Markov chains.Based on principles known from process algebra, the author systematically develops an algebra of interactive Markov chains. By presenting a number of distinguishing results, of both theoretical and practical nature, the author substantiates the claim that interactive Markov chains are more than just another formalism: Among other, an algebraic theory of interactive Markov chains is developed, devise algorithms to mechanize compositional aggregation are presented, and state spaces of several million states resulting from the study of an ordinary telefone system are analyzed.

This new volume of the long-established St. Flour Summer School of Probability includes the notes of the three major lecture courses by Erwin Bolthausen on "Large Deviations and Iterating Random Walks", by Edwin Perkins on "Dawson-Watanabe Superprocesses and Measure-Valued Diffusions", and by Aad van der Vaart on "Semiparametric Statistics".

The Workshop on Stochastic Theory and Control, sponsored by the NSF and KU, with co-technical sponsorship of the CSS, was held on October 18-20, 2001 at the University of Kansas in Lawrence, Kansas. A group of leading scholars in the field of stochastic theory and control, gathered at this event to discuss leading-edge topics of stochastic control, which includes risk sensitive control, adaptive control, mathematics of finance, estimation, identification, optimal control, nonlinear filtering, stochastic differential equations, stochastic partial differential equations, and stochastic theory and its applications. The workshop provided an opportunity for all of stochastic control researchers to network and discuss cutting-edge technologies and applications, teaching, and future directions of stochastic control.

Opening new directions in research in both discrete event dynamic systems as well as in stochastic control, this volume focuses on a wide class of control and of optimization problems over sequences of integer numbers. This is a counterpart of convex optimization in the setting of discrete optimization. The theory developed is applied to the control of stochastic discrete-event dynamic systems. Some applications are admission, routing, service allocation and vacation control in queuing networks. Pure and applied mathematicians will enjoy reading the book since it brings together many disciplines in mathematics: combinatorics, stochastic processes, stochastic control and optimization, discrete event dynamic systems, algebra.

Discrete-time Stochastic Systems gives a comprehensive introduction to the estimation and control of dynamic stochastic systems and provides complete derivations of key results such as the basic relations for Wiener filtering. The book covers both state-space methods and those based on the polynomial approach. Similarities and differences between these approaches are highlighted. Some non-linear aspects of stochastic systems (such as the bispectrum and extended Kalman filter) are also introduced and analysed. The books chief features are as follows: inclusion of the polynomial approach provides alternative and simpler computational methods than simple reliance on state-space methods; algorithms for analysis and design of stochastic systems allow for ease of implementation and experimentation by the reader; the highlighting of spectral factorization gives appropriate emphasis to this key concept often overlooked in the literature; explicit solutions of Wiener problems are handy schemes, well suited for computations compared with more commonly available but abstract formulations; complex-valued models that are directly applicable to many problems in signal processing and communications. Changes in the second edition include: additional information covering spectral factorisation and the innovations form; the chapter on optimal estimation being completely rewritten to focus on a posterior estimates rather than maximum likelihood; new material on fixed lag smoothing and algorithms for solving Riccati equations are improved and more up to date; new presentation of polynomial control and new derivation of linear-quadratic-Gaussian control. Discrete-time Stochastic Systems is primarily of benefit to students taking M.Sc. courses in stochastic estimation and control, electronic engineering and signal processing but may also be of assistance for self study and as a reference.

The aim of this book is to present a recently developed approach suitable for investigating a variety of qualitative aspects of order-preserving random dynamical systems and to give the background for further development of the theory. The main objects considered are equilibria and attractors. The effectiveness of this approach is demonstrated by analysing the long-time behaviour of some classes of random and stochastic ordinary differential equations which arise in many applications.

The following notes grew out oflectures held during the DMV-Seminar on Random Media in November 1999 at the Mathematics Research Institute of Oberwolfach, and in February-March 2000 at the Ecole Normale Superieure in Paris. In both places the atmosphere was very friendly and stimulating. The positive response of the audience was encouragement enough to write up these notes. I hope they will carryover the enjoyment of the live lectures. I whole heartedly wish to thank Profs. Matthias Kreck and Jean-Franc;ois Le Gall who were respon sible for these two very enjoyable visits, Laurent Miclo for his comments on an earlier version of these notes, and last but not least Erwin Bolthausen who was my accomplice during the DMV-Seminar. A Brief Introduction The main theme of this series of lectures are "Random motions in random me dia." The subject gathers a variety of probabilistic models often originated from physical sciences such as solid state physics, physical chemistry, oceanography, biophysics . . ., in which typically some diffusion mechanism takes place in an inho mogeneous medium. Randomness appears at two levels. It comes in the description of the motion of the particle diffusing in the medium, this is a rather traditional point of view for probability theory; but it also comes in the very description of the medium in which the diffusion takes place."

This book reviews recent theoretical, computational and experimental developments in mechanics of random and multiscale solid materials. The aim is to provide tools for better understanding and prediction of the effects of stochastic (non-periodic) microstructures on materials' mesoscopic and macroscopic properties. Particular topics involve a review of experimental techniques for the microstructure description, a survey of key methods of probability theory applied to the description and representation of microstructures by random modes, static and dynamic elasticity and non-linear problems in random media via variational principles, stochastic wave propagation, Monte Carlo simulation of random continuous and discrete media, fracture statistics models, and computational micromechanics.