Computational neuroscience is the theoretical study of the brain to uncover the principles and mechanisms that guide the development, organization, information processing, and mental functions of the nervous system. Although not a new area, it is only recently that enough knowledge has been gathered to establish computational neuroscience as a scientific discipline in its own right. Given the complexity of the field, and its increasing importance in progressing our understanding of how the brain works, there has long been a need for an introductory text on what is often assumed to be an impenetrable topic. The new edition of Fundamentals of Computational Neuroscience build on the success and strengths of the previous editions. It introduces the theoretical foundations of neuroscience with a focus on the nature of information processing in the brain. The book covers the introduction and motivation of simplified models of neurons that are suitable for exploring information processing in large brain-like networks. Additionally, it introduces several fundamental network architectures and discusses their relevance for information processing in the brain, giving some examples of models of higher-order cognitive functions to demonstrate the advanced insight that can be gained with such studies. Each chapter starts by introducing its topic with experimental facts and conceptual questions related to the study of brain function. An additional feature is the inclusion of simple Matlab programs that can be used to explore many of the mechanisms explained in the book. An accompanying webpage includes programs for download. The book will be the essential text for anyone in the brain sciences who wants to get to grips with this topic.

This text introduces engineering students to probability theory and stochastic processes. Along with thorough mathematical development of the subject, the book presents intuitive explanations of key points in order to give students the insights they need to apply math to practical engineering problems. The first seven chapters contain the core material that is essential to any introductory course. In one-semester undergraduate courses, instructors can select material from the remaining chapters to meet their individual goals. Graduate courses can cover all chapters in one semester.

The book provides a sound mathematical base for life insurance mathematics and applies the underlying concepts to concrete examples. Moreover the models presented make it possible to model life insurance policies by means of Markov chains. Two chapters covering ALM and abstract valuation concepts on the background of Solvency II complete this volume. Numerous examples and a parallel treatment of discrete and continuous approaches help the reader to implement the theory directly in practice.

The Routledge Companion to Intelligence Studies provides a broad overview of the growing field of intelligence studies. The recent growth of interest in intelligence and security studies has led to an increased demand for popular depictions of intelligence and reference works to explain the architecture and underpinnings of intelligence activity. Divided into five comprehensive sections, this Companion provides a strong survey of the cutting-edge research in the field of intelligence studies: Part I: The evolution of intelligence studies; Part II: Abstract approaches to intelligence; Part III: Historical approaches to intelligence; Part IV: Systems of intelligence; Part V: Contemporary challenges. With a broad focus on the origins, practices and nature of intelligence, the book not only addresses classical issues, but also examines topics of recent interest in security studies. The overarching aim is to reveal the rich tapestry of intelligence studies in both a sophisticated and accessible way. This Companion will be essential reading for students of intelligence studies and strategic studies, and highly recommended for students of defence studies, foreign policy, Cold War studies, diplomacy and international relations in general.

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

These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor Nelson traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations. He continues through recent dynamical theories of Brownian motion, and concludes with a discussion of the relevance of these theories to quantum field theory and quantum statistical mechanics.

The field of applied probability has changed profoundly in the past twenty years. The development of computational methods has greatly contributed to a better understanding of the theory. A First Course in Stochastic Models provides a self-contained introduction to the theory and applications of stochastic models. Emphasis is placed on establishing the theoretical foundations of the subject, thereby providing a framework in which the applications can be understood. Without this solid basis in theory no applications can be solved.

• Provides an introduction to the use of stochastic models through an integrated presentation of theory, algorithms and applications.

• Incorporates recent developments in computational probability.

• Includes a wide range of examples that illustrate the models and make the methods of solution clear.

• Features an abundance of motivating exercises that help the student learn how to apply the theory.

• Accessible to anyone with a basic knowledge of probability.

This book explains the notion of Brakke's mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke's mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke's existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard's regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process.

Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Holder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations."

A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.

The fourth edition of this successful text provides an introduction to probability and random processes, with many practical applications. It is aimed at mathematics undergraduates and postgraduates, and has four main aims. US BL To provide a thorough but straightforward account of basic probability theory, giving the reader a natural feel for the subject unburdened by oppressive technicalities. BE BL To discuss important random processes in depth with many examples.BE BL To cover a range of topics that are significant and interesting but less routine.BE BL To impart to the beginner some flavour of advanced work.BE UE OP The book begins with the basic ideas common to most undergraduate courses in mathematics, statistics, and science. It ends with material usually found at graduate level, for example, Markov processes, (including Markov chain Monte Carlo), martingales, queues, diffusions, (including stochastic calculus with Ito's formula), renewals, stationary processes (including the ergodic theorem), and option pricing in mathematical finance using the Black-Scholes formula. Further, in this new revised fourth edition, there are sections on coupling from the past, Levy processes, self-similarity and stability, time changes, and the holding-time/jump-chain construction of continuous-time Markov chains. Finally, the number of exercises and problems has been increased by around 300 to a total of about 1300, and many of the existing exercises have been refreshed by additional parts. The solutions to these exercises and problems can be found in the companion volume, One Thousand Exercises in Probability, third edition, (OUP 2020).CP

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs.

The goal of this textbook is to introduce students to the stochastic analysis tools that play an increasing role in the probabilistic approach to optimization problems, including stochastic control and stochastic differential games. While optimal control is taught in many graduate programs in applied mathematics and operations research, the author was intrigued by the lack of coverage of the theory of stochastic differential games. This is the first title in SIAM's Financial Mathematics book series and is based on the author's lecture notes. It will be helpful to students who are interested in stochastic differential equations (forward, backward, forward-backward); the probabilistic approach to stochastic control (dynamic programming and the stochastic maximum principle); and mean field games and control of McKean-Vlasov dynamics. The theory is illustrated by applications to models of systemic risk, macroeconomic growth, flocking/schooling, crowd behavior, and predatory trading, among others.

This is the first book to promote the use of stochastic, or random, processes to understand, model and predict our climate system. One of the most important applications of this technique is in the representation of comprehensive climate models of processes which, although crucial, are too small or fast to be explicitly modelled. The book shows how stochastic methods can lead to improvements in climate simulation and prediction, compared with more conventional bulk-formula parameterization procedures. Beginning with expositions of the relevant mathematical theory, the book moves on to describe numerous practical applications. It covers the complete range of time scales of climate variability, from seasonal to decadal, centennial, and millennial. With contributions from leading experts in climate physics, this book is invaluable to anyone working on climate models, including graduate students and researchers in the atmospheric and oceanic sciences, numerical weather forecasting, climate prediction, climate modelling, and climate change.

Focusing on recent advances in option pricing under the SABR model, this book shows how to price options under this model in an arbitrage-free, theoretically consistent manner. It extends SABR to a negative rates environment, and shows how to generalize it to a similar model with additional degrees of freedom, allowing simultaneous model calibration to swaptions and CMSs. Since the SABR model is used on practically every trading floor to construct interest rate options volatility cubes in an arbitrage-free manner, a careful treatment of it is extremely important. The book will be of interest to experienced industry practitioners, as well as to students and professors in academia. Aimed mainly at financial industry practitioners (for example quants and former physicists) this book will also be interesting to mathematicians who seek intuition in the mathematical finance.

This book focuses on quantitative approximation results for weak limit theorems when the target limiting law is infinitely divisible with finite first moment. Two methods are presented and developed to obtain such quantitative results. At the root of these methods stands a Stein characterizing identity discussed in the third chapter and obtained thanks to a covariance representation of infinitely divisible distributions. The first method is based on characteristic functions and Stein type identities when the involved sequence of random variables is itself infinitely divisible with finite first moment. In particular, based on this technique, quantitative versions of compound Poisson approximation of infinitely divisible distributions are presented. The second method is a general Stein's method approach for univariate selfdecomposable laws with finite first moment. Chapter 6 is concerned with applications and provides general upper bounds to quantify the rate of convergence in classical weak limit theorems for sums of independent random variables. This book is aimed at graduate students and researchers working in probability theory and mathematical statistics.

Josef Anton Strini analyzes a special stochastic optimal control problem. The problem under study arose from a dynamic cash management model in finance, where decisions about the dividend and financing policies of a firm have to be made. Additionally, using the dynamic programming approach, he extends the present discourse by the formal derivation of the Hamilton-Jacobi-Bellman equation and by examining the verification step carefully. Finally, the treatment is completed by solving the problem numerically.

This book provides a concise introduction to the behavior of mechanical structures and testing their stochastic stability under the influence of noise. It explains the physical effects of noise and in particular the concept of Gaussian white noise. In closing, the book explains how to model the effects of noise on mechanical structures, and how to nullify / compensate for it by designing effective controllers.

This two-volume set provides a comprehensive and self-contained approach to the dynamics, ergodic theory, and geometry of elliptic functions mapping the complex plane onto the Riemann sphere. Volume I discusses many fundamental results from ergodic theory and geometric measure theory in detail, including finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various conformal measures, conformal graph directed Markov systems and iterated functions systems, classical theory of elliptic functions. In Volume II, all these techniques, along with an introduction to topological dynamics of transcendental meromorphic functions, are applied to describe the beautiful and rich dynamics and fractal geometry of elliptic functions. Much of this material is appearing for the first time in book or even paper form. Both researchers and graduate students will appreciate the detailed explanations of essential concepts and full proofs provided in what is sure to be an indispensable reference.

The activity of neurons in the brain is noisy in that their firing times are random when they are firing at a given mean rate. This introduces a random or stochastic property into brain processing which we show in this book is fundamental to understanding many aspects of brain function, including probabilistic decision making, perception, memory recall, short-term memory, attention, and even creativity. In The Noisy Brain we show that in many of these processes, the noise caused by the random neuronal firing times is useful. However, this stochastic dynamics can be unstable or overstable, and we show that the stability of attractor networks in the brain in the face of noise may help to understand some important dysfunctions that occur in schizophrenia, normal aging, and obsessive-compulsive disorder. The Noisy Brain provides a unifying computational approach to brain function that links synaptic and biophysical properties of neurons through the firing of single neurons to the properties of the noise in large connected networks of noisy neurons to the levels of functional neuroimaging and behaviour. The book describes integrate-and-fire neuronal attractor networks with noise, and complementary mean-field analyses using approaches from theoretical physics. The book shows how they can be used to understand neuronal, functional neuroimaging, and behavioural data on decision-making, perception, memory recall, short-term memory, attention, and brain dysfunctions that occur in schizophrenia, normal aging, and obsessive-compulsive disorder. The Noisy Brain will be valuable for those in the fields of neuroscience, psychology, cognitive neuroscience, and biology from advanced undergraduate level upwards. It will also be of interest to those interested in neuroeconomics, animal behaviour, zoology, psychiatry, medicine, physics, and philosophy. The book has been written with modular chapters and sections, making it possible to select particular Chapters for course work. Advanced material on the physics of stochastic dynamics in the brain is contained in the Appendix.

Stochastic processes have a wide range of applications ranging from image processing, neuroscience, bioinformatics, financial management, and statistics. Mathematical, physical, and engineering systems use stochastic processes for modeling and reasoning phenomena. While comparing AI-stochastic systems with other counterpart systems, we are able to understand their significance, thereby applying new techniques to obtain new real-time results and solutions. Stochastic Processes and Their Applications in Artificial Intelligence opens doors for artificial intelligence experts to use stochastic processes as an effective tool in real-world problems in computational biology, speech recognition, natural language processing, and reinforcement learning. Covering key topics such as social media, big data, and artificial intelligence models, this reference work is ideal for mathematicians, industry professionals, researchers, scholars, academicians, practitioners, instructors, and students.

The first comprehensive graduate-level introduction to stochastic thermodynamics Stochastic thermodynamics is a well-defined subfield of statistical physics that aims to interpret thermodynamic concepts for systems ranging in size from a few to hundreds of nanometers, the behavior of which is inherently random due to thermal fluctuations. This growing field therefore describes the nonequilibrium dynamics of small systems, such as artificial nanodevices and biological molecular machines, which are of increasing scientific and technological relevance. This textbook provides an up-to-date pedagogical introduction to stochastic thermodynamics, guiding readers from basic concepts in statistical physics, probability theory, and thermodynamics to the most recent developments in the field. Gradually building up to more advanced material, the authors consistently prioritize simplicity and clarity over exhaustiveness and focus on the development of readers' physical insight over mathematical formalism. This approach allows the reader to grow as the book proceeds, helping interested young scientists to enter the field with less effort and to contribute to its ongoing vibrant development. Chapters provide exercises to complement and reinforce learning. Appropriate for graduate students in physics and biophysics, as well as researchers, Stochastic Thermodynamics serves as an excellent initiation to this rapidly evolving field. Emphasizes a pedagogical approach to the subject Highlights connections with the thermodynamics of information Pays special attention to molecular biophysics applications Privileges physical intuition over mathematical formalism Solutions manual available on request for instructors adopting the book in a course

Here is a work that adds much to the sum of our knowledge in a key area of science today. It is concerned with the estimation of discrete-time semi-Markov and hidden semi-Markov processes. A unique feature of the book is the use of discrete time, especially useful in some specific applications where the time scale is intrinsically discrete. The models presented in the book are specifically adapted to reliability studies and DNA analysis. The book is mainly intended for applied probabilists and statisticians interested in semi-Markov chains theory, reliability and DNA analysis, and for theoretical oriented reliability and bioinformatics engineers.

The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists.

"The book is a valuable completion of the literature in this field. It is written in an ambitious mathematical style and can be recommended to statisticians as well as biostatisticians."
--Biometrische Zeitschrift

"Not many books manage to combine convincingly topics from probability theory over mathematical statistics to applied statistics. This is one of them. The book has other strong points to recommend it: it is written with meticulous care, in a lucid style, general results being illustrated by examples from statistical theory and practice, and a bunch of exercises serve to further elucidate and elaborate on the text."
--Mathematical Reviews

"This book gives a thorough introduction to martingale and counting process methods in survival analysis thereby filling a gap in the literature."
--Zentralblatt fur Mathematik und ihre Grenzgebiete/Mathematics Abstracts

"The authors have performed a valuable service to researchers in providing this material in [a] self-contained and accessible form. . . This text [is] essential reading for the probabilist or mathematical statistician working in the area of survival analysis."
--Short Book Reviews, International Statistical Institute

Counting Processes and Survival Analysis explores the martingale approach to the statistical analysis of countingprocesses, with an emphasis on the application of those methods to censored failure time data. This approach has proven remarkably successful in yielding results about statistical methods for many problems arising in censored data. A thorough treatment of the calculus of martingales as well as the most important applications of these methods to censored data is offered. Additionally, the book examines classical problems in asymptotic distribution theory for counting process methods and newer methods for graphical analysis and diagnostics of censored data. Exercises are included to provide practice in applying martingale methods and insight into the calculus itself.

The Poisson "law of small numbers" is a central principle in modern theories of reliability, insurance, and the statistics of extremes. It also has ramifications in apparently unrelated areas, such as the description of algebraic and combinatorial structures, and the distribution of prime numbers. Yet despite its importance, the law of small numbers is only an approximation. In 1975, however, a new technique was introduced, the Stein-Chen method, which makes it possible to estimate the accuracy of the approximation in a wide range of situations. This book provides an introduction to the method, and a varied selection of examples of its application, emphasizing the flexibility of the technique when combined with a judicious choice of coupling. It also contains more advanced material, in particular on compound Poisson and Poisson process approximation, where the reader is brought to the boundaries of current knowledge. The study will be of special interest to postgraduate students and researchers in applied probability as well as computer scientists.