# Search Results for "foundations-of-infinitesimal-stochastic-analysis"

## Foundations of Infinitesimal Stochastic Analysis

**Author**: K.D. Stroyan,J.M. Bayod**Publisher:**Elsevier**ISBN:**0080960421**Category:**Computers**Page:**491**View:**3324

This book gives a complete and elementary account of fundamental results on hyperfinite measures and their application to stochastic processes, including the *-finite Stieltjes sum approximation of martingale integrals. Many detailed examples, not found in the literature, are included. It begins with a brief chapter on tools from logic and infinitesimal (or non-standard) analysis so that the material is accessible to beginning graduate students.

## An Infinitesimal Approach to Stochastic Analysis

**Author**: H. Jerome Keisler**Publisher:**American Mathematical Soc.**ISBN:**0821822977**Category:**Mathematics**Page:**184**View:**3521

## Nonstandard Methods in Stochastic Analysis and Mathematical Physics

**Author**: Sergio Albeverio,Jens Erik Fenstad,Raphael Høegh-Krohn,Tom Lindstrøm**Publisher:**Courier Dover Publications**ISBN:**0486468992**Category:**Mathematics**Page:**526**View:**8088

Two-part treatment begins with a self-contained introduction to the subject, followed by applications to stochastic analysis and mathematical physics. "A welcome addition." — Bulletin of the American Mathematical Society. 1986 edition.

## Hyperfinite Dirichlet Forms and Stochastic Processes

**Author**: Sergio Albeverio,Ruzong Fan,Frederik S. Herzberg**Publisher:**Springer Science & Business Media**ISBN:**9783642196591**Category:**Mathematics**Page:**284**View:**2721

This monograph treats the theory of Dirichlet forms from a comprehensive point of view, using "nonstandard analysis." Thus, it is close in spirit to the discrete classical formulation of Dirichlet space theory by Beurling and Deny (1958). The discrete infinitesimal setup makes it possible to study the diffusion and the jump part using essentially the same methods. This setting has the advantage of being independent of special topological properties of the state space and in this sense is a natural one, valid for both finite- and infinite-dimensional spaces. The present monograph provides a thorough treatment of the symmetric as well as the non-symmetric case, surveys the theory of hyperfinite Lévy processes, and summarizes in an epilogue the model-theoretic genericity of hyperfinite stochastic processes theory.

## An Introduction to Mathematical Analysis for Economic Theory and Econometrics

**Author**: Dean Corbae,Maxwell B. Stinchcombe,Juraj Zeman**Publisher:**Princeton University Press**ISBN:**1400833086**Category:**Business & Economics**Page:**688**View:**4355

Providing an introduction to mathematical analysis as it applies to economic theory and econometrics, this book bridges the gap that has separated the teaching of basic mathematics for economics and the increasingly advanced mathematics demanded in economics research today. Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman equip students with the knowledge of real and functional analysis and measure theory they need to read and do research in economic and econometric theory. Unlike other mathematics textbooks for economics, An Introduction to Mathematical Analysis for Economic Theory and Econometrics takes a unified approach to understanding basic and advanced spaces through the application of the Metric Completion Theorem. This is the concept by which, for example, the real numbers complete the rational numbers and measure spaces complete fields of measurable sets. Another of the book's unique features is its concentration on the mathematical foundations of econometrics. To illustrate difficult concepts, the authors use simple examples drawn from economic theory and econometrics. Accessible and rigorous, the book is self-contained, providing proofs of theorems and assuming only an undergraduate background in calculus and linear algebra. Begins with mathematical analysis and economic examples accessible to advanced undergraduates in order to build intuition for more complex analysis used by graduate students and researchers Takes a unified approach to understanding basic and advanced spaces of numbers through application of the Metric Completion Theorem Focuses on examples from econometrics to explain topics in measure theory

## Principles of Infinitesimal Stochastic and Financial Analysis

**Author**: Imme van den Berg**Publisher:**World Scientific**ISBN:**9789810243586**Category:**Mathematics**Page:**136**View:**2048

There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level. The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude -- in the sense of nonstandard analysis -- are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.

## Stochastic Calculus with Infinitesimals

**Author**: Frederik S. Herzberg**Publisher:**Springer**ISBN:**3642331491**Category:**Mathematics**Page:**112**View:**6783

Stochastic 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.

## Fundamentals of Stochastic Filtering

**Author**: Alan Bain,Dan Crisan**Publisher:**Springer Science & Business Media**ISBN:**0387768963**Category:**Mathematics**Page:**390**View:**6268

This book provides a rigorous mathematical treatment of the non-linear stochastic filtering problem using modern methods. Particular emphasis is placed on the theoretical analysis of numerical methods for the solution of the filtering problem via particle methods. The book should provide sufficient background to enable study of the recent literature. While no prior knowledge of stochastic filtering is required, readers are assumed to be familiar with measure theory, probability theory and the basics of stochastic processes. Most of the technical results that are required are stated and proved in the appendices. Exercises and solutions are included.

## Applied Nonstandard Analysis

**Author**: Martin Davis**Publisher:**Courier Corporation**ISBN:**0486152340**Category:**Mathematics**Page:**208**View:**6228

This applications-oriented text assumes no knowledge of mathematical logic in its development of nonstandard analysis techniques and their applications to elementary real analysis and topological and Hilbert space. 1977 edition.

## Model theory of stochastic processes

**Author**: Sergio Fajardo,H. Jerome Keisler**Publisher:**A K Peters Ltd**ISBN:**9781568811673**Category:**Mathematics**Page:**136**View:**5295

This book presents new research in probability theory using ideas from mathematical logic. It is a general study of stochastic processes on adapted probability spaces, employing the concept of similarity of stochastic processes based on the notion of adapted distribution. The authors use ideas from model theory and methods from nonstandard analysis. The construction of spaces with certain richness properties, defined by insights from model theory, becomes easy using nonstandard methods, but remains difficult or impossible without them.

## Probabilistic Analysis and Related Topics

**Author**: A. T. Bharucha-Reid**Publisher:**Elsevier**ISBN:**1483275469**Category:**Mathematics**Page:**270**View:**8078

Probabilistic Analysis and Related Topics, Volume 3 focuses on the continuity, integrability, and differentiability of random functions, including operator theory, measure theory, and functional and numerical analysis. The selection first offers information on the qualitative theory of stochastic systems and Langevin equations with multiplicative noise. Discussions focus on phase-space evolution via direct integration, phase-space evolution, linear and nonlinear systems, linearization, and generalizations. The text then ponders on the stability theory of stochastic difference systems and Markov properties for random fields. Topics include Markov property of solutions of stochastic partial differential equations; Markov property for generalized Gaussian random fields; Markov properties for generalized random fields; stochastic stability of nonlinear systems; and linear stochastic systems. The publication examines the method of random contractors and its applications to random nonlinear equations, including integral contractors and applications to random equations; random contractors with random nonlinear majorant functions; and random contractors and application to random nonlinear operator equations. The selection is a valuable reference for mathematicians and researchers interested in the general theory of random functions.

## Fundamentals of Probability with Stochastic Processes

**Author**: Saeed Ghahramani**Publisher:**CRC Press**ISBN:**042985627X**Category:**Mathematics**Page:**632**View:**3935

"The 4th edition of Ghahramani's book is replete with intriguing historical notes, insightful comments, and well-selected examples/exercises that, together, capture much of the essence of probability. Along with its Companion Website, the book is suitable as a primary resource for a first course in probability. Moreover, it has sufficient material for a sequel course introducing stochastic processes and stochastic simulation." --Nawaf Bou-Rabee, Associate Professor of Mathematics, Rutgers University Camden, USA "This book is an excellent primer on probability, with an incisive exposition to stochastic processes included as well. The flow of the text aids its readability, and the book is indeed a treasure trove of set and solved problems. Every sub-topic within a chapter is supplemented by a comprehensive list of exercises, accompanied frequently by self-quizzes, while each chapter ends with a useful summary and another rich collection of review problems." --Dalia Chakrabarty, Department of Mathematical Sciences, Loughborough University, UK "This textbook provides a thorough and rigorous treatment of fundamental probability, including both discrete and continuous cases. The book’s ample collection of exercises gives instructors and students a great deal of practice and tools to sharpen their understanding. Because the definitions, theorems, and examples are clearly labeled and easy to find, this book is not only a great course accompaniment, but an invaluable reference." --Joshua Stangle, Assistant Professor of Mathematics, University of Wisconsin – Superior, USA This one- or two-term calculus-based basic probability text is written for majors in mathematics, physical sciences, engineering, statistics, actuarial science, business and finance, operations research, and computer science. It presents probability in a natural way: through interesting and instructive examples and exercises that motivate the theory, definitions, theorems, and methodology. This book is mathematically rigorous and, at the same time, closely matches the historical development of probability. Whenever appropriate, historical remarks are included, and the 2096 examples and exercises have been carefully designed to arouse curiosity and hence encourage students to delve into the theory with enthusiasm. New to the Fourth Edition: 538 new examples and exercises have been added, almost all of which are of applied nature in realistic contexts Self-quizzes at the end of each section and self-tests at the end of each chapter allow students to check their comprehension of the material An all-new Companion Website includes additional examples, complementary topics not covered in the previous editions, and applications for more in-depth studies, as well as a test bank and figure slides. It also includes complete solutions to all self-test and self-quiz problems Saeed Ghahramani is Professor of Mathematics and Dean of the College of Arts and Sciences at Western New England University. He received his Ph.D. from the University of California at Berkeley in Mathematics and is a recipient of teaching awards from Johns Hopkins University and Towson University. His research focuses on applied probability, stochastic processes, and queuing theory.

## Non-standard Analysis

**Author**: Abraham Robinson**Publisher:**Princeton University Press**ISBN:**1400884225**Category:**Mathematics**Page:**308**View:**2288

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body.

## Truth, Possibility and Probability

*New Logical Foundations of Probability and Statistical Inference*

**Author**: R. Chuaqui**Publisher:**Elsevier**ISBN:**9780080872773**Category:**Mathematics**Page:**483**View:**689

Anyone involved in the philosophy of science is naturally drawn into the study of the foundations of probability. Different interpretations of probability, based on competing philosophical ideas, lead to different statistical techniques, and frequently to mutually contradictory consequences. This unique book presents a new interpretation of probability, rooted in the traditional interpretation that was current in the 17th and 18th centuries. Mathematical models are constructed based on this interpretation, and statistical inference and decision theory are applied, including some examples in artificial intelligence, solving the main foundational problems. Nonstandard analysis is extensively developed for the construction of the models and in some of the proofs. Many nonstandard theorems are proved, some of them new, in particular, a representation theorem that asserts that any stochastic process can be approximated by a process defined over a space with equiprobable outcomes.

## Large Deviations for Stochastic Processes

**Author**: Jin Feng,Thomas G. Kurtz**Publisher:**American Mathematical Soc.**ISBN:**0821841459**Category:**Mathematics**Page:**410**View:**7636

The book is devoted to the results on large deviations for a class of stochastic processes. Following an introduction and overview, the material is presented in three parts. Part 1 gives necessary and sufficient conditions for exponential tightness that are analogous to conditions for tightness in the theory of weak convergence. Part 2 focuses on Markov processes in metric spaces. For a sequence of such processes, convergence of Fleming's logarithmically transformed nonlinear semigroups is shown to imply the large deviation principle in a manner analogous to the use of convergence of linear semigroups in weak convergence. Viscosity solution methods provide applicable conditions for the necessary convergence. Part 3 discusses methods for verifying the comparison principle for viscosity solutions and applies the general theory to obtain a variety of new and known results on large deviations for Markov processes. In examples concerning infinite dimensional state spaces, new comparison principles are derived for a class of Hamilton-Jacobi equations in Hilbert spaces and in spaces of probability measures.

## The Malliavin Calculus and Related Topics

**Author**: David Nualart**Publisher:**Springer Science & Business Media**ISBN:**3540283293**Category:**Mathematics**Page:**382**View:**8446

The Malliavin calculus is an infinite-dimensional differential calculus on a Gaussian space, developed to provide a probabilistic proof to Hörmander's sum of squares theorem but has found a range of applications in stochastic analysis. This book presents the features of Malliavin calculus and discusses its main applications. This second edition includes recent applications in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.

## Radically Elementary Probability Theory. (AM-117)

**Author**: Edward Nelson**Publisher:**Princeton University Press**ISBN:**1400882141**Category:**Mathematics**Page:**107**View:**9382

Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.

## Recent Developments in Computational Finance

*Foundations, Algorithms and Applications*

**Author**: Thomas Gerstner,Peter E. Kloeden**Publisher:**World Scientific**ISBN:**9814436429**Category:**Business & Economics**Page:**472**View:**6128

Computational finance is an interdisciplinary field which joins financial mathematics, stochastics, numerics and scientific computing. Its task is to estimate as accurately and efficiently as possible the risks that financial instruments generate. This volume consists of a series of cutting-edge surveys of recent developments in the field written by leading international experts. These make the subject accessible to a wide readership in academia and financial businesses. The book consists of 13 chapters divided into 3 parts: foundations, algorithms and applications. Besides surveys of existing results, the book contains many new previously unpublished results.

## Applied Stochastic Processes and Control for Jump-Diffusions

*Modeling, Analysis, and Computation*

**Author**: Floyd B. Hanson**Publisher:**SIAM**ISBN:**9780898718638**Category:**Diffusion processes**Page:**443**View:**8962

This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump diffusions in continuous time. The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems.