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| martingale solved problems: Probability with Martingales David Williams, 1991-02-14 This is a masterly introduction to the modern, and rigorous, theory of probability. The author emphasises martingales and develops all the necessary measure theory. |
| martingale solved problems: Measures, Integrals and Martingales René L. Schilling, 2005-11-10 This is a concise and elementary introduction to contemporary measure and integration theory as it is needed in many parts of analysis and probability theory. Undergraduate calculus and an introductory course on rigorous analysis in R are the only essential prerequisites, making the text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included to consolidate what has already been learned and to discover variants and extensions to the main material. Hints and solutions can be found on the authors website, which can be reached at http: //www.motapa.de/measures_integrals_and_martingales/index.htm |
| martingale solved problems: Stochastic Calculus and Applications Samuel N. Cohen, Robert J. Elliott, 2015-11-18 Completely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance. Building upon the original release of this title, this text will be of great interest to research mathematicians and graduate students working in those fields, as well as quants in the finance industry. New features of this edition include: End of chapter exercises; New chapters on basic measure theory and Backward SDEs; Reworked proofs, examples and explanatory material; Increased focus on motivating the mathematics; Extensive topical index. Such a self-contained and complete exposition of stochastic calculus and applications fills an existing gap in the literature. The book can be recommended for first-year graduate studies. It will be useful for all who intend to work with stochastic calculus as well as with its applications.–Zentralblatt (from review of the First Edition) |
| martingale solved problems: Probability Rick Durrett, 2010-08-30 This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The fourth edition begins with a short chapter on measure theory to orient readers new to the subject. |
| martingale solved problems: Diffusions, Markov Processes, and Martingales: Itô calculus L. C. G. Rogers, David Williams, 2000 This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science. |
| martingale solved problems: Martingale Limit Theory and Its Application P. Hall, C. C. Heyde, 2014-07-10 Martingale Limit Theory and Its Application discusses the asymptotic properties of martingales, particularly as regards key prototype of probabilistic behavior that has wide applications. The book explains the thesis that martingale theory is central to probability theory, and also examines the relationships between martingales and processes embeddable in or approximated by Brownian motion. The text reviews the martingale convergence theorem, the classical limit theory and analogs, and the martingale limit theorems viewed as the rate of convergence results in the martingale convergence theorem. The book explains the square function inequalities, weak law of large numbers, as well as the strong law of large numbers. The text discusses the reverse martingales, martingale tail sums, the invariance principles in the central limit theorem, and also the law of the iterated logarithm. The book investigates the limit theory for stationary processes via corresponding results for approximating martingales and the estimation of parameters from stochastic processes. The text can be profitably used as a reference for mathematicians, advanced students, and professors of higher mathematics or statistics. |
| martingale solved problems: Martingale Methods in Statistics Yoichi Nishiyama, 2021-11-23 Martingale Methods in Statistics provides a unique introduction to statistics of stochastic processes written with the author’s strong desire to present what is not available in other textbooks. While the author chooses to omit the well-known proofs of some of fundamental theorems in martingale theory by making clear citations instead, the author does his best to describe some intuitive interpretations or concrete usages of such theorems. On the other hand, the exposition of relatively new theorems in asymptotic statistics is presented in a completely self-contained way. Some simple, easy-to-understand proofs of martingale central limit theorems are included. The potential readers include those who hope to build up mathematical bases to deal with high-frequency data in mathematical finance and those who hope to learn the theoretical background for Cox’s regression model in survival analysis. A highlight of the monograph is Chapters 8-10 dealing with Z-estimators and related topics, such as the asymptotic representation of Z-estimators, the theory of asymptotically optimal inference based on the LAN concept and the unified approach to the change point problems via Z-process method. Some new inequalities for maxima of finitely many martingales are presented in the Appendix. Readers will find many tips for solving concrete problems in modern statistics of stochastic processes as well as in more fundamental models such as i.i.d. and Markov chain models. |
| martingale solved problems: Counterexamples in Probability And Statistics A.F. Siegel, 2017-11-22 This volume contains six early mathematical works, four papers on fiducial inference, five on transformations, and twenty-seven on a miscellany of topics in mathematical statistics. Several previously unpublished works are included. |
| martingale solved problems: Theory of Martingales Robert Liptser, A.N. Shiryayev, 2012-12-06 One service mathematics has rc:ndered the 'Et moi, ', si j'avait su comment CD revenir, je n'y serais point alle. ' human race. It has put common SCIIJC back Jules Verne where it belongs. on the topmost shelf next to tbe dusty canister 1abdled 'discarded non- The series is divergent; tberefore we may be sense'. able to do sometbing witb it Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ... '; 'One service logic has rendered com puter science ... '; 'One service category theory has rendered mathematics ... '. All arguably true_ And all statements obtainable this way form part of the raison d'etre of this series_ This series, Mathematics and Its ApplicatiOns, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope_ At the time I wrote Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. |
| martingale solved problems: Multidimensional Diffusion Processes Daniel W. Stroock, S.R.S. Varadhan, 2007-02-03 This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. This approach was initiated by Stroock and Varadhan in their famous papers. (...) The proofs and techniques are presented in such a way that an adaptation in other contexts can be easily done. (...) The reader must be familiar with standard probability theory and measure theory which are summarized at the beginning of the book. This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view. Mathematische Operationsforschung und Statistik, 1981 |
| martingale solved problems: Sharp Martingale and Semimartingale Inequalities Adam Osękowski, 2012-08-14 This monograph is a presentation of a unified approach to a certain class of semimartingale inequalities, which can be regarded as probabilistic extensions of classical estimates for conjugate harmonic functions on the unit disc. The approach, which has its roots in the seminal works of Burkholder in the 80s, enables to deduce a given inequality for semimartingales from the existence of a certain special function with some convex-type properties. Remarkably, an appropriate application of the method leads to the sharp version of the estimate under investigation, which is particularly important for applications. These include the theory of quasiregular mappings (with deep implications to the geometric function theory); the boundedness of two-dimensional Hilbert transform and a more general class of Fourier multipliers; the theory of rank-one convex and quasiconvex functions; and more. The book is divided into a few separate parts. In the introductory chapter we present motivation for the results and relate them to some classical problems in harmonic analysis. The next part contains a general description of the method, which is applied in subsequent chapters to the study of sharp estimates for discrete-time martingales; discrete-time sub- and supermartingales; continuous time processes; the square and maximal functions. Each chapter contains additional bibliographical notes included for reference. |
| martingale solved problems: Introduction To Stochastic Calculus With Applications (2nd Edition) Fima C Klebaner, 2005-06-20 This book presents a concise treatment of stochastic calculus and its applications. It gives a simple but rigorous treatment of the subject including a range of advanced topics, it is useful for practitioners who use advanced theoretical results. It covers advanced applications, such as models in mathematical finance, biology and engineering.Self-contained and unified in presentation, the book contains many solved examples and exercises. It may be used as a textbook by advanced undergraduates and graduate students in stochastic calculus and financial mathematics. It is also suitable for practitioners who wish to gain an understanding or working knowledge of the subject. For mathematicians, this book could be a first text on stochastic calculus; it is good companion to more advanced texts by a way of examples and exercises. For people from other fields, it provides a way to gain a working knowledge of stochastic calculus. It shows all readers the applications of stochastic calculus methods and takes readers to the technical level required in research and sophisticated modelling.This second edition contains a new chapter on bonds, interest rates and their options. New materials include more worked out examples in all chapters, best estimators, more results on change of time, change of measure, random measures, new results on exotic options, FX options, stochastic and implied volatility, models of the age-dependent branching process and the stochastic Lotka-Volterra model in biology, non-linear filtering in engineering and five new figures.Instructors can obtain slides of the text from the author./a |
| martingale solved problems: Problems And Solutions In Stochastic Calculus With Applications Patrik Albin, Kais Hamza, Fima C Klebaner, 2024-08-27 Problems and Solutions in Stochastic Calculus with Applications exposes readers to simple ideas and proofs in stochastic calculus and its applications. It is intended as a companion to the successful original title Introduction to Stochastic Calculus with Applications (Third Edition) by Fima Klebaner. The current book is authored by three active researchers in the fields of probability, stochastic processes, and their applications in financial mathematics, mathematical biology, and more. The book features problems rooted in their ongoing research. Mathematical finance and biology feature pre-eminently, but the ideas and techniques can equally apply to fields such as engineering and economics.The problems set forth are accessible to students new to the subject, with most of the problems and their solutions centring on a single idea or technique at a time to enhance the ease of learning. While the majority of problems are relatively straightforward, more complex questions are also set in order to challenge the reader as their understanding grows. The book is suitable for either self-study or for instructors, and there are numerous opportunities to generate fresh problems by modifying those presented, facilitating a deeper grasp of the material. |
| martingale solved problems: Continuous Martingales and Brownian Motion Daniel Revuz, Marc Yor, 2013-03-09 From the reviews: This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion. The great strength of Revuz and Yor is the enormous variety of calculations carried out both in the main text and also (by implication) in the exercises. ... This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises, and throwing out challenging remarks about areas awaiting further research... Bull.L.M.S. 24, 4 (1992) Since the first edition in 1991, an impressive variety of advances has been made in relation to the material of this book, and these are reflected in the successive editions. |
| martingale solved problems: Optimal Stopping and Free-Boundary Problems Goran Peskir, Albert Shiryaev, 2006-11-10 This book discloses a fascinating connection between optimal stopping problems in probability and free-boundary problems. It focuses on key examples and the theory of optimal stopping is exposed at its basic principles in discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from change of time, space, and measure, to more recent ones such as local time-space calculus and nonlinear integral equations. A chapter on stochastic processes makes the material more accessible. The book will appeal to those wishing to master stochastic calculus via fundamental examples. Areas of application include financial mathematics, financial engineering, and mathematical statistics. |
| martingale solved problems: Stochastic Optimization Models In Finance (2006 Edition) William T Ziemba, Raymond G Vickson, 2006-09-11 A reprint of one of the classic volumes on portfolio theory and investment, this book has been used by the leading professors at universities such as Stanford, Berkeley, and Carnegie-Mellon. It contains five parts, each with a review of the literature and about 150 pages of computational and review exercises and further in-depth, challenging problems.Frequently referenced and highly usable, the material remains as fresh and relevant for a portfolio theory course as ever. |
| martingale solved problems: Martingales and Markov Chains Paolo Baldi, 2002-04-26 A thorough grounding in Markov chains and martingales is essential in dealing with many problems in applied probability, and is a gateway to the more complex situations encountered in the study of stochastic processes. Exercises are a fundamental and valuable training tool that deepen students' understanding of theoretical principles and prepare th |
| martingale solved problems: Ruin Probabilities Yuliya Mishura, Olena Ragulina, 2016-11-08 Ruin Probabilities: Smoothness, Bounds, Supermartingale Approach deals with continuous-time risk models and covers several aspects of risk theory. The first of them is the smoothness of the survival probabilities. In particular, the book provides a detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizon survival probabilities for different risk models. Next, it gives some possible applications of the results concerning the smoothness of the survival probabilities. Additionally, the book introduces the supermartingale approach, which generalizes the martingale one introduced by Gerber, to get upper exponential bounds for the infinite-horizon ruin probabilities in some generalizations of the classical risk model with risky investments. - Provides new original results - Detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizon survival probabilities, as well as possible applications of these results - An excellent supplement to current textbooks and monographs in risk theory - Contains a comprehensive list of useful references |
| martingale solved problems: Exercises and Solutions Manual for Integration and Probability Paul Malliavin, Gerard Letac, 1995-06-13 This book is designed to be an introduction to analysis with the proper mix of abstract theories and concrete problems. It starts with general measure theory, treats Borel and Radon measures (with particular attention paid to Lebesgue measure) and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the Fourier analysis of such. It continues with a Hilbertian treatment of the basic laws of probability including Doob's martingale convergence theorem and finishes with Malliavin's stochastic calculus of variations developed in the context of Gaussian measure spaces. This invaluable contribution to the existing literature gives the reader a taste of the fact that analysis is not a collection of independent theories but can be treated as a whole. |
| martingale solved problems: Markov Processes, Semigroups, and Generators Vassili N. Kolokoltsov, 2011 This work offers a highly useful, well developed reference on Markov processes, the universal model for random processes and evolutions. The wide range of applications, in exact sciences as well as in other areas like social studies, require a volume that offers a refresher on fundamentals before conveying the Markov processes and examples for |
| martingale solved problems: Martingales in Banach Spaces Gilles Pisier, 2016-06-06 This book focuses on the major applications of martingales to the geometry of Banach spaces, and a substantial discussion of harmonic analysis in Banach space valued Hardy spaces is also presented. It covers exciting links between super-reflexivity and some metric spaces related to computer science, as well as an outline of the recently developed theory of non-commutative martingales, which has natural connections with quantum physics and quantum information theory. Requiring few prerequisites and providing fully detailed proofs for the main results, this self-contained study is accessible to graduate students with a basic knowledge of real and complex analysis and functional analysis. Chapters can be read independently, with each building from the introductory notes, and the diversity of topics included also means this book can serve as the basis for a variety of graduate courses. |
| martingale solved problems: Nonlinear Markov Processes and Kinetic Equations Vassili N. Kolokoltsov, 2010-07-15 A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology. |
| martingale solved problems: Measure Theory. Applications to Stochastic Analysis G. Kallianpur, D. Kölzow, 2006-11-15 |
| martingale solved problems: Foundations of Modern Probability Olav Kallenberg, 2002-01-08 The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity. |
| martingale solved problems: Mathematics of Random Media Werner E. Kohler, Benjamin Steven White, In recent years, there has been remarkable growth in the mathematics of random media. The field has deep scientific and technological roots, as well as purely mathematical ones in the theory of stochastic processes. This collection of papers by leading researchers provides an overview of this rapidly developing field. The papers were presented at the 1989 AMS-SIAM Summer Seminar in Applied Mathematics, held at Virginia Polytechnic Institute and State University in Blacksburg, Virginia. In addition to new results on stochastic differential equations and Markov processes, fields whose elegant mathematical techniques are of continuing value in application areas, the conference was organized around four themes: Systems of interacting particles are normally viewed in connection with the fundamental problems of statistical mechanics, but have also been used to model diverse phenomena such as computer architectures and the spread of biological populations. Powerful mathematical techniques have been developed for their analysis, and a number of important systems are now well understood. Random perturbations of dynamical systems have also been used extensively as models in physics, chemistry, biology, and engineering. Among the recent unifying mathematical developments is the theory of large deviations, which enables the accurate calculation of the probabilities of rare events. For these problems, approaches based on effective but formal perturbation techniques parallel rigorous mathematical approaches from probability theory and partial differential equations. The book includes representative papers from forefront research of both types. Effective medium theory, otherwise known as the mathematical theory of homogenization, consists of techniques for predicting the macroscopic properties of materials from an understanding of their microstructures. For example, this theory is fundamental in the science of composites, where it is used for theoretical determination of electrical and mechanical properties. Furthermore, the inverse problem is potentially of great technological importance in the design of composite materials which have been optimized for some specific use. Mathematical theories of the propagation of waves in random media have been used to understand phenomena as diverse as the twinkling of stars, the corruption of data in geophysical exploration, and the quantum mechanics of disordered solids. Especially effective methods now exist for waves in randomly stratified, one-dimensional media. A unifying theme is the mathematical phenomenon of localization, which occurs when a wave propogating into a random medium is attenuated exponentially with propagation distance, with the attenuation caused solely by the mechanism of random multiple scattering. Because of the wide applicability of this field of research, this book would appeal to mathematicians, scientists, and engineers in a wide variety of areas, including probabilistic methods, the theory of disordered materials, systems of interacting particles, the design of materials, and dynamical systems driven by noise. In addition, graduate students and others will find this book useful as an overview of current research in random media. |
| martingale solved problems: Metastability Anton Bovier, Frank den Hollander, 2016-02-11 This monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes. The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets. The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour. The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability. |
| martingale solved problems: Lévy Matters VI Franziska Kühn, 2017-10-05 Presenting some recent results on the construction and the moments of Lévy-type processes, the focus of this volume is on a new existence theorem, which is proved using a parametrix construction. Applications range from heat kernel estimates for a class of Lévy-type processes to existence and uniqueness theorems for Lévy-driven stochastic differential equations with Hölder continuous coefficients. Moreover, necessary and sufficient conditions for the existence of moments of Lévy-type processes are studied and some estimates on moments are derived. Lévy-type processes behave locally like Lévy processes but, in contrast to Lévy processes, they are not homogeneous in space. Typical examples are processes with varying index of stability and solutions of Lévy-driven stochastic differential equations. This is the sixth volume in a subseries of the Lecture Notes in Mathematics called Lévy Matters. Each volume describes a number of important topics in the theory or applications of Lévy processes and pays tribute to the state of the art of this rapidly evolving subject, with special emphasis on the non-Brownian world. |
| martingale solved problems: Problems and Solutions in Mathematical Finance, Volume 1 Eric Chin, Sverrir lafsson, Dian Nel, 2014-11-10 Mathematical finance requires the use of advanced mathematical techniques drawn from the theory of probability, stochastic processes and stochastic differential equations. These areas are generally introduced and developed at an abstract level, making it problematic when applying these techniques to practical issues in finance. Problems and Solutions in Mathematical Finance Volume I: Stochastic Calculus is the first of a four-volume set of books focusing on problems and solutions in mathematical finance. This volume introduces the reader to the basic stochastic calculus concepts required for the study of this important subject, providing a large number of worked examples which enable the reader to build the necessary foundation for more practical orientated problems in the later volumes. Through this application and by working through the numerous examples, the reader will properly understand and appreciate the fundamentals that underpin mathematical finance. Written mainly for students, industry practitioners and those involved in teaching in this field of study, Stochastic Calculus provides a valuable reference book to complement one’s further understanding of mathematical finance. |
| martingale solved problems: Stochastic Reachability Analysis of Hybrid Systems Luminita Manuela Bujorianu, 2012-04-23 Stochastic reachability analysis (SRA) is a method of analyzing the behavior of control systems which mix discrete and continuous dynamics. For probabilistic discrete systems it has been shown to be a practical verification method but for stochastic hybrid systems it can be rather more. As a verification technique SRA can assess the safety and performance of, for example, autonomous systems, robot and aircraft path planning and multi-agent coordination but it can also be used for the adaptive control of such systems. Stochastic Reachability Analysis of Hybrid Systems is a self-contained and accessible introduction to this novel topic in the analysis and development of stochastic hybrid systems. Beginning with the relevant aspects of Markov models and introducing stochastic hybrid systems, the book then moves on to coverage of reachability analysis for stochastic hybrid systems. Following this build up, the core of the text first formally defines the concept of reachability in the stochastic framework and then treats issues representing the different faces of SRA: • stochastic reachability based on Markov process theory; • martingale methods; • stochastic reachability as an optimal stopping problem; and • dynamic programming. The book is rounded off by an appendix providing mathematical underpinning on subjects such as ordinary differential equations, probabilistic measure theory and stochastic modeling, which will help the non-expert-mathematician to appreciate the text. Stochastic Reachability Analysis of Hybrid Systems characterizes a highly interdisciplinary area of research and is consequently of significant interest to academic researchers and graduate students from a variety of backgrounds in control engineering, applied mathematics and computer science. The Communications and Control Engineering series reports major technological advances which have potential for great impact in the fields of communication and control. It reflects research in industrial and academic institutions around the world so that the readership can exploit new possibilities as they become available. |
| martingale solved problems: Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory Harold Joseph Kushner, 1984 Control and communications engineers, physicists, and probability theorists, among others, will find this book unique. It contains a detailed development of approximation and limit theorems and methods for random processes and applies them to numerous problems of practical importance. In particular, it develops usable and broad conditions and techniques for showing that a sequence of processes converges to a Markov diffusion or jump process. This is useful when the natural physical model is quite complex, in which case a simpler approximation la diffusion process, for example) is usually made. The book simplifies and extends some important older methods and develops some powerful new ones applicable to a wide variety of limit and approximation problems. The theory of weak convergence of probability measures is introduced along with general and usable methods (for example, perturbed test function, martingale, and direct averaging) for proving tightness and weak convergence. Kushner's study begins with a systematic development of the method. It then treats dynamical system models that have state-dependent noise or nonsmooth dynamics. Perturbed Liapunov function methods are developed for stability studies of nonMarkovian problems and for the study of asymptotic distributions of non-Markovian systems. Three chapters are devoted to applications in control and communication theory (for example, phase-locked loops and adoptive filters). Smallnoise problems and an introduction to the theory of large deviations and applications conclude the book. Harold J. Kushner is Professor of Applied Mathematics and Engineering at Brown University and is one of the leading researchers in the area of stochastic processes concerned with analysis and synthesis in control and communications theory. This book is the sixth in The MIT Press Series in Signal Processing, Optimization, and Control, edited by Alan S. Willsky. |
| martingale solved problems: The Splendors and Miseries of Martingales Laurent Mazliak, Glenn Shafer, 2022-10-17 Over the past eighty years, martingales have become central in the mathematics of randomness. They appear in the general theory of stochastic processes, in the algorithmic theory of randomness, and in some branches of mathematical statistics. Yet little has been written about the history of this evolution. This book explores some of the territory that the history of the concept of martingales has transformed. The historian of martingales faces an immense task. We can find traces of martingale thinking at the very beginning of probability theory, because this theory was related to gambling, and the evolution of a gambler’s holdings as a result of following a particular strategy can always be understood as a martingale. More recently, in the second half of the twentieth century, martingales became important in the theory of stochastic processes at the very same time that stochastic processes were becoming increasingly important in probability, statistics and more generally in various applied situations. Moreover, a history of martingales, like a history of any other branch of mathematics, must go far beyond an account of mathematical ideas and techniques. It must explore the context in which the evolution of ideas took place: the broader intellectual milieux of the actors, the networks that already existed or were created by the research, even the social and political conditions that favored or hampered the circulation and adoption of certain ideas. This books presents a stroll through this history, in part a guided tour, in part a random walk. First, historical studies on the period from 1920 to 1950 are presented, when martingales emerged as a distinct mathematical concept. Then insights on the period from 1950 into the 1980s are offered, when the concept showed its value in stochastic processes, mathematical statistics, algorithmic randomness and various applications. |
| martingale solved problems: Stochastic Models Donald Andrew Dawson, Luis G. Gorostiza, B. Gail Ivanoff, 2000 This book presents the refereed proceedings of the International Conference on Stochastic Models held in Ottawa (ON, Canada) in honor of Professor Donald A. Dawson. Contributions to the volume were written by students and colleagues of Professor Dawson, many of whom are eminent researchers in their own right. A main theme of the book is the development and study of the Dawson-Watanabe superprocess, a fundamental building block in modelling interaction particle systems undergoing reproduction and movement. The volume also contains an excellent review article by Professor Dawson and a complete list of his work. This comprehensive work offers a wide assortment of articles on Markov processes, branching processes, mathematical finance, filtering, queueing networks, time series, and statistics. It should be of interest to a broad mathematical audience. |
| martingale solved problems: Random Evolutions and Their Applications Anatoly Swishchuk, 2012-12-06 The main purpose of this handbook is to summarize and to put in order the ideas, methods, results and literature on the theory of random evolutions and their applications to the evolutionary stochastic systems in random media, and also to present some new trends in the theory of random evolutions and their applications. In physical language, a random evolution ( RE ) is a model for a dynamical sys tem whose state of evolution is subject to random variations. Such systems arise in all branches of science. For example, random Hamiltonian and Schrodinger equations with random potential in quantum mechanics, Maxwell's equation with a random refractive index in electrodynamics, transport equations associated with the trajec tory of a particle whose speed and direction change at random, etc. There are the examples of a single abstract situation in which an evolving system changes its mode of evolution or law of motion because of random changes of the environment or in a medium. So, in mathematical language, a RE is a solution of stochastic operator integral equations in a Banach space. The operator coefficients of such equations depend on random parameters. Of course, in such generality , our equation includes any homogeneous linear evolving system. Particular examples of such equations were studied in physical applications many years ago. A general mathematical theory of such equations has been developed since 1969, the Theory of Random Evolutions. |
| martingale solved problems: Nonlinear Problems in Aviation and Aerospace S. Sivasundaram, 2000-01-10 The study of nonlinear phenomena in aviation and aerospace includes developments in computer technology and the use of nonlinear mathematical models. Nonlinearities are a feature of aircraft dynamics and flight control systems and need to respond to achieve stability and performance. This multiauthor volume comprises selected papers from the conference Nonlinear Problems in Aviation and Aerospace at Embry-Riddle Aeronautical University and additional invited papers from many distinguished scientists. Coverage includes orbit determination of a tethered satellite system using laser and radar tracking, and intelligent control of agile aircraft, flight control with and without control surfaces. |
| martingale solved problems: Derivation and Martingales Charles A. Hayes, C.Y. Pauc, 2012-12-06 In Part I of this report the pointwise derivation of scalar set functions is investigated, first along the lines of R. DE POSSEL (abstract derivation basis) and A. P. MORSE (blankets); later certain concrete situations (e. g. , the interval basis) are studied. The principal tool is a Vitali property, whose precise form depends on the derivation property studied. The halo (defined at the beginning of Part I, Ch. IV) properties can serve to establish a Vitali property, or sometimes produce directly a derivation property. The main results established are the theorem of JESSEN-MARCINKIEWICZ-ZYGMUND (Part I, Ch. V) and the theorem of A. P. MORSE on the universal derivability of star blankets (Ch. VI) . . In Part II, points are at first discarded; the setting is somatic. It opens by treating an increasing stochastic basis with directed index sets (Th. I. 3) on which premartingales, semimartingales and martingales are defined. Convergence theorems, due largely to K. KRICKEBERG, are obtained using various types of convergence: stochastic, in the mean, in Lp-spaces, in ORLICZ spaces, and according to the order relation. We may mention in particular Th. II. 4. 7 on the stochastic convergence of a submartingale of bounded variation. To each theorem for martingales and semi-martingales there corresponds a theorem in the atomic case in the theory of cell (abstract interval) functions. The derivates concerned are global. Finally, in Ch. |
| martingale solved problems: Introduction To Stochastic Calculus With Applications (3rd Edition) Fima C Klebaner, 2012-03-21 This book presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering. In finance, the stochastic calculus is applied to pricing options by no arbitrage. In biology, it is applied to populations' models, and in engineering it is applied to filter signal from noise. Not everything is proved, but enough proofs are given to make it a mathematically rigorous exposition.This book aims to present the theory of stochastic calculus and its applications to an audience which possesses only a basic knowledge of calculus and probability. It may be used as a textbook by graduate and advanced undergraduate students in stochastic processes, financial mathematics and engineering. It is also suitable for researchers to gain working knowledge of the subject. It contains many solved examples and exercises making it suitable for self study.In the book many of the concepts are introduced through worked-out examples, eventually leading to a complete, rigorous statement of the general result, and either a complete proof, a partial proof or a reference. Using such structure, the text will provide a mathematically literate reader with rapid introduction to the subject and its advanced applications. The book covers models in mathematical finance, biology and engineering. For mathematicians, this book can be used as a first text on stochastic calculus or as a companion to more rigorous texts by a way of examples and exercises./a |
| martingale solved problems: Mathematics of Finance George Yin, Qing Zhang, 2004 Contains papers based on talks given at the first AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance held at Snowbird. This book includes such topics as modeling, estimation, optimization, control, and risk assessment and management. It is suitable for students interested in mathematical finance. |
| martingale solved problems: Ecole d'Ete de Probabilites de Saint-Flour XXI - 1991 Donald A. Dawson, Bernard Maisonneuve, Joel Spencer, 2006-11-14 CONTENTS: D.D. Dawson: Measure-valued Markov Processes.- B. Maisonneuve: Processus de Markov: Naissance, Retournement, Regeneration.- J. Spencer: Nine lectures on Random Graphs. |
| martingale solved problems: PDE and Martingale Methods in Option Pricing Andrea Pascucci, 2011-04-15 This book offers an introduction to the mathematical, probabilistic and numerical methods used in the modern theory of option pricing. The text is designed for readers with a basic mathematical background. The first part contains a presentation of the arbitrage theory in discrete time. In the second part, the theories of stochastic calculus and parabolic PDEs are developed in detail and the classical arbitrage theory is analyzed in a Markovian setting by means of of PDEs techniques. After the martingale representation theorems and the Girsanov theory have been presented, arbitrage pricing is revisited in the martingale theory optics. General tools from PDE and martingale theories are also used in the analysis of volatility modeling. The book also contains an Introduction to Lévy processes and Malliavin calculus. The last part is devoted to the description of the numerical methods used in option pricing: Monte Carlo, binomial trees, finite differences and Fourier transform. |
| martingale solved problems: An Introduction to the Analysis of Paths on a Riemannian Manifold Daniel W. Stroock, 2000 Hoping to make the text more accessible to readers not schooled in the probabalistic tradition, Stroock (affiliation unspecified) emphasizes the geometric over the stochastic analysis of differential manifolds. Chapters deconstruct Brownian paths, diffusions in Euclidean space, intrinsic and extrinsic Riemannian geometry, Bocher's identity, and the bundle of orthonormal frames. The volume humbly concludes with an admission of defeat in regard to recovering the Li-Yau basic differential inequality. Annotation copyrighted by Book News, Inc., Portland, OR. |
What is a martingale? - Quantitative Finance Stack Exchange
Feb 9, 2011 · A martingale can be viewed as a fair game (a game in which there is no arbitrage strategy) A (centered) random walk is a martingale (think of it as the total Gain of the fair game) …
black scholes - Why is $C(t,S_t)/B_t$ a martingale? - Quantitative ...
Jun 3, 2015 · Obviously, it'll create an arbitrage opportunity. Mathematically, this is simply a martingale under the risk-neutral measure. Now, we know the option price must be a martingale …
Strictly local martingales: what is the intuition behind them?
A strict local martingale is a local martingale which is not a true martigale. In fact, a positive strict local martingale is a supermartingale -- i.e. the expectation decreases with horizon. In quant …
Success with Martingale & Hedge - Possible? | Forex Factory
May 17, 2018 · So to give some extra life to martingale, I thought to try hedge in crucial moments. Last year, I was using martingale EA Quanttec with the very dangerous setting; Distance 10 pips …
No Loss Martingale Strategy - Forex Factory
Sep 15, 2015 · Martingale, with its 'death sequence', is a completely unnecessary risk. To state the obvious, that's why ...
Meriam EA - How to Make Martingale EAs Safe - Forex Factory
Jan 30, 2020 · Martingale, Anti-martingale, and Compounding 40 replies. Martingale, Reverse Martingale, Modified Martingale, Maths 1 reply. Martingale Calculation EA (also reverse …
How to use Itô's formula to deduce that a stochastic process is a ...
The question is, how could I deduce that this stochastic process is a martingale or not using Itô's formula? The only thing I know is: Looking at the stochastic integral $ \int K dM $ where $ …
Trapping System: Tunnel Martingale - Forex Factory
Jan 30, 2015 · A bro asked me to build EA which is based on martingale series, with the series of BUY and SELL at the upper price and lower price divided by gap. So it will form like a tunnel. …
Martingale EA - Forex Factory
Dec 15, 2023 · Martingale, Anti-martingale, and Compounding 40 replies. Martingale, Reverse Martingale, Modified Martingale, Maths 1 reply. Martingale Calculation EA (also reverse …
Did Martingale work for someone? - Forex Factory
Jun 27, 2011 · Martingale, stops all of the small SL hits just before it turns which cost you. Martingale or accepting SL hits, both seems to take out a account in the same average amount of …
What is a martingale? - Quantitative Finance Stack Exchange
Feb 9, 2011 · A martingale can be viewed as a fair game (a game in which there is no arbitrage strategy) A (centered) random walk is a martingale (think of it as the total Gain of the fair …
black scholes - Why is $C(t,S_t)/B_t$ a martingale? - Quantitative ...
Jun 3, 2015 · Obviously, it'll create an arbitrage opportunity. Mathematically, this is simply a martingale under the risk-neutral measure. Now, we know the option price must be a …
Strictly local martingales: what is the intuition behind them?
A strict local martingale is a local martingale which is not a true martigale. In fact, a positive strict local martingale is a supermartingale -- i.e. the expectation decreases with horizon. In quant …
Success with Martingale & Hedge - Possible? | Forex Factory
May 17, 2018 · So to give some extra life to martingale, I thought to try hedge in crucial moments. Last year, I was using martingale EA Quanttec with the very dangerous setting; Distance 10 …
No Loss Martingale Strategy - Forex Factory
Sep 15, 2015 · Martingale, with its 'death sequence', is a completely unnecessary risk. To state the obvious, that's why ...
Meriam EA - How to Make Martingale EAs Safe - Forex Factory
Jan 30, 2020 · Martingale, Anti-martingale, and Compounding 40 replies. Martingale, Reverse Martingale, Modified Martingale, Maths 1 reply. Martingale Calculation EA (also reverse …
How to use Itô's formula to deduce that a stochastic process is a ...
The question is, how could I deduce that this stochastic process is a martingale or not using Itô's formula? The only thing I know is: Looking at the stochastic integral $ \int K dM $ where $ …
Trapping System: Tunnel Martingale - Forex Factory
Jan 30, 2015 · A bro asked me to build EA which is based on martingale series, with the series of BUY and SELL at the upper price and lower price divided by gap. So it will form like a tunnel. …
Martingale EA - Forex Factory
Dec 15, 2023 · Martingale, Anti-martingale, and Compounding 40 replies. Martingale, Reverse Martingale, Modified Martingale, Maths 1 reply. Martingale Calculation EA (also reverse …
Did Martingale work for someone? - Forex Factory
Jun 27, 2011 · Martingale, stops all of the small SL hits just before it turns which cost you. Martingale or accepting SL hits, both seems to take out a account in the same average …
