An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine

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An Introduction to Continuous-Time Stochastic Processes

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Measure Theory, Probability, and Stochastic Processes Springer International Publishing AG

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This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix. Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the author's more advanced textbook in the same series (GTM 274).

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Introduction to Stochastic Control Theory Dover Publications Inc.

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PrefaceAcknowledgments1. Stochastic Control 1. Introduction 2. Theory of Feedback Control 3. How to Characterize Disturbances 4. Stochastic Control Theory 5. Outline of the Contents of the Book 6. Bibliography and Comments2. Stochastic Processes 1. Introduction 2. The Concept of a Stochastic Process 3. Some Special Stochastic Processes 4. The Covariance Function 5. The Concept of Spectral Density 6. Analysis of Stochastic Processes 7. Bibliography and Comments3. Stochastic State Models 1. Introduction 2. Discrete Time Systems 3. Solution of Stochastic Difference Equations 4. Continuous Time Systems 5. Stochastic Integrals 6. Linear Stochastic Differential Equations 7. Nonlinear Stochastic Differential Equations 8. Stochastic Calculus--The Ito Differentiation Rule 9. Modeling of Physical Processes by Stochastic Differential Equations 10. Sampling a Stochastic Differential Equation 11. Bibliography and Comments4. Analysis of Dynamical Systems Whose Inputs are Stochastic Processes 1. Introduction 2. Discrete Time Systems 3. Spectral Factorization of Discrete Time Processes 4. Analysis of Continuous Time Systems Whose Input Signals are Stochastic Processes 5. Spectral Factorization of Continuous Time Processes 6. Bibliography and Comments5. Parametric Optimization 1. Introduction 2. Evaluation of Loss Functions for Discrete Time Systems 3. Evaluation of Loss Functions for Continuous Time Systems 4. Reconstruction of State Variables for Discrete Time Systems 5. Reconstruction of State Variables for Continuous Time Systems 6. Bibliography and Comments6. Minimal Variance Control Strategies 1. Introduction 2. A Simple Example 3. Optimal Prediction of Discrete Time Stationary Processes 4. Minimal Variance Control Strategies 5. Sensitivity of the Optimal System 6. An Industrial Application 7. Bibliography and Comments7. Prediction and Filtering Theory 1. Introduction 2. Formulation of Prediction and Estimation Problems 3. Preliminaries 4. State Estimation for Discrete Time Systems 5. Duality 6. State Estimation for Continuous Time Processes 7. Bibliography and Comments8. Linear Stochastic Control Theory 1. Introduction 2. Formulation 3. Preliminaries 4. Complete State Information 5. Incomplete State Information 1 6. Incomplete State Information 2 7. Continuous Time Problems 8. Bibliography and CommentsIndex

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Introduction to Stochastic Processes in Physics Johns Hopkins University Press

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This book provides an accessible introduction to stochastic processes in physics and describes the basic mathematical tools of the trade: probability, random walks, and Wiener and Ornstein-Uhlenbeck processes. It includes end-of-chapter problems and emphasizes applications. An Introduction to Stochastic Processes in Physics builds directly upon early-twentieth-century explanations of the "peculiar character in the motions of the particles of pollen in water" as described, in the early nineteenth century, by the biologist Robert Brown. Lemons has adopted Paul Langevin's 1908 approach of applying Newton's second law to a "Brownian particle on which the total force included a random component" to explain Brownian motion. This method builds on Newtonian dynamics and provides an accessible explanation to anyone approaching the subject for the first time. Students will find this book a useful aid to learning the unfamiliar mathematical aspects of stochastic processes while applying them to physical processes that he or she has already encountered.

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Essentials of Stochastic Processes Springer International Publishing AG

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In its revised new edition, this book covers Markov chains in discrete and continuous time, Poisson processes, renewal processes, martingales and mathematical finance. Offers many examples and more than 300 carefully chosen exercises for better understanding.

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Stochastic Processes 2e John Wiley & Sons Inc

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A nonmeasure theoretic introduction to stochastic processes. Considers its diverse range of applications and provides readers with probabilistic intuition and insight in thinking about problems. This revised edition contains additional material on compound Poisson random variables including an identity which can be used to efficiently compute moments; a new chapter on Poisson approximations; and coverage of the mean time spent in transient states as well as examples relating to the Gibba s sampler, the Metropolis algorithm and mean cover time in star graphs. Numerous exercises and problems have been added throughout the text.

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Introduction to Stochastic Calculus for Finance Springer-Verlag Berlin and Heidelberg GmbH & Co....

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The large number of already available textbooks on stochastic calculus with specific applications to finance requires a justification for another contribution to this subject. The justifcation is mainly pedagogical. These lecture notes start with an elementary approach to stochastic calculus due to Föllmer, who showed that one can develop Ito's calculus "pathwise" as an exercise in real analysis. The text opens to students interested in finance a quick (but by no means "dirty") road to the tools required for advanced finance in continuous time, including option pricing by martingale methods, term structure models in a HJM-framework and the Libor market model. The reader is supposed only to be familiar with elementary real analysis (e.g. Taylor's Theorem) and basic probability theory. The text is also useful for mathematicians interested in the methods of modern mathematical finance without prior knowledge of advanced stochastic analysis.

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Stochastic Calculus for Finance II Springer-Verlag New York Inc.

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Stochastic Calculus for Finance evolved from the first ten years of the Carnegie Mellon Professional Master's program in Computational Finance. The content of this book has been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise statements of results, plausibility arguments, and even some proofs, but more importantly intuitive explanations developed and refine through classroom experience with this material are provided. The book includes a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties. Advanced topics include foreign exchange models, forward measures, and jump-diffusion processes.§This book is being published in two volumes. This second volume develops stochastic calculus, martingales, risk-neutral pricing, exotic options and term structure models, all in continuous time.§Masters level students and researchers in mathematical finance and financial engineering will find this book useful.§Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education. "A wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions. In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach....It should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance." --SIAM This text has grown out of a two-semester course sequence in the Carnegie Mellon Master's program in Computational Finance. It contains numerous examples, exercises, and references. It assumes the reader is familiar with differential and integral calculus and basic concepts from calculus-based probability. It does not assume familiarity with measure-theoretic probability, but rather informally develops the necessary tools from this subject within the text.

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Levy Processes and Infinitely Divisible Distributions Cambridge University Press

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Lévy processes are rich mathematical objects and constitute perhaps the most basic class of stochastic processes with a continuous time parameter. This book is intended to provide the reader with comprehensive basic knowledge of Lévy processes, and at the same time serve as an introduction to stochastic processes in general. No specialist knowledge is assumed and proofs are given in detail. Systematic study is made of stable and semi-stable processes, and the author gives special emphasis to the correspondence between Lévy processes and infinitely divisible distributions. All serious students of random phenomena will find that this book has much to offer.

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Stationary Processes and Discrete Parameter Markov Processes Springer International Publishing AG

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This textbook explores two distinct stochastic processes that evolve at random: weakly stationary processes and discrete parameter Markov processes. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study.After recapping the essentials from Fourier analysis, the book begins with an introduction to the spectral representation of a stationary process. Topics in ergodic theory follow, including Birkhoff's Ergodic Theorem and an introduction to dynamical systems. From here, the Markov property is assumed and the theory of discrete parameter Markov processes is explored on a general state space. Chapters cover a variety of topics, including birth-death chains, hitting probabilities and absorption, the representation of Markov processes as iterates of random maps, and large deviation theory for Markov processes. A chapter on geometric rates of convergence to equilibrium includes a splitting condition that captures the recurrence structure of certain iterated maps in a novel way. A selection of special topics concludes the book, including applications of large deviation theory, the FKG inequalities, coupling methods, and the Kalman filter.Featuring many short chapters and a modular design, this textbook offers an in-depth study of stationary and discrete-time Markov processes. Students and instructors alike will appreciate the accessible, example-driven approach and engaging exercises throughout. A single, graduate-level course in probability is assumed.

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Probability and Random Processes Oxford University Press

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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 Itô'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, Lévy 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

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Probability and Random Processes Oxford University Press

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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 Itô'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, Lévy 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

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Brownian Motion and Stochastic Calculus Springer-Verlag New York Inc.

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Designed as a text for graduate courses in stochastic processes. Written for readers familiar with measure-theoretic probability and discrete-time processes who wish 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. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and this in turn permits a presentation of recent advances in financial economics (options pricing and consumption/investment optimization). 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 t ext is complemented by a large number of problems and exercises.

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Brownian Motion, Martingales, and Stochastic Calculus Springer International Publishing AG

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This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô's formula, the optional stopping theorem and Girsanov's theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. § Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. § Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.

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