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Stochastic Analysis and Related Topics
by Laurent Decreusefond Jamal NajimSince the early eighties, Ali Süleyman Üstünel has been one of the main contributors to the field of Malliavin calculus. In a workshop held in Paris, June 2010 several prominent researchers gave exciting talks in honor of his 60th birthday. The present volume includes scientific contributions from this workshop. Probability theory is first and foremost aimed at solving real-life problems containing randomness. Markov processes are one of the key tools for modeling that plays a vital part concerning such problems. Contributions on inventory control, mutation-selection in genetics and public-private partnerships illustrate several applications in this volume. Stochastic differential equations, be they partial or ordinary, also play a key role in stochastic modeling. Two of the contributions analyze examples that share a focus on probabilistic tools, namely stochastic analysis and stochastic calculus. Three other papers are devoted more to the theoretical development of these aspects. The volume addresses graduate students and researchers interested in stochastic analysis and its applications.
Stochastic Analysis for Finance with Simulations
by Geon Ho Ho ChoeThis book is an introduction to stochastic analysis and quantitative finance; it includes both theoretical and computational methods. Topics covered are stochastic calculus, option pricing, optimal portfolio investment, and interest rate models. Also included are simulations of stochastic phenomena, numerical solutions of the Black-Scholes-Merton equation, Monte Carlo methods, and time series. Basic measure theory is used as a tool to describe probabilistic phenomena. The level of familiarity with computer programming is kept to a minimum. To make the book accessible to a wider audience, some background mathematical facts are included in the first part of the book and also in the appendices. This work attempts to bridge the gap between mathematics and finance by using diagrams, graphs and simulations in addition to rigorous theoretical exposition. Simulations are not only used as the computational method in quantitative finance, but they can also facilitate an intuitive and deeper understanding of theoretical concepts. Stochastic Analysis for Finance with Simulations is designed for readers who want to have a deeper understanding of the delicate theory of quantitative finance by doing computer simulations in addition to theoretical study. It will particularly appeal to advanced undergraduate and graduate students in mathematics and business, but not excluding practitioners in finance industry.
Stochastic Analysis for Gaussian Random Processes and Fields: With Applications (Chapman & Hall/CRC Monographs on Statistics and Applied Probability)
by Vidyadhar S. Mandrekar Leszek GawareckiStochastic Analysis for Gaussian Random Processes and Fields: With Applications presents Hilbert space methods to study deep analytic properties connecting probabilistic notions. In particular, it studies Gaussian random fields using reproducing kernel Hilbert spaces (RKHSs).The book begins with preliminary results on covariance and associated RKHS
Stochastic Analysis for Poisson Point Processes
by Giovanni Peccati Matthias ReitznerStochastic geometry is the branchof mathematics that studies geometric structures associated with randomconfigurations, such as random graphs, tilings and mosaics. Due to its closeties with stereology and spatial statistics, the results in this area arerelevant for a large number of important applications, e. g. to the mathematicalmodeling and statistical analysis of telecommunication networks, geostatisticsand image analysis. In recent years - due mainly to the impetus of the authorsand their collaborators - a powerful connection has been established betweenstochastic geometry and the Malliavin calculus of variations, which is acollection of probabilistic techniques based on the properties ofinfinite-dimensional differential operators. This has led in particular to thediscovery of a large number of new quantitative limit theorems forhigh-dimensional geometric objects. This unique book presents anorganic collection of authoritative surveys written by the principal actors in thisrapidly evolving field, offering a rigorous yet lively presentation of its manyfacets.
Stochastic Analysis in Production Process and Ecology Under Uncertainty
by Bogusław BiedaThe monograph addresses a problem of stochastic analysis based on the uncertainty assessment by simulation and application of this method in ecology and steel industry under uncertainty. The first chapter defines the Monte Carlo (MC) method and random variables in stochastic models. Chapter two deals with the contamination transport in porous media. Stochastic approach for Municipal Solid Waste transit time contaminants modeling using MC simulation has been worked out. The third chapter describes the risk analysis of the waste to energy facility proposal for Konin city, including the financial aspects. Environmental impact assessment of the ArcelorMittal Steel Power Plant, in Kraków - in the chapter four - is given. Thus, four scenarios of the energy mix production processes were studied. Chapter five contains examples of using ecological Life Cycle Assessment (LCA) - a relatively new method of environmental impact assessment - which help in preparing pro-ecological strategy, and which can lead to reducing the amount of wastes produced in the ArcelorMittal Steel Plant production processes. Moreover, real input and output data of selected processes under uncertainty, mainly used in the LCA technique, have been examined. The last chapter of this monograph contains final summary. The log-normal probability distribution, widely used in risk analysis and environmental management, in order to develop a stochastic analysis of the LCA, as well as uniform distribution for stochastic approach of pollution transport in porous media has been proposed. The distributions employed in this monograph are assembled from site-specific data, data existing in the most current literature, and professional judgment.
Stochastic Analysis of Biochemical Systems
by David F. Anderson Thomas G. KurtzThis book focuses on counting processes and continuous-time Markov chains motivated by examples and applications drawn from chemical networks in systems biology. The book should serve well as a supplement for courses in probability and stochastic processes. While the material is presented in a manner most suitable for students who have studied stochastic processes up to and including martingales in continuous time, much of the necessary background material is summarized in the Appendix. Students and Researchers with a solid understanding of calculus, differential equations and elementary probability and who are well-motivated by the applications will find this book of interest. David F. Anderson is Associate Professor in the Department of Mathematics at the University of Wisconsin and Thomas G. Kurtz is Emeritus Professor in the Departments of Mathematics and Statistics at that university. Their research is focused on probability and stochastic processes with applications in biology and other areas of science and technology. These notes are based in part on lectures given by Professor Anderson at the University of Wisconsin - Madison and by Professor Kurtz at Goethe University Frankfurt.
Stochastic Approaches to Electron Transport in Micro- and Nanostructures (Modeling and Simulation in Science, Engineering and Technology)
by Ivan Dimov Siegfried Selberherr Mihail NedjalkovThe book serves as a synergistic link between the development of mathematical models and the emergence of stochastic (Monte Carlo) methods applied for the simulation of current transport in electronic devices. Regarding the models, the historical evolution path, beginning from the classical charge carrier transport models for microelectronics to current quantum-based nanoelectronics, is explicatively followed. Accordingly, the solution methods are elucidated from the early phenomenological single particle algorithms applicable for stationary homogeneous physical conditions up to the complex algorithms required for quantum transport, based on particle generation and annihilation. The book fills the gap between monographs focusing on the development of the theory and the physical aspects of models, their application, and their solution methods and monographs dealing with the purely theoretical approaches for finding stochastic solutions of Fredholm integral equations.
Stochastic Approximation: A Dynamical Systems Viewpoint (Texts and Readings in Mathematics #48)
by Vivek S. BorkarThis book serves as an advanced text for a graduate course on stochastic algorithms for the students of probability and statistics, engineering, economics and machine learning. This second edition gives a comprehensive treatment of stochastic approximation algorithms based on the ordinary differential equation (ODE) approach which analyses the algorithm in terms of a limiting ODE. It has a streamlined treatment of the classical convergence analysis and includes several recent developments such as concentration bounds, avoidance of traps, stability tests, distributed and asynchronous schemes, multiple time scales, general noise models, etc., and a category-wise exposition of many important applications. It is also a useful reference for researchers and practitioners in the field.
Stochastic Approximation: Second Edition (Texts and Readings in Mathematics #48)
by Vivek S. BorkarThis book serves as an advanced text for a graduate course on stochastic algorithms for graduate students in probability and statistics, engineering, economics and machine learning. This second edition gives a comprehensive treatment of stochastic approximation algorithms based on the “ordinary differential equation (ODE) approach” which analyses the algorithm in terms of a limiting ODE. It has a streamlined treatment of the classical convergence analysis and includes several recent developments such as concentration bounds, avoidance of traps, stability tests, distributed and asynchronous schemes, multiple time scales, general noise models, etc., and a category-wise exposition of many important applications. It is also a useful reference for researchers and practitioners in the field.
Stochastic Averaging and Stochastic Extremum Seeking
by Shu-Jun Liu Miroslav KrsticStochastic Averaging and Extremum Seeking treats methods inspired by attempts to understand the seemingly non-mathematical question of bacterial chemotaxis and their application in other environments. The text presents significant generalizations on existing stochastic averaging theory developed from scratch and necessitated by the need to avoid violation of previous theoretical assumptions by algorithms which are otherwise effective in treating these systems. Coverage is given to four main topics. Stochastic averaging theorems are developed for the analysis of continuous-time nonlinear systems with random forcing, removing prior restrictions on nonlinearity growth and on the finiteness of the time interval. The new stochastic averaging theorems are usable not only as approximation tools but also for providing stability guarantees. Stochastic extremum-seeking algorithms are introduced for optimization of systems without available models. Both gradient- and Newton-based algorithms are presented, offering the user the choice between the simplicity of implementation (gradient) and the ability to achieve a known, arbitrary convergence rate (Newton). The design of algorithms for non-cooperative/adversarial games is described. The analysis of their convergence to Nash equilibria is provided. The algorithms are illustrated on models of economic competition and on problems of the deployment of teams of robotic vehicles. Bacterial locomotion, such as chemotaxis in E. coli, is explored with the aim of identifying two simple feedback laws for climbing nutrient gradients. Stochastic extremum seeking is shown to be a biologically-plausible interpretation for chemotaxis. For the same chemotaxis-inspired stochastic feedback laws, the book also provides a detailed analysis of convergence for models of nonholonomic robotic vehicles operating in GPS-denied environments. The book contains block diagrams and several simulation examples, including examples arising from bacterial locomotion, multi-agent robotic systems, and economic market models. Stochastic Averaging and Extremum Seeking will be informative for control engineers from backgrounds in electrical, mechanical, chemical and aerospace engineering and to applied mathematicians. Economics researchers, biologists, biophysicists and roboticists will find the applications examples instructive.
Stochastic Averaging: Methods and Applications, Volume 1
by Wei-Qiu Zhu Mao-Lin Deng Guo-Qiang CaiThe stochastic averaging methods are among the most effective and widely applied approximate methods for studying nonlinear stochastic dynamics. Upon an overview of global research on the subject, the book highlights a comprehensive summary of research results obtained by the group led by Professor Weiqiu Zhu at Zhejiang University in China and the group led by Professors Y. K. Lin and G. Q. Cai at Florida Atlantic University in the USA over the past three decades. The books are structured to progress logically from foundational principles to simple problems and then to increasingly complex applications. To facilitate understanding and mastery of the methods, the books offer essential preliminary knowledge and a wealth of examples. The book comprises two volumes. Volume 1 introduces the basic principles of stochastic averaging methods and their applications to single-degree-of-freedom systems under various random excitations. It also covers stochastic averaging methods for quasi-Hamiltonian systems subjected to different random excitations, including Gaussian white noise, combined Gaussian and Poisson white noises, and fractional Gaussian noise. Volume 2 explores stochastic averaging methods for quasi-integrable Hamiltonian systems under colored noise excitation, quasi-integrable Hamiltonian systems with genetic effects under Gaussian white noise and colored noise excitations, and quasi-generalized Hamiltonian systems under Gaussian white noise excitation. Additionally, it covers applications of these methods in ecosystems and some other natural science and engineering scenarios. These books serve as both introductory texts and valuable reference resources for readers in higher education and research institutions who are interested in or actively engaged in research involving nonlinear stochastic dynamics. The fields covered include mechanics, physics, chemistry, biology, ecology, astronautics and aeronautics, oceanography, civil engineering, mechanical engineering, and electrical engineering.
Stochastic Benchmarking: Theory and Applications (International Series in Operations Research & Management Science #317)
by Vincent Charles Alireza Amirteimoori Biresh K. Sahoo Saber MehdizadehThis book introduces readers to benchmarking techniques in the stochastic environment, primarily stochastic data envelopment analysis (DEA), and provides stochastic models in DEA for the possibility of variations in inputs and outputs. It focuses on the application of theories and interpretations of the mathematical programs, which are combined with economic and organizational thinking. The book’s main purpose is to shed light on the advantages of the different methods in deterministic and stochastic environments and thoroughly prepare readers to properly use these methods in various cases. Simple examples, along with graphical illustrations and real-world applications in industry, are provided for a better understanding. The models introduced here can be easily used in both theoretical and real-world evaluations. This book is intended for graduate and PhD students, advanced consultants, and practitioners with an interest in quantitative performance evaluation.
Stochastic Biomathematical Models
by Mostafa Bachar Susanne Ditlevsen Jerry J. BatzelStochastic biomathematical models are becoming increasingly important as new light is shed on the role of noise in living systems. In certain biological systems, stochastic effects may even enhance a signal, thus providing a biological motivation for the noise observed in living systems. Recent advances in stochastic analysis and increasing computing power facilitate the analysis of more biophysically realistic models, and this book provides researchers in computational neuroscience and stochastic systems with an overview of recent developments. Key concepts are developed in chapters written by experts in their respective fields. Topics include: one-dimensional homogeneous diffusions and their boundary behavior, large deviation theory and its application in stochastic neurobiological models, a review of mathematical methods for stochastic neuronal integrate-and-fire models, stochastic partial differential equation models in neurobiology, and stochastic modeling of spreading cortical depression.
Stochastic Calculus
by Paolo BaldiThis book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications. It is the only textbook on the subject to include more than two hundred exercises with complete solutions. After explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used. Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. Including full mathematical statements and rigorous proofs, this book is completely self-contained and suitable for lecture courses as well as self-study.
Stochastic Calculus and Applications
by Robert J. Elliott Samuel N. CohenCompletely 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)
Stochastic Calculus and Differential Equations for Physics and Finance
by Joseph L. MccauleyStochastic calculus provides a powerful description of a specific class of stochastic processes in physics and finance. However, many econophysicists struggle to understand it. This book presents the subject simply and systematically, giving graduate students and practitioners a better understanding and enabling them to apply the methods in practice. The book develops Ito calculus and Fokker-Planck equations as parallel approaches to stochastic processes, using those methods in a unified way. The focus is on nonstationary processes, and statistical ensembles are emphasized in time series analysis. Stochastic calculus is developed using general martingales. Scaling and fat tails are presented via diffusive models. Fractional Brownian motion is thoroughly analyzed and contrasted with Ito processes. The Chapman-Kolmogorov and Fokker-Planck equations are shown in theory and by example to be more general than a Markov process. The book also presents new ideas in financial economics and a critical survey of econometrics.
Stochastic Calculus for Finance
by Janusz Traple Ekkehard Kopp Marek CapinskiThis book focuses specifically on the key results in stochastic processes that have become essential for finance practitioners to understand. The authors study the Wiener process and Itô integrals in some detail, with a focus on results needed for the Black-Scholes option pricing model. After developing the required martingale properties of this process, the construction of the integral and the Itô formula (proved in detail) become the centrepiece, both for theory and applications, and to provide concrete examples of stochastic differential equations used in finance. Finally, proofs of the existence, uniqueness and the Markov property of solutions of (general) stochastic equations complete the book. Using careful exposition and detailed proofs, this book is a far more accessible introduction to Itô calculus than most texts. Students, practitioners and researchers will benefit from its rigorous, but unfussy, approach to technical issues. Solutions to the exercises are available online.
Stochastic Calculus in Infinite Dimensions and SPDEs (SpringerBriefs in Mathematics)
by Dan Crisan Daniel GoodairIntroducing a groundbreaking framework for stochastic partial differential equations (SPDEs), this work presents three significant advancements over the traditional variational approach. Firstly, Stratonovich SPDEs are explicitly addressed. Widely used in physics, Stratonovich SPDEs have typically been converted to Ito form for mathematical treatment. While this conversion is understood heuristically, a comprehensive treatment in infinite dimensions has been lacking, primarily due to insufficient rigorous results on martingale properties. Secondly, the framework incorporates differential noise, assuming the noise operator is only bounded from a smaller Hilbert space into a larger one, rather than within the same space. This necessitates additional regularity in the Ito form to solve the original Stratonovich SPDE. This aspect has been largely overlooked, despite the increasing popularity of gradient-dependent Stratonovich noise in fluid dynamics and regularisation by noise studies. Lastly, the framework departs from the explicit duality structure (Gelfand Triple), which is typically expected in the study of analytically strong solutions. This extension builds on the classical variational framework established by Röckner and Pardoux, advancing it in all three key aspects. Explore this innovative approach that not only addresses existing challenges but also opens new avenues for research and application in SPDEs.
Stochastic Calculus with Infinitesimals
by Frederik S. HerzbergStochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.
Stochastic Calculus: A Practical Introduction (Probability and Stochastics Series #6)
by Richard DurrettThis compact yet thorough text zeros in on the parts of the theory that are particularly relevant to applications . It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential equations. It solves stochastic differential equations by a variety of methods and studies in detail the one-dimensional case. The book concludes with a treatment of semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions. The presentation is unparalleled in its clarity and simplicity. Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together the material you need to effectively and efficiently impart the practical background they need.
Stochastic Chemical Kinetics
by Gábor Lente Péter ÉrdiThis volume reviews the theory and simulation methods of stochastic kinetics by integrating historical and recent perspectives, presents applications, mostly in the context of systems biology and also in combustion theory. In recent years, due to the development in experimental techniques, such as optical imaging, single cell analysis, and fluorescence spectroscopy, biochemical kinetic data inside single living cells have increasingly been available. The emergence of systems biology brought renaissance in the application of stochastic kinetic methods.
Stochastic Choice Theory (Econometric Society Monographs)
by Tomasz StrzaleckiModels of stochastic choice are studied in decision theory, discrete choice econometrics, behavioral economics and psychology. Numerous experiments show that perception of stimuli is not deterministic, but stochastic (randomly determined). A growing body of evidence indicates that the same is true of economic choices. Whether trials are separated by days or minutes, the fraction of choice reversals is substantial. Stochastic Choice Theory offers a systematic introduction to these models, unifying insights from various fields. It explores mathematical models of stochastic choice, which have a variety of applications in game theory, industrial organization, labor economics, marketing, and experimental economics. Offering a systematic introduction to the field, this book builds up from scratch without any prior knowledge requirements and surveys recent developments, bringing readers to the frontier of research.
Stochastic Communities: A Mathematical Theory of Biodiversity
by A. DewdneyStochastic Communities presents a theory of biodiversity by analyzing the distribution of abundances among species in the context of a community. The basis of this theory is a distribution called the "J distribution." This distribution is a pure hyperbola and mathematically implied by the "stochastic species hypothesis" assigning equal probabilities of birth and death within the population of each species over varying periods of time. The J distribution in natural communities has strong empirical support resulting from a meta-study and strong theoretical support from a theorem that is mathematically implied by the stochastic species hypothesis.
Stochastic Control Theory
by Makiko NisioThis book offers a systematic introduction to the optimal stochastic control theory via the dynamic programming principle, which is a powerful tool to analyze control problems. First we consider completely observable control problems with finite horizons. Using a time discretization we construct a nonlinear semigroup related to the dynamic programming principle (DPP), whose generator provides the Hamilton-Jacobi-Bellman (HJB) equation, and we characterize the value function via the nonlinear semigroup, besides the viscosity solution theory. When we control not only the dynamics of a system but also the terminal time of its evolution, control-stopping problems arise. This problem is treated in the same frameworks, via the nonlinear semigroup. Its results are applicable to the American option price problem. Zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games are studied via a nonlinear semigroup related to DPP (the min-max principle, to be precise). Using semi-discretization arguments, we construct the nonlinear semigroups whose generators provide lower and upper Isaacs equations. Concerning partially observable control problems, we refer to stochastic parabolic equations driven by colored Wiener noises, in particular, the Zakai equation. The existence and uniqueness of solutions and regularities as well as Itô's formula are stated. A control problem for the Zakai equations has a nonlinear semigroup whose generator provides the HJB equation on a Banach space. The value function turns out to be a unique viscosity solution for the HJB equation under mild conditions. This edition provides a more generalized treatment of the topic than does the earlier book Lectures on Stochastic Control Theory (ISI Lecture Notes 9), where time-homogeneous cases are dealt with. Here, for finite time-horizon control problems, DPP was formulated as a one-parameter nonlinear semigroup, whose generator provides the HJB equation, by using a time-discretization method. The semigroup corresponds to the value function and is characterized as the envelope of Markovian transition semigroups of responses for constant control processes. Besides finite time-horizon controls, the book discusses control-stopping problems in the same frameworks.
Stochastic Control in Discrete and Continuous Time
by Atle SeierstadThis book provides a comprehensive introduction to stochastic control problems in discrete and continuous time. The material is presented logically, beginning with the discrete-time case using few mathematical tools before proceeding to the stochastic continuous-time models requiring more advanced mathematics. Topics covered include stochastic maximum principles for discrete time, continuous time, and for problems with terminal conditions. A nonstandard treatment of piecewise deterministic problems, related to some control problems, is also presented. Numerous illustrative examples and exercises are included to enhance the understanding of the reader. By interlinking many fields in stochastic control, the material gives the student the opportunity to see the connections between different fields and the underlying ideas that unify them. This text will be of benefit to students in economics, engineering, applied mathematics and related fields. Prerequisites include a course in calculus and elementary probability theory.