Stochastics Books
Taylor & Francis Ltd An Advanced Course in Probability and Stochastic
Book SynopsisAn Advanced Course in Probability and Stochastic Processes provides a modern and rigorous treatment of probability theory and stochastic processes at an upper undergraduate and graduate level. Starting with the foundations of measure theory, this book introduces the key concepts of probability theory in an accessible way, providing full proofs and extensive examples and illustrations. Fundamental stochastic processes such as Gaussian processes, Poisson random measures, Lévy processes, Markov processes, and Itô processes are presented and explored in considerable depth, showcasing their many interconnections. Special attention is paid to martingales and the Wiener process and their central role in the treatment of stochastic integrals and stochastic calculus. This book includes many exercises, designed to test and challenge the reader and expand their skillset. An Advanced Course in Probability and Stochastic Processes is meant for students and researchers who have a soTable of Contents1. Measure Theory 2. Probability 3. Convergence 4. Conditioning 5. Martingales 6. Wiener and Brownian Motion Processes 7. Itô Calculus Appendix A. Selected Solutions Appendix B. Function Spaces Appendix C. Existence of the Lebesgue Measure Index
£111.89
Taylor & Francis Ltd Introduction to Stochastic Calculus Applied to
Book SynopsisSince the publication of the first edition of this book, the area of mathematical finance has grown rapidly, with financial analysts using more sophisticated mathematical concepts, such as stochastic integration, to describe the behavior of markets and to derive computing methods. Maintaining the lucid style of its popular predecessor, Introduction to Stochastic Calculus Applied to Finance, Second Edition incorporates some of these new techniques and concepts to provide an accessible, up-to-date initiation to the field. New to the Second EditionComplements on discrete models, including Rogers'' approach to the fundamental theorem of asset pricing and super-replication in incomplete markets Discussions on local volatility, Dupire''s formula, the change of numéraire techniques, forward measures, and the forward Libor model A new chapter on credit risk modeling An extension of the chapter on simulTrade ReviewThe second edition of this book provides a concise and accessible introduction to the probabilistic techniques needed to understand the most widely used financial models. This edition incorporates many new techniques and concepts to be used to describe the behavior of financial markets. … the solutions obtained using SciLab for computer experiments are available at http://cermics.enpc.fr/~bl/scilab/ These experiments were well designed by the authors based on their teaching and research experience and were found to be effective in communicating these concepts and ideas and enhancing the understanding of readers. … a solid introduction to stochastic approaches used in the financial world. The authors cover many key finance topics … . The book can be used as a reference text by researchers and graduate students in financial mathematics. It also is ideal reading material for practicing financial analysts and consultants using mathematical models for finance.—Technometrics, May 2009, Vol. 51, No. 2 Table of ContentsDiscrete-Time Models. Optimal Stopping Problem and American Options. Brownian Motion and Stochastic Differential Equations. The Black-Scholes Model. Option Pricing and Partial Differential Equations. Interest Rate Models. Asset Models with Jumps. Credit Risk Models. Simulation and Algorithms for Financial Models. Appendix. Bibliography. Index.
£43.99
Taylor & Francis Ltd Stochastic Modelling for Systems Biology Third
Book SynopsisSince the first edition of Stochastic Modelling for Systems Biology, there have been many interesting developments in the use of likelihood-free methods of Bayesian inference for complex stochastic models. Having been thoroughly updated to reflect this, this third edition covers everything necessary for a good appreciation of stochastic kinetic modelling of biological networks in the systems biology context. New methods and applications are included in the book, and the use of R for practical illustration of the algorithms has been greatly extended. There is a brand new chapter on spatially extended systems, and the statistical inference chapter has also been extended with new methods, including approximate Bayesian computation (ABC). Stochastic Modelling for Systems Biology, Third Edition is now supplemented by an additional software library, written in Scala, described in a new appendix to the book.New in the Third Edition New chapter on sTrade Review"...stochastic modeling has drawn the attention of many researchers in biology and physiology. A textbook, with much elaboration, is highly valuable to understanding the underlying mathematical and computational methods in biological stochastic modeling. Prof Wilkinson has designed the content of this book to fill a gap in the educational text/reference books available for students/researchers learning about stochastic modeling in biological systems... This third edition book almost covers all of the material necessary for students studying stochastic kinetics modelling. The exercises in every chapter certainly illustrate the theory and concept of the book. Appendices A and B elaborate on all of the SBML code and other software associated with the book. The codes are also complemented by links to the author’s webpage and a GitHub repository. The author must be appreciated for adding so many references for further reading. The content of the book is designed for a one-semester graduate-level course in stochastic modeling in biology. Thus, this book is targeted at master and graduate students in interdisciplinary subjects such as applied mathematics, computational biology, bioinformatics, biophysics, Biochemistry, and biomedical engineering."- Chitaranjan Mahapatra, Appeared in ISCB News, January 2020 Table of ContentsIntroduction to biological modelling Representation of biochemical networks Probability models Stochastic simulation Markov processes Chemical and biochemical kinetics Case studies Beyond the Gillespie algorithm Spatially extended systems Bayesian inference and MCMC Inference for stochastic kinetic models Conclusions Appendices
£111.89
Taylor & Francis Ltd Performance Analysis and Synthesis for
Book SynopsisThe book addresses the system performance with a focus on the network-enhanced complexities and developing the engineering-oriented design framework of controllers and filters with potential applications in system sciences, control engineering and signal processing areas. Therefore, it provides a unified treatment on the analysis and synthesis for discrete-time stochastic systems with guarantee of certain performances against network-enhanced complexities with applications in sensor networks and mobile robotics. Such a result will be of great importance in the development of novel control and filtering theories including industrial impact.Key Features Provides original methodologies and emerging concepts to deal with latest issues in the control and filtering with an emphasis on a variety of network-enhanced complexities Gives results of stochastic control and filtering distributed control and filtering, and security control of complex nTable of Contents1 Introduction. 2 Finite-Horizon H∞ Control with Randomly Occurring Non-linearities and Fading Measurements. 3. Finite-Horizon H∞ Consensus Control for Multi-Agent Systems with Missing Measurements. 4 Finite-Horizon Distributed H∞ State Estimation with Stochastic Parameters through Sensor Networks. 5 Finite-Horizon Dissipative Control for State-Saturated Discrete Time-Varying Systems with Missing Measurements. 6 Finite-Horizon H∞ Filtering for State-Saturated Discrete Time-Varying Systems with Packet Dropouts. 7 Finite-Horizon Envelope-Constrained H∞ Filtering with Fading Measurements. 8 Distributed Filtering under Uniform Quantizations and Deception Attacks through Sensor Networks. 9 Event-Triggered Distributed H∞ State Estimation with Packet Dropouts through Sensor Networks. 10 Event-Triggered Consensus Control for Multi-Agent Systems in the Framework of Input-to-State Stability in Probability. 11 Event-Triggered Security Control for Discrete-Time Stochastic Systems subject to Cyber-Attacks. 12 Event-Triggered Consensus Control for Multi-Agent Systems subject to Cyber-Attacks in the Framework of Observers.
£166.25
Cambridge University Press Stochastic Integration with Jumps 89 Encyclopedia of Mathematics and its Applications Series Number 89
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£70.82
Cambridge University Press Random Fields on the Sphere Representation Limit Theorems and Cosmological Applications 389 London Mathematical Society Lecture Note Series Series Number 389
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£70.29
Cambridge University Press Positive Harmonic Functions and Diffusion 45 Cambridge Studies in Advanced Mathematics Series Number 45
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£133.95
Cambridge University Press Stochastic Approximation
Book SynopsisSimple, compact toolkit for designing and analyzing algorithms, with concrete examples from control and communications engineering, artificial intelligence, economic modelling.Trade Review'I highly recommend [this book] to all readers interested in the theory of recursive algorithms and its applications in practice.' Mathematical Reviews'This simple compact toolkit for designing and analyzing stochastic approximation algorithms requires only basic literacy in probability and differential equations … Ideal for graduate students, researchers and practitioners in electrical engineering and computer science, especially those working in control, communications, signal processing and machine learning, this book is also relevant to economics, probability and statistics.' L'Enseignement MathématiqueTable of ContentsPreface; 1. Introduction; 2. Basic convergence analysis; 3. Stability criteria; 4. Lock-in probability; 5. Stochastic recursive inclusions; 6. Multiple timescales; 7. Asynchronous schemes; 8. A limit theorem for fluctuations; 9. Constant stepsize algorithms; 10. Applications; 11. Appendices; References; Index.
£55.09
Cambridge University Press Stochastic Approximation Cambridge Tracts in Mathematics
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£36.09
Cambridge University Press General Irreducible Markov Chains and NonNegative Operators
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£42.74
Cambridge University Press A Guide to FirstPassage Processes
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£105.45
Cambridge University Press An Introduction to Computational Stochastic PDEs 50 Cambridge Texts in Applied Mathematics Series Number 50
Book SynopsisThis book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of-the-art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB codes included (and downloadable) allows readers to perform computations themselves and solve the teTrade Review'This book gives both accessible and extensive coverage on stochastic partial differential equations and their numerical solutions. It offers a well-elaborated background needed for solving numerically stochastic PDEs, both parabolic and elliptic. For the numerical solutions it presents not only proofs of convergence results of different numerical methods but also actual implementations, here in Matlab, with technical details included … With numerical implementations hard to find elsewhere in the literature, and a nice presentation of new research findings together with rich references, the book is a welcome companion for anyone working on numerical solutions of stochastic PDEs, and may also be suitable for use in a course on computational stochastic PDEs.' Roger Pettersson, Mathematical ReviewsTable of ContentsPart I. Deterministic Differential Equations: 1. Linear analysis; 2. Galerkin approximation and finite elements; 3. Time-dependent differential equations; Part II. Stochastic Processes and Random Fields: 4. Probability theory; 5. Stochastic processes; 6. Stationary Gaussian processes; 7. Random fields; Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs); 9. Elliptic PDEs with random data; 10. Semilinear stochastic PDEs.
£52.24
Cambridge University Press Markov Chains and Stochastic Stability Cambridge Mathematical Library
Book SynopsisMeyn and Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.Trade Review'This second edition remains true to the remarkable standards of scholarship established by the first edition … it will no doubt be a very welcome addition to the literature.' Peter W. Glynn, Prologue to the Second EditionTable of ContentsList of figures; Prologue to the second edition Peter W. Glynn; Preface to the second edition Sean Meyn; Preface to the first edition; Part I. Communication and Regeneration: 1. Heuristics; 2. Markov models; 3. Transition probabilities; 4. Irreducibility; 5. Pseudo-atoms; 6. Topology and continuity; 7. The nonlinear state space model; Part II. Stability Structures: 8. Transience and recurrence; 9. Harris and topological recurrence; 10. The existence of Π; 11. Drift and regularity; 12. Invariance and tightness; Part III. Convergence: 13. Ergodicity; 14. f-Ergodicity and f-regularity; 15. Geometric ergodicity; 16. V-Uniform ergodicity; 17. Sample paths and limit theorems; 18. Positivity; 19. Generalized classification criteria; 20. Epilogue to the second edition; Part IV. Appendices: A. Mud maps; B. Testing for stability; C. Glossary of model assumptions; D. Some mathematical background; Bibliography; Indexes.
£72.19
Cambridge University Press Levy Processes and Stochastic Calculus 116 Cambridge Studies in Advanced Mathematics Series Number 116
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£76.94
Cambridge University Press Stochastic Physics and Climate Modelling
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£128.25
Cambridge University Press Stochastic Calculus and Differential Equations for Physics and Finance
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£118.75
Cambridge University Press Ergodic Control of Diffusion Processes 143 Encyclopedia of Mathematics and its Applications Series Number 143
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£94.99
Cambridge University Press Derivatives in Financial Markets with Stochastic Volatility
Book SynopsisThis book, first published in 2000, addresses financial mathematics of pricing and hedging derivative securities in uncertain and changing market volatility. The mathematics is introduced through examples and illustrated with simulations, and the modeling approach described is validated and tested on market data. The material is suitable for a one-semester course for graduate students.Trade Review'… provides a good overview to the theoretical and practical problems when dealing with stochastic volatility'. Ralf Korn, Mathematical Methods of Operations Research'… something genuinely new … explained with admirable clarity in this extremely well-written book … [which] is short and to the point, and the production quality is high. Buy it.' Mark Davis, Risk Magazine'… well written and makes ideal reading for a graduate course on mathematical finance. The authors took great care in making their ideas clear. I support this text strongly and recommend it for the intended audience.' P. A. L. Embrechts, Publication of the International Statistical Institute'Thanks to a well-written first chapter on the Black-Scholes theory of derivative pricing, the book is essentially self-contained if one has some basic knowledge in stochastic methods and arbitrage pricing. Its style is largely informal which makes it also accessible to practitioners in the finance industry.' M. Schweizer, Zentralblatt für Mathematik'… an excellent book that succeeds admirably in all its aims. It can satisfy both practitioners and researchers at the same time. It is very well written and it is concise and informative.' Angelos Dassios, The Statistician'I consider this book to be an outstanding achievement. the theory is practically very relevant and scientifically on a high level. The book also serves as a good introduction into the basic ideas of Mathematical Finance, putting emphasis on the techniques of partial differential equations. It can therefore also be recommended to readers with little knowledge about Mathematical Finance.' Monatshefte für MathematikTable of Contents1. The Black-Scholes theory of derivative pricing; 2. Introduction to stochastic volatility models; 3. Scales in mean-reverting stochastic volatility; 4. Tools for estimating the rate of mean-reversion; 5. Symptotics for pricing European derivatives; 6. Implementation and stability; 7. Hedging strategies; 8. Application to exotic derivatives; 9. Application to American derivatives; 10. Generalizations; 11. Applications to interest rates models.
£95.95
Cambridge University Press Graph Directed Markov Systems Geometry and Dynamics of Limit Sets 148 Cambridge Tracts in Mathematics Series Number 148
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£105.45
Cambridge University Press Statistical Analysis of Stochastic Processes in Time
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£80.74
Cambridge University Press A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends
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£95.00
Cambridge University Press Stochastic Partial Differential Equations with Lvy Noise An Evolution Equation Approach 113 Encyclopedia of Mathematics and its Applications Series Number 113
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£125.40
Cambridge University Press Partial Differential Equations for Probabilists
Book SynopsisThis book provides probabilists with sufficient background to begin applying PDEs to probability theory and probability theory to PDEs. It covers the theory of linear and second order PDEs of parabolic and elliptic type. While most of the techniques described have antecedents in probability theory, the book does cover a few purely analytic techniques.Trade Review'The book will capture your attention with elegant proofs presented in an almost perfectly self-contained manner, with abundant talk in a lecturer's tone by the author himself, but with a little bit of an aficionado's taste. The book, arranged idiosyncratically, has such a strong impact that, at the next moment, you may find yourself carried away in looking for mathematical treasures scattered here and there in each chapter. The reviewer recommends the present book with confidence to anyone who in interested in PDE and probability theory. At least you should always keep this at your side if you are a probabilist at all.' Isamu Doku, Mathematical ReviewsTable of Contents1. Kolmogorov's forward, basic results; 2. Non-elliptic regularity results; 3. Preliminary elliptic regularity results; 4. Nash theory; 5. Localization; 6. On a manifold; 7. Subelliptic estimates and Hörmander's theorem.
£54.15
Cambridge University Press Finite Markov Chains and Algorithmic Applications 52 London Mathematical Society Student Texts Series Number 52
Book SynopsisBased on a lecture course given at Chalmers University of Technology, this 2002 book is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before applying it to study a range of randomized algorithms with important applications in optimization and other problems in computing. Amongst the algorithms covered are the Markov chain Monte Carlo method, simulated annealing, and the recent Propp-Wilson algorithm. This book will appeal not only to mathematicians, but also to students of statistics and computer science. The subject matter is introduced in a clear and concise fashion and the numerous exercises included will help students to deepen their understanding.Trade Review'Has climbing up onto the MCMC juggernaut seemed to require just too much effort? This delightful little monograph provides an effortless way in. The chapters are bite-sized with helpful, do-able exercises (by virtue of strategically placed hints) that complement the text.' Publication of the International Statistical Institute'… a very nice introduction to the modern theory of Markov chain simulation algorithms.' R. E. Maiboroda, Zentralblatt MATH' … extremely elegant. I am sure that students will find great pleasure in using the book - and that teachers will have the same pleasure in using it to prepare a course on the subject.' Mathematics of Computation'This elegant little book is a beautiful introduction to the theory of simulation algorithms, using (discrete) Markov chains (on finite state spaces) … highly recommended to anyone interested in the theory of Markov chain simulation algorithms.' Nieuw Archief voor WiskundeTable of Contents1. Basics of probability theory; 2. Markov chains; 3. Computer simulation of Markov chains; 4. Irreducible and aperiodic Markov chains; 5. Stationary distributions; 6. Reversible Markov chains; 7. Markov chain Monte Carlo; 8. Fast convergence of MCMC algorithms; 9. Approximate counting; 10. Propp-Wilson algorithm; 11. Sandwiching; 12. Propp-Wilson with read once randomness; 13. Simulated annealing; 14. Further reading.
£37.37
Cambridge University Press An Introduction to Computational Stochastic PDEs 50 Cambridge Texts in Applied Mathematics Series Number 50
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£105.45
Cambridge University Press Compound Renewal Processes
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£104.50
Cambridge University Press State Estimation for Robotics
Book SynopsisThis book is intended for students and practitioners of robotics working with noisy sensor data to estimate state variables. New edition highlights include a new chapter on variational inference and new sections on adaptive covariance estimation and on inertial navigation as well as a primer on matrix calculus.Trade Review'This book provides a timely, concise, and well-scoped introduction to state estimation for robotics. It complements existing textbooks by giving a balanced presentation of estimation theoretic and geometric tools and discusses how these tools can be used to solve common estimation problems arising in robotics. It also strikes an excellent balance between theory and motivating examples.' Luca Carlone, IEEE Control Systems MagazineTable of ContentsAcronyms and abbreviations; Notation; Foreword to first edition; Foreword to second edition; 1. Introduction; Part I. Estimation Machinery: 2. Primer on probability theory; 3. Linear-Gaussian estimation; 4. Nonlinear non-Gaussian estimation; 5. Handling nonidealities in estimation; 6. Variational inference; Part II. Three-Dimensional Machinery: 7. Primer on three-dimensional geometry; 8. Matrix lie groups; Part III. Applications: 9. Pose estimation problems; 10. Pose-and-point estimation problems; 11. Continuous-time estimation; Appendix A: matrix primer; Appendix B: rotation and pose extras; Appendix C: miscellaneous extras; Appendix D: solutions to exercises; References; Index.
£66.49
Cambridge University Press Generalized Normalizing Flows via Markov Chains
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£17.00
Cambridge University Press Stochastic Geometry for Wireless Networks
Book SynopsisCovering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry enables the effective analysis of wireless network performance across all possible network configurations, promoting good design choices for future wireless architectures and protocols that reduce interference effects.Trade Review'This book is a welcome addition to the rapidly developing area of applications of stochastic geometric models to telecommunications.' Ilya S. Molchanov, American Mathematical SocietyTable of ContentsPart I. Point Process Theory: 1. Introduction; 2. Description of point processes; 3. Point process models; 4. Sums and products over point processes; 5. Interference and outage in wireless networks; 6. Moment measures of point processes; 7. Marked point processes; 8. Conditioning and Palm theory; Part II. Percolation, Connectivity and Coverage: 9. Introduction; 10. Bond and site percolation; 11. Random geometric graphs and continuum percolation; 12. Connectivity; 13. Coverage; Appendix: introduction to R.
£85.49
Cambridge University Press Mathematics of TwoDimensional Turbulence 194 Cambridge Tracts in Mathematics Series Number 194
Book SynopsisThis book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D NavierâStokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) â proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.Table of Contents1. Preliminaries; 2. Two-dimensional Navier–Stokes equations; 3. Uniqueness of stationary measure and mixing; 4. Ergodicity and limiting theorems; 5. Inviscid limit; 6. Miscellanies; 7. Appendix; 8. Solutions to some exercises.
£66.49
Cambridge University Press Nonhomogeneous Random Walks Lyapunov Function Methods for NearCritical Stochastic Systems 209 Cambridge Tracts in Mathematics Series Number 209
Book SynopsisStochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototyTrade Review'This is another impressive volume in the prestigious `Cambridge Tracts in Mathematics' series … The authors of this book are well-known for their long standing and well-recognized contributions to this area of research. Besides their own results published over the last two decades, the authors cover all significant achievements up to date … It is remarkable to see detailed `Bibliographical notes' at the end of each chapter. The authors have done a great job by providing valuable information about the historical development of any topic treated in this book. We find extremely interesting facts, stories and references. All this makes the book more than interesting to read and use.' Jordan M. Stoyanov, Zentralblatt MATH'This book gives a comprehensive account of the study of random walks with spatially non-homogeneous transition kernels. The main theme is to study recurrence versus transience and moments of passage times, as well as path asymptotics, by constructing suitable Lyapunov functions, which define semi-martingales when composed with the random walk. Of special interest are the Lamperti processes, which are stochastic processes on [0, ∞) with drift vanishing asymptotically on the order of 1/x as the space variable x tends to infinity. … Each chapter ends with detailed bibliographical notes.' Rongfeng Sun, Mathematical ReviewsTable of Contents1. Introduction; 2. Semimartingale approach and Markov chains; 3. Lamperti's problem; 4. Many-dimensional random walks; 5. Heavy tails; 6. Further applications; 7. Markov chains in continuous time; Glossary of named assumptions; Bibliography; Index.
£128.25
Cambridge University Press Stochastic Equations in Infinite Dimensions 152 Encyclopedia of Mathematics and its Applications Series Number 152
Book SynopsisNow in its second edition, this book gives a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. Thoroughly updated, it also includes two brand new chapters surveying recent developments in the area.Trade ReviewReview of the first edition: 'The exposition is excellent and readable throughout, and should help bring the theory to a wider audience.' Daniel L. Ocone, Stochastics and Stochastic ReportsReview of the first edition: '… a welcome contribution to the rather new area of infinite dimensional stochastic evolution equations, which is far from being complete, so it should provide both a useful background and motivation for further research.' Yuri Kifer, The Annals of ProbabilityReview of the first edition: '… an excellent book which covers a large part of stochastic evolution equations with clear proofs and a very interesting analysis of their properties … In my opinion this book will become an indispensable tool for everybody working on stochastic evolution equations and related areas.' P. Kotelenez, American Mathematical SocietyTable of ContentsPreface; Introduction; Part I. Foundations: 1. Random variables; 2. Probability measures; 3. Stochastic processes; 4. Stochastic integral; Part II. Existence and Uniqueness: 5. Linear equations with additive noise; 6. Linear equations with multiplicative noise; 7. Existence and uniqueness for nonlinear equations; 8. Martingale solutions; 9. Markov property and Kolmogorov equation; 10. Absolute continuity and Girsanov theorem; 11. Large time behavior of solutions; 12. Small noise asymptotic; 13. Survey of specific equations; 14. Some recent developments; Appendix A. Linear deterministic equations; Appendix B. Some results on control theory; Appendix C. Nuclear and Hilbert–Schmidt operators; Appendix D. Dissipative mappings; Bibliography; Index.
£125.48
Cambridge University Press Probability on Graphs
Book SynopsisThis introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.Table of ContentsPreface; 1. Random walks on graphs; 2. Uniform spanning tree; 3. Percolation and self-avoiding walk; 4. Association and influence; 5. Further percolation; 6. Contact process; 7. Gibbs states; 8. Random-cluster model; 9. Quantum Ising model; 10. Interacting particle systems; 11. Random graphs; 12. Lorentz gas; References; Index.
£35.14
Cambridge University Press Applied Stochastic Differential Equations
Book SynopsisStochastic differential equations are differential equations whose solutions are stochastic processes. They exhibit appealing mathematical properties that are useful in modeling uncertainties and noisy phenomena in many disciplines. This book is motivated by applications of stochastic differential equations in target tracking and medical technology and, in particular, their use in methodologies such as filtering, smoothing, parameter estimation, and machine learning. It builds an intuitive hands-on understanding of what stochastic differential equations are all about, but also covers the essentials of Itô calculus, the central theorems in the field, and such approximation schemes as stochastic RungeKutta. Greater emphasis is given to solution methods than to analysis of theoretical properties of the equations. The book''s practical approach assumes only prior understanding of ordinary differential equations. The numerous worked examples and end-of-chapter exercises include application-Trade Review'Stochastic differential equations have long been used by physicists and engineers, especially in filtering and prediction theory, and more recently have found increasing application in the life sciences, finance and an ever-increasing range of fields. The authors provide intended users with an intuitive, readable introduction and overview without going into technical mathematical details from the often-demanding theory of stochastic analysis, yet clearly pointing out the pitfalls that may arise if its distinctive differences are disregarded. A large part of the book deals with underlying ideas and methods, such as analytical, approximative and computational, which are illustrated through many insightful examples. Linear systems, especially with additive noise and Gaussian solutions, are emphasized, though nonlinear systems are not neglected, and a large number of useful results and formulas are given. The latter part of the book provides an up to date survey and comparison of filtering and parameter estimation methods with many representative algorithms, and culminates with their application to machine learning.' Peter Kloeden, Johann Wolfgang Goethe-Universität Frankfurt am Main'Overall, this is a very well-written and excellent introductory monograph to SDEs, covering all important analytical properties of SDEs, and giving an in-depth discussion of applied methods useful in solving various real-life problems.' Igor Cialenco, MathSciNet'Chapters are rich in examples, numerical simulations, illustrations, derivations and computational assignment' Martin Ondreját, the European Mathematical Society and the Heidelberg Academy of Sciences and HumanitiesTable of Contents1. Introduction; 2. Some background on ordinary differential equations; 3. Pragmatic introduction to stochastic differential equations; 4. Ito calculus and stochastic differential equations; 5. Probability distributions and statistics of SDEs; 6. Statistics of linear stochastic differential equations; 7. Useful theorems and formulas for SDEs; 8. Numerical simulation of SDEs; 9. Approximation of nonlinear SDEs; 10. Filtering and smoothing theory; 11. Parameter estimation in SDE models; 12. Stochastic differential equations in machine learning; 13. Epilogue.
£35.14
John Wiley & Sons Inc Levy Processes in Credit Risk
Book SynopsisLevy Processes in Credit Risk is an introductory guide to using Levy processes for credit risk modelling, covering all types of credit derivatives: from the single name vanillas such as CDSs right through to structured credit risk products such as CPPIs and CPDOs.Trade Review"This text introduces into the use of Levy processes in credit risk modeling. After a general overview of credit risk and standard credit derivatives, the authors provide a short introduction into Levy processes in general. This material is then used to study single-name credit derivatives. Following this, the authors introduce into firm-value Levy models, including the Merton model, Black-Cox model, Levy first passage model, variance gamma model and the one sided Levy default model. The problem of calibration is discussed. After that, the authors introduce intensity Levy models such as the Jarrow and Turnbull model, the Cox model and the intensity-OU model. Multivariate credit products, collateralized debt obligations and multivariate index modeling are discussed in the following. In the final part of their book, the authors study credit CPPIs and CPDOs as well as asset-backed securities." (Zentralblatt MATH, 2010) Table of ContentsPreface. Acknowledgements. PART I: INTRODUCTION. 1 An Introduction to Credit Risk. 1.1 Credit Risk. 1.1.1 Historical and Risk-Neutral Probabilities. 1.1.2 Bond Prices and Default Probability. 1.2 Credit Risk Modelling. 1.3 Credit Derivatives. 1.4 Modelling Assumptions. 1.4.1 Probability Space and Filtrations. 1.4.2 The Risk-Free Asset. 2 An Introduction to Lévy Processes. 2.1 Brownian Motion. 2.2 Lévy Processes. 2.3 Examples of Lévy Processes. 2.3.1 Poisson Process. 2.3.2 Compound Poisson Process. 2.3.3 The Gamma Process. 2.3.4 Inverse Gaussian Process. 2.3.5 The CMY Process. 2.3.6 The Variance Gamma Process. 2.4 Ornstein–Uhlenbeck Processes. 2.4.1 The Gamma-OU Process. 2.4.2 The Inverse Gaussian-OU Process. PART II: SINGLE-NAME MODELLING. 3 Single-Name Credit Derivatives. 3.1 Credit Default Swaps. 3.1.1 Credit Default Swaps Pricing. 3.1.2 Calibration Assumptions. 3.2 Credit Default Swap Forwards. 3.2.1 Credit Default Swap Forward Pricing. 3.3 Constant Maturity Credit Default Swaps. 3.3.1 Constant Maturity Credit Default Swaps Pricing. 3.4 Options on CDS. 4 Firm-Value Lévy Models. 4.1 The Merton Model. 4.2 The Black–Cox Model with Constant Barrier. 4.3 The Lévy First-Passage Model. 4.4 The Variance Gamma Model. 4.4.1 Sensitivity to the Parameters. 4.4.2 Calibration on CDS Term Structure Curve. 4.5 One-Sided Lévy Default Model. 4.5.1 Wiener–Hopf Factorization and Default Probabilities. 4.5.2 Illustration of the Pricing of Credit Default Swaps. 4.6 Dynamic Spread Generator. 4.6.1 Generating Spread Paths. 4.6.2 Pricing of Options on CDSs. 4.6.3 Black’s Formulas and Implied Volatility. Appendix: Solution of the PDIE. 5 IntensityLévy Models. 5.1 Intensity Models for Credit Risk. 5.1.1 Jarrow–Turnbull Model. 5.1.2 Cox Models. 5.2 The Intensity-OU Model. 5.3 Calibration of the Model on CDS Term Structures. PART III: MULTIVARIATE MODELLING. 6 Multivariate Credit Products. 6.1 CDOs. 6.2 Credit Indices. 7 Collateralized Debt Obligations. 7.1 Introduction. 7.2 The Gaussian One-Factor Model. 7.3 Generic One-Factor Lévy Model. 7.4 Examples of Lévy Models. 7.5 Lévy Base Correlation. 7.5.1 The Concept of Base Correlation. 7.5.2 Pricing Non-Standard Tranches. 7.5.3 Correlation Mapping for Bespoke CDOs. 7.6 Delta-Hedging CDO tranches. 7.6.1 Hedging with the CDS Index. 7.6.2 Delta-Hedging with a Single-Name CDS. 7.6.3 Mezz-Equity hedging. 8 Multivariate Index Modelling. 8.1 Black’s Model. 8.2 VG Credit Spread Model. 8.3 Pricing Swaptions using FFT. 8.4 Multivariate VG Model. PART IV: EXOTIC STRUCTURED CREDIT RISK PRODUCTS. 9 Credit CPPIs and CPDOs. 9.1 Introduction. 9.2 CPPIs. 9.3 Gap Risk. 9.4 CPDOs. 10 Asset-Backed Securities. 10.1 Introduction. 10.2 Default Models. 10.2.1 Generalized Logistic Default Model. 10.2.2 Lévy Portfolio Default Model. 10.2.3 Normal One-Factor Default Model. 10.2.4 Generic One-Factor Lévy Default Model. 10.3 Prepayment Models. 10.3.1 Constant Prepayment Model. 10.3.2 Lévy Portfolio Prepayment Model. 10.3.3 Normal One-Factor Prepayment Model. 10.4 Numerical Results. Bibliography. Index.
£110.42
John Wiley & Sons Inc VarianceConstrained MultiObjective Stochastic
Book Synopsis Unifies existing and emerging concepts concerning multi-objective control and stochastic control with engineering-oriented phenomena Establishes a unified theoretical framework for control and filtering problems for a class of discrete-time nonlinear stochastic systems with consideration to performance Includes case studies of several nonlinear stochastic systems Investigates the phenomena of incomplete information, including missing/degraded measurements, actuator failures and sensor saturations Considers both time-invariant systems and time-varying systems Exploits newly developed techniques to handle the emerging mathematical and computational challenges Table of ContentsPreface vii Acknowledgements ix List of Abbreviations xi 1 Introduction 1 1.1 Analysis and Synthesis of Nonlinear Stochastic Systems 2 1.1.1 Nonlinear Systems 3 1.1.2 Stochastic Systems 4 1.2 Multi-Objective Control and Filtering with Variance Constraints 5 1.2.1 Covariance Control Theory 5 1.2.2 Multiple Performance Requirements 7 1.2.3 Design Techniques for Nonlinear Stochastic Systems with Variance Constraints 9 1.2.4 A Special Case of Multi-Objective Design: Mixed H2/H1 Control/Filtering 11 1.3 Outline 12 2 Robust H1 Control with Variance Constraints 17 2.1 Problem Formulation 18 2.2 Stability, H1 Performance and Variance Analysis 20 2.2.1 Stability, H1 Performance Analysis 21 2.2.2 Variance Analysis 23 2.3 Robust Controller Design 27 2.4 Numerical Example 30 2.5 Summary 33 3 Robust Mixed H2=H1 Filtering 41 3.1 System Description and Problem Formulation 42 3.2 Algebraic Characterizations for Robust H2=H1 Filtering 44 3.2.1 Robust H2 Filtering 44 3.2.2 Robust H1 Filtering 50 3.3 Robust H2=H1 Filter Design Techniques 51 3.4 An Illustrative Example 60 3.5 Summary 62 4 Filtering with Missing Measurements 63 4.1 Problem Formulation 64 4.2 Stability and Variance Analysis 67 4.3 Robust Filter Design 71 4.4 Numerical Example 75 4.5 Summary 78 5 Robust Fault-Tolerant Control 87 5.1 Problem Formulation 88 5.2 Stability and Variance Analysis 90 5.3 Robust Controller Design 92 5.4 Numerical Example 98 5.5 Summary 103 6 Robust H2 SMC 105 6.1 The System Model 106 6.2 Robust H2 Sliding Mode Control 107 6.2.1 Switching Surface 107 6.2.2 Performances of the Sliding Motion 108 6.2.3 Computational Algorithm 114 6.3 Sliding Mode Controller 115 6.4 Numerical Example 116 6.5 Summary 118 7 Dissipative Control with Degraded Measurements 125 7.1 Problem Formulation 126 7.2 Stability, Dissipativity and Variance Analysis 129 7.3 Observer-Based Controller Design 134 7.3.1 Solvability of Multi-Objective Control Problem 134 7.3.2 Computational Algorithm 139 7.4 Numerical Example 140 7.5 Summary 142 8 Variance-Constrained H1 Control with Multiplicative Noises 145 8.1 Problem Formulation 146 8.2 Stability, H1 Performance, Variance Analysis 147 8.2.1 Stability 148 8.2.2 H1 performance 150 8.2.3 Variance analysis 152 8.3 Robust State Feedback Controller Design 153 8.4 A Numerical Example 156 8.5 Summary 157 9 Robust Finite-Horizon H1 Control 159 9.1 Problem Formulation 160 9.2 Performance Analysis 162 9.2.1 H1 Performance 162 9.2.2 Variance Analysis 164 9.3 Robust Finite Horizon Controller Design 167 9.4 Numerical Example 171 9.5 Summary 173 10 Finite-Horizon Filtering with Degraded Measurements 177 10.1 Problem Formulation 178 10.2 Performance Analysis 181 10.2.1 H1 Performance Analysis 181 10.2.2 System Covariance Analysis 186 10.3 Robust Filter Design 187 10.4 Numerical Example 190 10.5 Summary 191 11 Mixed H2=H1 Control with Randomly Occurring Nonlinearities: the Finite-Horizon Case 197 11.1 Problem Formulation 199 11.2 H1 Performance 200 11.3 Mixed H2=H1 Controller Design 204 11.3.1 State-Feedback Controller Design 204 11.3.2 Computational Algorithm 207 11.4 Numerical Example 207 11.5 Summary 211 12 Finite-Horizon H2=H1 Control of MJSs with Sensor Failures 213 12.1 Problem Formulation 214 12.2 H1 Performance 216 12.3 Mixed H2=H1 Controller Design 220 12.3.1 Controller Design 220 12.3.2 Computational Algorithm 224 12.4 Numerical Example 224 12.5 Summary 227 13 Finite-Horizon Control with ROSF 229 13.1 Problem Formulation 230 13.2 H1 And Covariance Performances Analysis 234 13.2.1 H1 Performance 234 13.2.2 Covariance Analysis 238 13.3 Robust Finite-Horizon Controller Design 240 13.3.1 Controller Design 240 13.3.2 Computational Algorithm 243 13.4 Numerical Example 243 13.5 Summary 244 14 Finite-Horizon H2=H1 Control with Actuator Failures 247 14.1 Problem Formulation 248 14.2 H1 Performance 251 14.3 Multi-Objective Controller Design 253 14.3.1 Controller Design 253 14.3.2 Computational Algorithm 256 14.4 Numerical Example 257 14.5 Summary 259 15 Conclusions and Future Topics 261 References 263
£100.65
ISTE Ltd and John Wiley & Sons Inc Continuous Semi-Markov Processes
Book SynopsisThis title considers the special of random processes known as semi-Markov processes. These possess the Markov property with respect to any intrinsic Markov time such as the first exit time from an open set or a finite iteration of these times. The class of semi-Markov processes includes strong Markov processes, Lévy and Smith stepped semi-Markov processes, and some other subclasses. Extensive coverage is devoted to non-Markovian semi-Markov processes with continuous trajectories and, in particular, to semi-Markov diffusion processes. Readers looking to enrich their knowledge on Markov processes will find this book a valuable resource.Table of ContentsIntroduction 9 Chapter 1. Stepped Semi-Markov Processes 17 1.1. Random sequence 17 1.2. Markov chain 20 1.3. Two-dimensional Markov chain 25 1.4. Semi-Markov process 29 1.5. Stationary distributions 32 Chapter 2. Sequences of First Exit Times and Regeneration Times 37 2.1. Basic maps 38 2.2. Markov times 42 2.3. Deducing sequences 46 2.4. Correct exit and continuity 55 2.5. Time of regeneration 63 Chapter 3. General Semi-Markov Processes 71 3.1. Definition of a semi-Markov process 72 3.2. Transition function of a SM process 79 3.3. Operators and SM walk 82 3.4. Operators and SM process 91 3.5. Criterion of Markov property for SM processes 103 3.6. Intervals of constancy 110 Chapter 4. Construction of Semi-Markov Processes using Semi-Markov Transition Functions 115 4.1. Realization of an innite systemof pairs 116 4.2. Extension of a measure 121 4.3. Construction of a measure 124 4.4. Construction of a projective system of measures 127 4.5. Semi-Markov processes 133 Chapter 5. Semi-Markov Processes of Diffusion Type 137 5.1. One-dimensional semi-Markov processes of diffusion type 138 5.1.1. Differential equation 138 5.1.2. Construction SM process 143 5.1.3. Some properties of the process 159 5.2. Multi-dimensional processes of diffusion type 168 5.2.1. Differential equations of elliptic type 168 5.2.2. Neighborhood of arbitrary form 171 5.2.3. Neighborhood of spherical form 178 5.2.4. Characteristic operator 188 Chapter 6. Time Change and Semi-Markov Processes 197 6.1. Time change and trajectories 198 6.2. Intrinsic time and traces 206 6.3. Canonical time change 211 6.4. Coordination of function and time change 223 6.5. Random time changes 228 6.6. Additive functionals 233 6.7. Distribution of a time run along the trace 242 6.8. Random curvilinear integrals 252 6.9. Characteristic operator and integral 264 6.10. Stochastic integral 268 6.10.1. Semi-martingale and martingale 268 6.10.2. Stochastic integral 275 6.10.3. Ito-Dynkin’s formula 277 Chapter 7. Limit Theorems for Semi-Markov Processes 281 7.1. Weak compactness and weak convergence 281 7.2. Weak convergence of semi-Markov processes 289 Chapter 8. Representation of a Semi-Markov Process as a Transformed Markov Process 299 8.1. Construction by operator 300 8.2. Comparison of processes 302 8.3. Construction by parameters of Lévy formula 307 8.4. Stationary distribution 311 Chapter 9. Semi-Markov Model of Chromatography 325 9.1. Chromatography 326 9.2. Model of liquid column chromatography 328 9.3. Some monotone Semi-Markov processes 332 9.4. Transfer with diffusion 337 9.5. Transfer with nal absorption 346 Bibliography 361 Index 369
£163.35
ISTE Ltd and John Wiley & Sons Inc Switching Processes in Queueing Models
Book SynopsisSwitching processes, invented by the author in 1977, is the main tool used in the investigation of traffic problems from automotive to telecommunications. The title provides a new approach to low traffic problems based on the analysis of flows of rare events and queuing models. In the case of fast switching, averaging principle and diffusion approximation results are proved and applied to the investigation of transient phenomena for wide classes of overloading queuing networks. The book is devoted to developing the asymptotic theory for the class of switching queuing models which covers models in a Markov or semi-Markov environment, models under the influence of flows of external or internal perturbations, unreliable and hierarchic networks, etc.Table of ContentsPreface 13 Definitions 17 Chapter 1. Switching Stochastic Models 19 1.1. Random processes with discrete component 19 1.1.1.Markov and semi-Markov processes 21 1.1.2. Processes with independent increments and Markov switching 21 1.1.3. Processes with independent increments and semi-Markov switching 23 1.2. Switching processes 24 1.2.1. Definition of switching processes 24 1.2.2. Recurrent processes of semi-Markov type (simple case) 26 1.2.3.RPSMwithMarkov switching 26 1.2.4. General case of RPSM 27 1.2.5. Processes with Markov or semi-Markov switching 27 1.3. Switching stochastic models 28 1.3.1. Sums of random variables 29 1.3.2. Random movements 29 1.3.3. Dynamic systems in a random environment 30 1.3.4. Stochastic differential equations in a random environment 30 1.3.5. Branching processes 31 1.3.6. State-dependent flows 32 1.3.7. Two-level Markov systems with feedback 32 1.4. Bibliography 33 Chapter 2. Switching Queueing Models 37 2.1. Introduction 37 2.2. Queueing systems 38 2.2.1. Markov queueing models 38 2.2.1.1. A state-dependent system MQ/MQ/1/∞ 39 2.2.1.2. Queueing system MM,Q/MM,Q/1/m 40 2.2.1.3. System MQ,B/MQ,B/1/∞ 41 2.2.2.Non-Markov systems 42 2.2.2.1. Semi-Markov system SM/MSM,Q/1 42 2.2.2.2. System MSM,Q/MSM,Q/1/∞ 43 2.2.2.3. System MSM,Q/MSM,Q/1/V 44 2.2.3. Models with dependent arrival flows 45 2.2.4. Polling systems 46 2.2.5. Retrial queueing systems 47 2.3. Queueing networks 48 2.3.1. Markov state-dependent networks 49 2.3.1.1. Markov network (MQ/MQ/m/∞)r 49 2.3.1.2. Markov networks (MQ,B/MQ,B/m/∞)r with batches 50 2.3.2.Non-Markov networks 50 2.3.2.1. State-dependent semi-Markov networks 50 2.3.2.2. Semi-Markov networks with random batches 52 2.3.2.3. Networks with state-dependent input 53 2.4.Bibliography 54 Chapter 3. Processes of Sums of Weakly-dependent Variables 57 3.1. Limit theorems for processes of sums of conditionally independent random variables 57 3.2. Limit theorems for sums with Markov switching 65 3.2.1. Flows of rare events 67 3.2.1.1. Discrete time 67 3.2.1.2. Continuous time 69 3.3. Quasi-ergodic Markov processes 70 3.4. Limit theorems for non-homogenous Markov processes 73 3.4.1. Convergence to Gaussian processes 74 3.4.2. Convergence to processes with independent increments 78 3.5. Bibliography 81 Chapter 4. Averaging Principle and Diffusion Approximation for Switching Processes 83 4.1. Introduction 83 4.2. Averaging principle for switching recurrent sequences 84 4.3. Averaging principle and diffusion approximation for RPSMs 88 4.4. Averaging principle and diffusion approximation for recurrent processes of semi-Markov type (Markov case) 95 4.4.1. Averaging principle and diffusion approximation for SMP 105 4.5. Averaging principle for RPSM with feedback 106 4.6. Averaging principle and diffusion approximation for switching processes 108 4.6.1. Averaging principle and diffusion approximation for processes with semi-Markov switching 112 4.7. Bibliography 113 Chapter 5. Averaging and Diffusion Approximation in Overloaded Switching Queueing Systems and Networks 117 5.1. Introduction 117 5.2. Markov queueing models 120 5.2.1. System MQ,B/MQ,B/1/∞ 121 5.2.2. System MQ/MQ/1/∞ 124 5.2.3. Analysis of the waiting time 129 5.2.4. An output process 131 5.2.5. Time-dependent system MQ,t/MQ,t/1/∞ 132 5.2.6. Asystemwith impatient calls 134 5.3. Non-Markov queueing models 135 5.3.1. System GI/MQ/1/∞ 135 5.3.2. Semi-Markov system SM/MSM,Q/1/∞ 136 5.3.3. System MSM,Q/MSM,Q/1/∞ 138 5.3.4. System SMQ/MSM,Q/1/∞ 139 5.3.5. System GQ/MQ/1/∞ 142 5.3.6. A system with unreliable servers 143 5.3.7. Polling systems 145 5.4. Retrial queueing systems 146 5.4.1. Retrial system MQ/G/1/w.r 147 5.4.2. System M¯ /G¯/1/w.r 150 5.4.3. Retrial system M/M/m/w.r 154 5.5. Queueing networks 159 5.5.1. State-dependent Markov network (MQ/MQ/1/∞)r 159 5.5.2. Markov state-dependent networks with batches 161 5.6. Non-Markov queueing networks 164 5.6.1. A network (MSM,Q/MSM,Q/1/∞)r with semi-Markov switching 164 5.6.2. State-dependent network with recurrent input 169 5.7. Bibliography 172 Chapter 6. Systems in Low Traffic Conditions 175 6.1. Introduction 175 6.2. Analysis of the first exit time from the subset of states 176 6.2.1. Definition of S-set 176 6.2.2. An asymptotic behavior of the first exit time 177 6.2.3. State space forming a monotone structure 180 6.2.4. Exit time as the time of first jump of the process of sums with Markov switching 182 6.3. Markov queueing systems with fast service 183 6.3.1. M/M/s/m systems 183 6.3.1.1. System MM/M/l/m in a Markov environment 185 6.3.2. Semi-Markov queueing systems with fast service 188 6.4. Single-server retrial queueing model 190 6.4.1. Case 1: fast service 191 6.4.1.1. State-dependent case 194 6.4.2. Case 2: fast service and large retrial rate 195 6.4.3. State-dependent model in a Markov environment 197 6.5. Multiserver retrial queueing models 201 6.6. Bibliography 204 Chapter 7. Flows of Rare Events in Low and Heavy Traffic Conditions 207 7.1. Introduction 207 7.2. Flows of rare events in systems with mixing 208 7.3. Asymptotically connected sets (Vn-S-sets) 211 7.3.1. Homogenous case 211 7.3.2. Non-homogenous case 214 7.4. Heavy traffic conditions 215 7.5. Flows of rare events in queueing models 216 7.5.1. Light traffic analysis in models with finite capacity 216 7.5.2. Heavy traffic analysis 218 7.6. Bibliography 219 Chapter 8. Asymptotic Aggregation of State Space 221 8.1. Introduction 221 8.2. Aggregation of finite Markov processes (stationary behavior) 223 8.2.1. Discrete time 223 8.2.2. Hierarchic asymptotic aggregation 225 8.2.3. Continuous time 227 8.3. Convergence of switching processes 228 8.4. Aggregation of states in Markov models 231 8.4.1. Convergence of the aggregated process to a Markov process (finite state space) 232 8.4.2. Convergence of the aggregated process with a general state space 236 8.4.3. Accumulating processes in aggregation scheme 237 8.4.4. MP aggregation in continuous time 238 8.5. Asymptotic behavior of the first exit time from the subset of states (non-homogenous in time case) 240 8.6. Aggregation of states of non-homogenous Markov processes 243 8.7. Averaging principle for RPSM in the asymptotically aggregated Markov environment 246 8.7.1. Switching MP with a finite state space 247 8.7.2. Switching MP with a general state space 250 8.7.3. Averaging principle for accumulating processes in the asymptotically aggregated semi-Markov environment 251 8.8. Diffusion approximation for RPSM in the asymptotically aggregated Markov environment 252 8.9. Aggregation of states in Markov queueing models 255 8.9.1. System MQ/MQ/r/∞ with unreliable servers in heavy traffic 255 8.9.2. System MM,Q/MM,Q/1/∞ in heavy traffic 256 8.10. Aggregation of states in semi-Markov queueing models 258 8.10.1. System SM/MSM,Q/1/∞ 258 8.10.2. System MSM,Q/MSM,Q/1/∞ 259 8.11. Analysis of flows of lost calls 260 8.12. Bibliography 263 Chapter 9. Aggregation in Markov Models with Fast Markov Switching 267 9.1. Introduction 267 9.2. Markov models with fast Markov switching 269 9.2.1.Markov processes with Markov switching 269 9.2.2. Markov queueing systems with Markov type switching 271 9.2.3. Averaging in the fast Markov type environment 272 9.2.4. Approximation of a stationary distribution 274 9.3. Proofs of theorems 275 9.3.1. Proof of Theorem 9.1 275 9.3.2. Proof of Theorem 9.2 277 9.3.3. Proof of Theorem 9.3 279 9.4. Queueing systems with fast Markov type switching 279 9.4.1. System MM,Q/MM,Q/1/N 279 9.4.1.1. Averaging of states of the environment 279 9.4.1.2. The approximation of a stationary distribution 280 9.4.2. Batch system BMM,Q/BMM,Q/1/N 281 9.4.3. System M/M/s/mwith unreliable servers 282 9.4.4. Priority model MQ/MQ/m/s,N 283 9.5. Non-homogenous in time queueing models 285 9.5.1. SystemMM,Q,t/MM,Q,t/s/m with fast switching – averaging of states 286 9.5.2. System MM,Q/MM,Q/s/m with fast switching – aggregation of states 287 9.6. Numerical examples 288 9.7. Bibliography 289 Chapter 10. Aggregation in Markov Models with Fast Semi-Markov Switching 291 10.1. Markov processes with fast semi-Markov switches 292 10.1.1.Averaging of a semi-Markov environment 292 10.1.2. Asymptotic aggregation of a semi-Markov environment 300 10.1.3. Approximation of a stationary distribution 305 10.2. Averaging and aggregation in Markov queueing systems with semi-Markov switching 309 10.2.1.Averaging of states of the environment 309 10.2.2. Asymptotic aggregation of states of the environment 310 10.2.3. The approximation of a stationary distribution 311 10.3. Bibliography 313 Chapter 11. Other Applications of Switching Processes 315 11.1. Self-organization in multicomponent interacting Markov systems 315 11.2. Averaging principle and diffusion approximation for dynamic systems with stochastic perturbations 319 11.2.1. Recurrent perturbations 319 11.2.2. Semi-Markov perturbations 321 11.3. Random movements 324 11.3.1. Ergodic case 324 11.3.2. Case of the asymptotic aggregation of state space 325 11.4. Bibliography 326 Chapter 12. Simulation Examples 329 12.1. Simulation of recurrent sequences 329 12.2. Simulation of recurrent point processes 331 12.3. Simulation ofRPSM 332 12.4. Simulation of state-dependent queueing models 334 12.5. Simulation of the exit time from a subset of states of a Markov chain 337 12.6. Aggregation of states in Markov models 340 Index 343
£150.05
ISTE Ltd and John Wiley & Sons Inc Introduction to Stochastic Models
Book SynopsisThis book provides a pedagogical examination of the way in which stochastic models are encountered in applied sciences and techniques such as physics, engineering, biology and genetics, economics and social sciences. It covers Markov and semi-Markov models, as well as their particular cases: Poisson, renewal processes, branching processes, Ehrenfest models, genetic models, optimal stopping, reliability, reservoir theory, storage models, and queuing systems. Given this comprehensive treatment of the subject, students and researchers in applied sciences, as well as anyone looking for an introduction to stochastic models, will find this title of invaluable use.Table of ContentsPreface ix Chapter 1. Introduction to Stochastic Processes 1 1.1. Sequences of random variables 1 1.2. The notion of stochastic process 10 1.3. Martingales 13 1.4. Markov chains 17 1.5. State classification 24 1.6. Continuous-time Markov processes 27 1.7. Semi-Markov processes 33 Chapter 2. Simple Stochastic Models 37 2.1. Urn models 37 2.2. Random walks 39 2.3. Brownian motion 44 2.4. Poisson processes 50 2.5. Birth and death processes 59 Chapter 3. Elements of Markov Modeling 61 3.1. Markov models: ideas, history, applications 61 3.2. The discrete-time Ehrenfest model 63 3.3. Markov models in genetics 79 3.4. Markov storage models 110 3.5. Reliability of Markov models 124 Chapter 4. Renewal Models 149 4.1. Fundamental concepts and examples 149 4.2. Waiting times 155 4.3. Modified renewal processes 159 4.4. Replacement models 161 4.5. Renewal reward processes 165 4.6. The risk problem of an insurance company 168 4.7. Counter models 171 4.8. Alternating renewal processes 180 4.9. Superposition of renewal processes 182 4.10. Regenerative processes 186 Chapter 5. Semi-Markov Models 189 5.1. Introduction 189 5.2. Markov renewal processes 190 5.3. First-passage times and state classification 196 5.4. Reliability 200 5.5. Reservoir models 207 5.6. Queues 218 5.7. Digital communication channels 222 Chapter 6. Branching Models 227 6.1. The Bienaymé-Galton-Watson model 227 6.2. Generalizations of the B-G-W model 271 6.3. Continuous-time models 302 Chapter 7. Optimal Stopping Models 315 7.1. The classic optimal stopping problem 315 7.2. Renewal with binary decision 333 Bibliography 343 Notation 367 Index 369
£132.00
ISTE Ltd and John Wiley & Sons Inc Discrete-time Asset Pricing Models in Applied
Book SynopsisStochastic finance and financial engineering have been rapidly expanding fields of science over the past four decades, mainly due to the success of sophisticated quantitative methodologies in helping professionals manage financial risks. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending, modeling and hedging this kind of risk. These two volumes aim to provide a foundation course on applied stochastic finance. They are designed for three groups of readers: firstly, students of various backgrounds seeking a core knowledge on the subject of stochastic finance; secondly financial analysts and practitioners in the investment, banking and insurance industries; and finally other professionals who are interested in learning advanced mathematical and stochastic methods, which are basic knowledge in many areas, through finance. Volume 1 starts with the introduction of the basic financial instruments and the fundamental principles of financial modeling and arbitrage valuation of derivatives. Next, we use the discrete-time binomial model to introduce all relevant concepts. The mathematical simplicity of the binomial model also provides us with the opportunity to introduce and discuss in depth concepts such as conditional expectations and martingales in discrete time. However, we do not expand beyond the needs of the stochastic finance framework. Numerous examples, each highlighted and isolated from the text for easy reference and identification, are included. The book concludes with the use of the binomial model to introduce interest rate models and the use of the Markov chain model to introduce credit risk. This volume is designed in such a way that, among other uses, makes it useful as an undergraduate course.Table of ContentsPreface xi Chapter 1. Probability and Random Variables 1 1.1. Introductory notes 1 1.2. Probability space 2 1.3. Conditional probability and independence 8 1.4. Random variables 12 1.5. Expectation and variance of a random variable 24 1.6. Jointly distributed random variables 28 1.7. Moment generating functions 32 1.8. Probability inequalities and limit theorems 37 1.9.Multivariate normal distribution 44 Chapter 2. An Introduction to Financial Instruments and Derivatives 49 2.1. Introduction 49 2.2. Bonds and basic interest rates 50 2.3. Forward contracts 58 2.4. Futures contracts 60 2.5.Swaps 60 2.6.Options 62 2.7. Types of market participants 67 2.8.Arbitrage relationships between call and put options 67 2.9.Exercises 69 Chapter 3. Conditional Expectation and Markov Chains 71 3.1. Introduction 71 3.2. Conditional expectation: the discrete case 72 3.3. Applications of conditional expectations 75 3.4. Properties of the conditional expectation 81 3.5. Markov chains 85 3.6. Exercises 131 4.1. Introductory notes 137 Chapter 4. The No-Arbitrage Binomial Pricing Model 137 4.2. Binomial model 138 4.3. Stochastic evolution of the asset prices 141 4.4. Binomial approximation to the lognormal distribution 143 4.5. One-period European call option 145 4.6. Two-period European call option 150 4.7. Multiperiod binomial model 153 4.8. The evolution of the asset prices as a Markov chain 154 4.9.Exercises 158 Chapter 5. Martingales 163 5.1. Introductory notes 163 5.2.Martingales 164 5.3. Optional sampling theorem 169 5.4. Submartingales, supermartingales and martingales convergence theorem 178 5.5.Martingale transforms 182 5.6. Uniform integrability and Doob’s decomposition 184 5.7.The snell envelope 187 5.8.Exercises 190 Chapter 6. Equivalent Martingale Measures, No-Arbitrage and Complete Markets 195 6.1. Introductory notes 195 6.2. Equivalent martingale measure and the Randon-Nikodým derivative process 196 6.3. Finite general markets 204 6.4. Fundamental theorem of asset pricing 215 6.5.Completemarkets andmartingale representation 222 6.6. Finding the equivalent martingale measure 228 6.7.Exercises 238 Chapter 7. American Derivative Securities 241 7.1. Introductory notes 241 7.2.A three-periodAmerican put option 242 7.3. Hedging strategy for an American put option 249 7.4.The algorithm of the American put option 254 7.5.Optimal time for the holder to exercise 255 7.6. American derivatives in general markets 262 7.7. Extending the concept of self-financing strategies 266 7.8.Exercises 269 Chapter 8. Fixed-Income Markets and Interest Rates 273 8.1. Introductory notes 273 8.2. The zero coupon bonds of all maturities 274 8.3. Arbitrage-free family of bond prices 278 8.4. Interest rate process and the term structure of bond prices 282 8.5. The evolution of the interest rate process 290 8.6. Binomial model with normally distributed spread of interest rates 293 8.7. Binomial model with lognormally distributed spread of interest rates 296 8.8. Option arbitrage pricing on zero coupon bonds 298 8.9. Fixed income derivatives 302 8.10. T-period equivalent forward measure 308 8.11. Futures contracts 317 8.12.Exercises 319 Chapter 9. Credit Risk 323 9.1. Introductory notes 323 9.2. Credit ratings and corporate bonds 324 9.3. Credit risk methodologies 326 9.4. Arbitrage pricing of defaultable bonds 327 9.5. Migration process as a Markov chain 330 9.6. Estimation of the real world transition probabilities 334 9.7. Term structure of credit spread and model calibration 337 9.8. Migration process under the real-world probability measure 341 9.9.Exercises 352 Chapter 10. The Heath-Jarrow-Morton Model 355 10.1. Introductory notes 355 10.2. Heath-Jarrow-Morton model 356 10.3. Hedging strategies for zero coupon bonds 362 10.4.Exercises 364 References 365 Appendices 374 Index 395
£150.05
ISTE Ltd and John Wiley & Sons Inc Discrete Stochastic Processes and Optimal
Book SynopsisOptimal filtering applied to stationary and non-stationary signals provides the most efficient means of dealing with problems arising from the extraction of noise signals. Moreover, it is a fundamental feature in a range of applications, such as in navigation in aerospace and aeronautics, filter processing in the telecommunications industry, etc. This book provides a comprehensive overview of this area, discussing random and Gaussian vectors, outlining the results necessary for the creation of Wiener and adaptive filters used for stationary signals, as well as examining Kalman filters which are used in relation to non-stationary signals. Exercises with solutions feature in each chapter to demonstrate the practical application of these ideas using MATLAB.Table of ContentsPreface ix Introduction xi Chapter 1. Random Vectors 1 1.1. Definitions and general properties. 1 1.2. Spaces L1 (dP) and L2 (dP) 20 1.3. Mathematical expectation and applications 23 1.4. Second order random variables and vectors. 39 1.5. Linear independence of vectors of L2 (dP) 46 1.6. Conditional expectation (concerning random vectors with density function) 51 1.7. Exercises for Chapter 1 56 Chapter 2. Gaussian Vectors 63 2.1. Some reminders regarding random Gaussian vectors 63 2.2. Definition and characterization of Gaussian vectors 66 2.3. Results relative to independence 68 2.4. Affine transformation of a Gaussian vector 72 2.5. The existence of Gaussian vectors. 74 2.6. Exercises for Chapter 2 84 Chapter 3. Introduction to Discrete Time Processes 93 3.1. Definition 93 3.2. WSS processes and spectral measure 105 3.3. Spectral representation of a WSS process 109 3.4. Introduction to digital filtering 114 3.5. Important example: autoregressive process 127 3.6. Exercises for Chapter 3 132 Chapter 4. Estimation 139 4.1. Position of the problem 139 4.2. Linear estimation 142 4.3. Best estimate – conditional expectation 154 4.4. Example: prediction of an autoregressive process AR (1) 162 4.5. Multivariate processes 163 4.6. Exercises for Chapter 4 172 Chapter 5. The Wiener Filter 177 5.1. Introduction 177 5.2. Resolution and calculation of the FIR filter 179 5.3. Evaluation of the least error 181 5.4. Resolution and calculation of the IIR filter 183 5.5. Evaluation of least mean square error 187 5.6. Exercises for Chapter 5 188 Chapter 6. Adaptive Filtering: Algorithm of the Gradient and the LMS 195 6.1. Introduction 195 6.2. Position of problem 198 6.3. Data representation 200 6.4. Minimization of the cost function 202 6.5. Gradient algorithm 209 6.6. Geometric interpretation 212 6.7. Stability and convergence 216 6.8. Estimation of gradient and LMS algorithm 221 6.9. Example of the application of the LMS algorithm 224 6.10. Exercises for Chapter 6 233 Chapter 7. The Kalman Filter 235 7.1. Position of problem 235 7.2. Approach to estimation 239 7.3. Kalman filtering 243 7.4. Exercises for Chapter 7 261 7.5. Appendices 267 7.6. Examples treated using Matlab software 273 Table of Symbols and Notations 281 Bibliography 283 Index 285
£132.00
ISTE Ltd and John Wiley & Sons Inc Stochastic Methods for Pension Funds
Book SynopsisQuantitative finance has become these last years a extraordinary field of research and interest as well from an academic point of view as for practical applications. At the same time, pension issue is clearly a major economical and financial topic for the next decades in the context of the well-known longevity risk. Surprisingly few books are devoted to application of modern stochastic calculus to pension analysis. The aim of this book is to fill this gap and to show how recent methods of stochastic finance can be useful for to the risk management of pension funds. Methods of optimal control will be especially developed and applied to fundamental problems such as the optimal asset allocation of the fund or the cost spreading of a pension scheme. In these various problems, financial as well as demographic risks will be addressed and modelled.Table of ContentsPreface xiii Chapter 1. Introduction: Pensions in Perspective 1 1.1. Pension issues 1 1.2. Pension scheme 7 1.3. Pension and risks 11 1.4. The multi-pillar philosophy 14 Chapter 2. Classical Actuarial Theory of Pension Funding 15 2.1. General equilibrium equation of a pension scheme 15 2.2. General principles of funding mechanisms for DB Schemes 21 2.3. Particular funding methods 22 Chapter 3. Deterministic and Stochastic Optimal Control 31 3.1. Introduction 31 3.2. Deterministic optimal control 31 3.3. Necessary conditions for optimality 33 3.4. The maximum principle 42 3.5. Extension to the one-dimensional stochastic optimal control 45 3.6. Examples 52 Chapter 4. Defined Contribution and Defined Benefit Pension Plans 55 4.1. Introduction 55 4.2. The defined benefit method 56 4.3. The defined contribution method 57 4.4. The notional defined contribution (NDC) method 58 4.5. Conclusions 93 Chapter 5. Fair and Market Values and Interest Rate Stochastic Models 95 5.1. Fair value 95 5.2. Market value of financial flows 96 5.3. Yield curve 97 5.4. Yield to maturity for a financial investment and for a bond 99 5.5. Dynamic deterministic continuous time model for an instantaneous interest rate 100 5.6. Stochastic continuous time dynamic model for an instantaneous interest rate 104 5.7. Zero-coupon pricing under the assumption of no arbitrage 114 5.8. Market evaluation of financial flows 130 5.9. Stochastic continuous time dynamic model for asset values 132 5.10. VaR of one asset 136 Chapter 6. Risk Modeling and Solvency for Pension Funds 149 6.1. Introduction 149 6.2. Risks in defined contribution 149 6.3. Solvency modeling for a DC pension scheme 150 6.4. Risks in defined benefit 170 6.5. Solvency modeling for a DB pension scheme 171 Chapter 7. Optimal Control of a Defined Benefit Pension Scheme 181 7.1. Introduction 181 7.2. A first discrete time approach: stochastic amortization strategy 181 7.3. Optimal control of a pension fund in continuous time 194 Chapter 8. Optimal Control of a Defined Contribution Pension Scheme 207 8.1. Introduction 207 8.2. Stochastic optimal control of annuity contracts 208 8.3. Stochastic optimal control of DC schemes with guarantees and under stochastic interest rates 223 Chapter 9. Simulation Models 231 9.1. Introduction231 9.2. The direct method 233 9.3. The Monte Carlo models 250 9.4. Salary lines construction 252 Chapter 10. Discrete Time Semi-Markov Processes (SMP) and Reward SMP 277 10.1. Discrete time semi-Markov processes 277 10.2. DTSMP numerical solutions 280 10.3. Solution of DTHSMP and DTNHSMP in the transient case: a transportation example 284 10.4. Discrete time reward processes 294 10.5. General algorithms for DTSMRWP 304 Chapter 11. Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management 307 11.1. Application to pension funds evolution 307 11.2. Generalized non-homogeneous semi-Markov model for manpower management 338 11.3. Algorithms 347 APPENDICES 359 Appendix 1. Basic Probabilistic Tools for Stochastic Modeling 361 Appendix 2. Itô Calculus and Diffusion Processes 397 Bibliography 437 Index 449
£158.60
ISTE Ltd and John Wiley & Sons Inc Introduction to Stochastic Analysis: Integrals
Book SynopsisThis is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes. The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô’s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.Trade Review“Thus, the book is a welcome addition in the effort to make stochastic integration and SDE as accessible as possible to the greater public interested in or in need of using them.” (Mathematical Reviews, 1 February 2013) “If I have a chance to teach (again) a course in stochastic financial modelling, I will definitely choose this to be among two or three sources to use. I have all the reasons to strongly recommend it to anybody in the area of modern stochastic modelling.” (Zentralblatt MATH, 1 December 2012)Table of ContentsPreface 9 Notation 13 Chapter 1. Introduction: Basic Notions of Probability Theory 17 1.1. Probability space 17 1.2. Random variables 21 1.3. Characteristics of a random variable 21 1.4. Types of random variables 23 1.5. Conditional probabilities and distributions 26 1.6. Conditional expectations as random variables 27 1.7. Independent events and random variables 29 1.8. Convergence of random variables 29 1.9. Cauchy criterion 31 1.10. Series of random variables 31 1.11. Lebesgue theorem 32 1.12. Fubini theorem 32 1.13. Random processes 33 1.14. Kolmogorov theorem 34 Chapter 2. Brownian Motion 35 2.1. Definition and properties 35 2.2. White noise and Brownian motion 45 2.3. Exercises 49 Chapter 3. Stochastic Models with Brownian Motion and White Noise 51 3.1. Discrete time 51 3.2. Continuous time 55 Chapter 4. Stochastic Integral with Respect to Brownian Motion 59 4.1. Preliminaries. Stochastic integral with respect to a step process 59 4.2. Definition and properties 69 4.3. Extensions 81 4.4. Exercises 85 Chapter 5. Itô’s Formula 87 5.1. Exercises 94 Chapter 6. Stochastic Differential Equations 97 6.1. Exercises 105 Chapter 7. Itô Processes 107 7.1. Exercises 121 Chapter 8. Stratonovich Integral and Equations 125 8.1. Exercises 136 Chapter 9. Linear Stochastic Differential Equations 137 9.1. Explicit solution of a linear SDE 137 9.2. Expectation and variance of a solution of an LSDE 141 9.3. Other explicitly solvable equations 145 9.4. Stochastic exponential equation 147 9.5. Exercises 153 Chapter 10. Solutions of SDEs as Markov Diffusion Processes 155 10.1. Introduction 155 10.2. Backward and forward Kolmogorov equations 161 10.3. Stationary density of a diffusion process 172 10.4. Exercises 176 Chapter 11. Examples 179 11.1. Additive noise: Langevin equation 180 11.2. Additive noise: general case 180 11.3. Multiplicative noise: general remarks 184 11.4. Multiplicative noise: Verhulst equation 186 11.5. Multiplicative noise: genetic model 189 Chapter 12. Example in Finance: Black–Scholes Model 195 12.1. Introduction: what is an option? 195 12.2. Self-financing strategies 197 12.3. Option pricing problem: the Black–Scholes model 204 12.4. Black–Scholes formula 206 12.5. Risk-neutral probabilities: alternative derivation of Black–Scholes formula 210 12.6. Exercises 214 Chapter 13. Numerical Solution of Stochastic Differential Equations 217 13.1. Memories of approximations of ordinary differential equations 218 13.2. Euler approximation 221 13.3. Higher-order strong approximations 224 13.4. First-order weak approximations 231 13.5. Higher-order weak approximations 238 13.6. Example: Milstein-type approximations 241 13.7. Example: Runge–Kutta approximations 244 13.8. Exercises 249 Chapter 14. Elements of Multidimensional Stochastic Analysis 251 14.1. Multidimensional Brownian motion 251 14.2. Itô’s formula for a multidimensional Brownian motion 252 14.3. Stochastic differential equations 253 14.4. Itô processes 254 14.5. Itô’s formula for multidimensional Itô processes 256 14.6. Linear stochastic differential equations 256 14.7. Diffusion processes 257 14.8. Approximations of stochastic differential equations 259 Solutions, Hints, and Answers 261 Bibliography 271 Index 273
£132.00
Springer Nature Switzerland AG Random Graphs, Phase Transitions, and the Gaussian Free Field: PIMS-CRM Summer School in Probability, Vancouver, Canada, June 5–30, 2017
Book SynopsisThe 2017 PIMS-CRM Summer School in Probability was held at the Pacific Institute for the Mathematical Sciences (PIMS) at the University of British Columbia in Vancouver, Canada, during June 5-30, 2017. It had 125 participants from 20 different countries, and featured two main courses, three mini-courses, and twenty-nine lectures. The lecture notes contained in this volume provide introductory accounts of three of the most active and fascinating areas of research in modern probability theory, especially designed for graduate students entering research: Scaling limits of random trees and random graphs (Christina Goldschmidt) Lectures on the Ising and Potts models on the hypercubic lattice (Hugo Duminil-Copin) Extrema of the two-dimensional discrete Gaussian free field (Marek Biskup) Each of these contributions provides a thorough introduction that will be of value to beginners and experts alike.Table of ContentsScaling Limits of Random Trees and Random Graphs (C. Goldschmidt).- Lectures on the Ising and Potts Models on the Hypercubic Lattice (H. Duminil-Copin).- Extrema of the Two-Dimensional Discrete Gaussian Free Field (M. Biskup).
£151.99
