Stochastic Integral And Differential Equations In Mathematical Modelling

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Stochastic Integral And Differential Equations In Mathematical Modelling

Santanu Saha Ray
Stochastic Integral And Differential Equations In Mathematical Modelling Book PDF

Author : Santanu Saha Ray
Publisher : World Scientific
Page : 319 pages
File Size : 46,8 Mb
Release : 2023-04-25
Category : Mathematics
ISBN : 9781800613591

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Stochastic Integral And Differential Equations In Mathematical Modelling by Santanu Saha Ray Pdf

The modelling of systems by differential equations usually requires that the parameters involved be completely known. Such models often originate from problems in physics or economics where we have insufficient information on parameter values. One important class of stochastic mathematical models is stochastic partial differential equations (SPDEs), which can be seen as deterministic partial differential equations (PDEs) with finite or infinite dimensional stochastic processes — either with colour noise or white noise. Though white noise is a purely mathematical construction, it can be a good model for rapid random fluctuations.Stochastic Integral and Differential Equations in Mathematical Modelling concerns the analysis of discrete-time approximations for stochastic differential equations (SDEs) driven by Wiener processes. It also provides a theoretical basis for working with SDEs and stochastic processes.This book is written in a simple and clear mathematical logical language, with basic definitions and theorems on stochastic calculus provided from the outset. Each chapter contains illustrated examples via figures and tables. The reader can also construct new wavelets by using the procedure presented in the book. Stochastic Integral and Differential Equations in Mathematical Modelling fulfils the existing gap in the literature for a comprehensive account of this subject area.

Modeling with Itô Stochastic Differential Equations

E. Allen
Modeling with It Stochastic Differential Equations Book PDF

Author : E. Allen
Publisher : Springer Science & Business Media
Page : 239 pages
File Size : 46,8 Mb
Release : 2007-03-08
Category : Mathematics
ISBN : 9781402059537

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Modeling with Itô Stochastic Differential Equations by E. Allen Pdf

This book explains a procedure for constructing realistic stochastic differential equation models for randomly varying systems in biology, chemistry, physics, engineering, and finance. Introductory chapters present the fundamental concepts of random variables, stochastic processes, stochastic integration, and stochastic differential equations. These concepts are explained in a Hilbert space setting which unifies and simplifies the presentation.

Stochastic Models for Fractional Calculus

Mark M. Meerschaert,Alla Sikorskii
Stochastic Models for Fractional Calculus Book PDF

Author : Mark M. Meerschaert,Alla Sikorskii
Publisher : Walter de Gruyter GmbH & Co KG
Page : 421 pages
File Size : 49,7 Mb
Release : 2019-10-21
Category : Mathematics
ISBN : 9783110559149

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Stochastic Models for Fractional Calculus by Mark M. Meerschaert,Alla Sikorskii Pdf

Fractional calculus is a rapidly growing field of research, at the interface between probability, differential equations, and mathematical physics. It is used to model anomalous diffusion, in which a cloud of particles spreads in a different manner than traditional diffusion. This monograph develops the basic theory of fractional calculus and anomalous diffusion, from the point of view of probability. In this book, we will see how fractional calculus and anomalous diffusion can be understood at a deep and intuitive level, using ideas from probability. It covers basic limit theorems for random variables and random vectors with heavy tails. This includes regular variation, triangular arrays, infinitely divisible laws, random walks, and stochastic process convergence in the Skorokhod topology. The basic ideas of fractional calculus and anomalous diffusion are closely connected with heavy tail limit theorems. Heavy tails are applied in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering. The goal of this book is to prepare graduate students in probability for research in the area of fractional calculus, anomalous diffusion, and heavy tails. Many interesting problems in this area remain open. This book will guide the motivated reader to understand the essential background needed to read and unerstand current research papers, and to gain the insights and techniques needed to begin making their own contributions to this rapidly growing field.

Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

Carlos A. Braumann
Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance Book PDF

Author : Carlos A. Braumann
Publisher : John Wiley & Sons
Page : 304 pages
File Size : 41,5 Mb
Release : 2019-03-08
Category : Mathematics
ISBN : 9781119166078

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Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance by Carlos A. Braumann Pdf

A comprehensive introduction to the core issues of stochastic differential equations and their effective application Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance offers a comprehensive examination to the most important issues of stochastic differential equations and their applications. The author — a noted expert in the field — includes myriad illustrative examples in modelling dynamical phenomena subject to randomness, mainly in biology, bioeconomics and finance, that clearly demonstrate the usefulness of stochastic differential equations in these and many other areas of science and technology. The text also features real-life situations with experimental data, thus covering topics such as Monte Carlo simulation and statistical issues of estimation, model choice and prediction. The book includes the basic theory of option pricing and its effective application using real-life. The important issue of which stochastic calculus, Itô or Stratonovich, should be used in applications is dealt with and the associated controversy resolved. Written to be accessible for both mathematically advanced readers and those with a basic understanding, the text offers a wealth of exercises and examples of application. This important volume: Contains a complete introduction to the basic issues of stochastic differential equations and their effective application Includes many examples in modelling, mainly from the biology and finance fields Shows how to: Translate the physical dynamical phenomenon to mathematical models and back, apply with real data, use the models to study different scenarios and understand the effect of human interventions Conveys the intuition behind the theoretical concepts Presents exercises that are designed to enhance understanding Offers a supporting website that features solutions to exercises and R code for algorithm implementation Written for use by graduate students, from the areas of application or from mathematics and statistics, as well as academics and professionals wishing to study or to apply these models, Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance is the authoritative guide to understanding the issues of stochastic differential equations and their application.

Stochastic Calculus and Stochastic Models

E. J. McShane
Stochastic Calculus and Stochastic Models Book PDF

Author : E. J. McShane
Publisher : Academic Press
Page : 252 pages
File Size : 48,5 Mb
Release : 2014-07-10
Category : Mathematics
ISBN : 9781483218779

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Stochastic Calculus and Stochastic Models by E. J. McShane Pdf

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Stochastic Calculus and Stochastic Models focuses on the properties, functions, and applications of stochastic integrals. The publication first ponders on stochastic integrals, existence of stochastic integrals, and continuity, chain rule, and substitution. Discussions focus on differentiation of a composite function, continuity of sample functions, existence and vanishing of stochastic integrals, canonical form, elementary properties of integrals, and the Itô-belated integral. The book then examines stochastic differential equations, including existence of solutions of stochastic differential equations, linear differential equations and their adjoints, approximation lemma, and the Cauchy-Maruyama approximation. The manuscript takes a look at equations in canonical form, as well as justification of the canonical extension in stochastic modeling; rate of convergence of approximations to solutions; comparison of ordinary and stochastic differential equations; and invariance under change of coordinates. The publication is a dependable reference for mathematicians and researchers interested in stochastic integrals.

Modeling with Itô Stochastic Differential Equations

E. Allen
Modeling with It Stochastic Differential Equations Book PDF

Author : E. Allen
Publisher : Springer
Page : 0 pages
File Size : 53,6 Mb
Release : 2007-03-09
Category : Mathematics
ISBN : 1402059523

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Modeling with Itô Stochastic Differential Equations by E. Allen Pdf

This book explains a procedure for constructing realistic stochastic differential equation models for randomly varying systems in biology, chemistry, physics, engineering, and finance. Introductory chapters present the fundamental concepts of random variables, stochastic processes, stochastic integration, and stochastic differential equations. These concepts are explained in a Hilbert space setting which unifies and simplifies the presentation.

An Introduction to Differential Equations

Anil G Ladde,G S Ladde
An Introduction to Differential Equations Book PDF

Author : Anil G Ladde,G S Ladde
Publisher : World Scientific Publishing Company
Page : 636 pages
File Size : 45,6 Mb
Release : 2013-01-11
Category : Mathematics
ISBN : 9789814397391

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An Introduction to Differential Equations by Anil G Ladde,G S Ladde Pdf

Volume 1: Deterministic Modeling, Methods and Analysis For more than half a century, stochastic calculus and stochastic differential equations have played a major role in analyzing the dynamic phenomena in the biological and physical sciences, as well as engineering. The advancement of knowledge in stochastic differential equations is spreading rapidly across the graduate and postgraduate programs in universities around the globe. This will be the first available book that can be used in any undergraduate/graduate stochastic modeling/applied mathematics courses and that can be used by an interdisciplinary researcher with a minimal academic background. An Introduction to Differential Equations: Volume 2 is a stochastic version of Volume 1 (“An Introduction to Differential Equations: Deterministic Modeling, Methods and Analysis”). Both books have a similar design, but naturally, differ by calculi. Again, both volumes use an innovative style in the presentation of the topics, methods and concepts with adequate preparation in deterministic Calculus. Errata Errata (32 KB)

Stochastic Integration and Differential Equations

Philip Protter
Stochastic Integration and Differential Equations Book PDF

Author : Philip Protter
Publisher : Springer
Page : 430 pages
File Size : 50,9 Mb
Release : 2013-12-21
Category : Mathematics
ISBN : 9783662100615

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Stochastic Integration and Differential Equations by Philip Protter Pdf

It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

Stochastic Partial Differential Equations

Helge Holden,Bernt Oksendal,Jan Uboe,Tusheng Zhang
Stochastic Partial Differential Equations Book PDF

Author : Helge Holden,Bernt Oksendal,Jan Uboe,Tusheng Zhang
Publisher : Springer Science & Business Media
Page : 238 pages
File Size : 44,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781468492156

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Stochastic Partial Differential Equations by Helge Holden,Bernt Oksendal,Jan Uboe,Tusheng Zhang Pdf

This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy". We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve th,~se SPDEs explicitly, or at least provide algorithms or approximations for the solutions.

An Introduction to Continuous-Time Stochastic Processes

Vincenzo Capasso,David Bakstein
An Introduction to Continuous Time Stochastic Processes Book PDF

Author : Vincenzo Capasso,David Bakstein
Publisher : Springer Nature
Page : 560 pages
File Size : 46,7 Mb
Release : 2021-06-18
Category : Mathematics
ISBN : 9783030696535

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An Introduction to Continuous-Time Stochastic Processes by Vincenzo Capasso,David Bakstein Pdf

This textbook, now in its fourth edition, offers a rigorous and self-contained introduction to the theory of continuous-time stochastic processes, stochastic integrals, and stochastic differential equations. Expertly balancing theory and applications, it features concrete examples of modeling real-world problems from biology, medicine, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required. Unlike other books on stochastic methods that specialize in a specific field of applications, this volume examines the ways in which similar stochastic methods can be applied across different fields. Beginning with the fundamentals of probability, the authors go on to introduce the theory of stochastic processes, the Itô Integral, and stochastic differential equations. The following chapters then explore stability, stationarity, and ergodicity. The second half of the book is dedicated to applications to a variety of fields, including finance, biology, and medicine. Some highlights of this fourth edition include a more rigorous introduction to Gaussian white noise, additional material on the stability of stochastic semigroups used in models of population dynamics and epidemic systems, and the expansion of methods of analysis of one-dimensional stochastic differential equations. An Introduction to Continuous-Time Stochastic Processes, Fourth Edition is intended for graduate students taking an introductory course on stochastic processes, applied probability, stochastic calculus, mathematical finance, or mathematical biology. Prerequisites include knowledge of calculus and some analysis; exposure to probability would be helpful but not required since the necessary fundamentals of measure and integration are provided. Researchers and practitioners in mathematical finance, biomathematics, biotechnology, and engineering will also find this volume to be of interest, particularly the applications explored in the second half of the book.

Stochastic Analysis and Diffusion Processes

Gopinath Kallianpur,P Sundar
Stochastic Analysis and Diffusion Processes Book PDF

Author : Gopinath Kallianpur,P Sundar
Publisher : OUP Oxford
Page : 365 pages
File Size : 55,7 Mb
Release : 2014-01-09
Category : Mathematics
ISBN : 9780191631443

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Stochastic Analysis and Diffusion Processes by Gopinath Kallianpur,P Sundar Pdf

Stochastic Analysis and Diffusion Processes presents a simple, mathematical introduction to Stochastic Calculus and its applications. The book builds the basic theory and offers a careful account of important research directions in Stochastic Analysis. The breadth and power of Stochastic Analysis, and probabilistic behavior of diffusion processes are told without compromising on the mathematical details. Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. The book proceeds to construct stochastic integrals, establish the Itô formula, and discuss its applications. Next, attention is focused on stochastic differential equations (SDEs) which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of SDEs and form the main theme of this book. The Stroock-Varadhan martingale problem, the connection between diffusion processes and partial differential equations, Gaussian solutions of SDEs, and Markov processes with jumps are presented in successive chapters. The book culminates with a careful treatment of important research topics such as invariant measures, ergodic behavior, and large deviation principle for diffusions. Examples are given throughout the book to illustrate concepts and results. In addition, exercises are given at the end of each chapter that will help the reader to understand the concepts better. The book is written for graduate students, young researchers and applied scientists who are interested in stochastic processes and their applications. The reader is assumed to be familiar with probability theory at graduate level. The book can be used as a text for a graduate course on Stochastic Analysis.

Parameter Estimation in Stochastic Differential Equations

Jaya P. N. Bishwal
Parameter Estimation in Stochastic Differential Equations Book PDF

Author : Jaya P. N. Bishwal
Publisher : Springer
Page : 268 pages
File Size : 46,9 Mb
Release : 2007-09-26
Category : Mathematics
ISBN : 9783540744481

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Parameter Estimation in Stochastic Differential Equations by Jaya P. N. Bishwal Pdf

Parameter estimation in stochastic differential equations and stochastic partial differential equations is the science, art and technology of modeling complex phenomena. The subject has attracted researchers from several areas of mathematics. This volume presents the estimation of the unknown parameters in the corresponding continuous models based on continuous and discrete observations and examines extensively maximum likelihood, minimum contrast and Bayesian methods.

Probability and Partial Differential Equations in Modern Applied Mathematics

Edward C. Waymire
Probability and Partial Differential Equations in Modern Applied Mathematics Book PDF

Author : Edward C. Waymire
Publisher : Springer Science & Business Media
Page : 265 pages
File Size : 41,6 Mb
Release : 2010-06-14
Category : Mathematics
ISBN : 9780387293714

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Probability and Partial Differential Equations in Modern Applied Mathematics by Edward C. Waymire Pdf

"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations. This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

Stochastic Integration in Banach Spaces

Vidyadhar Mandrekar,Barbara Rüdiger
Stochastic Integration in Banach Spaces Book PDF

Author : Vidyadhar Mandrekar,Barbara Rüdiger
Publisher : Springer
Page : 213 pages
File Size : 53,9 Mb
Release : 2014-12-03
Category : Mathematics
ISBN : 9783319128535

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Stochastic Integration in Banach Spaces by Vidyadhar Mandrekar,Barbara Rüdiger Pdf

Considering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integration theory, existence and uniqueness results and stability theory. The results will be of particular interest to natural scientists and the finance community. Readers should ideally be familiar with stochastic processes and probability theory in general, as well as functional analysis and in particular the theory of operator semigroups. ​

Stochastic Differential Equations in Infinite Dimensions

Leszek Gawarecki,Vidyadhar Mandrekar
Stochastic Differential Equations in Infinite Dimensions Book PDF

Author : Leszek Gawarecki,Vidyadhar Mandrekar
Publisher : Springer Science & Business Media
Page : 300 pages
File Size : 54,6 Mb
Release : 2010-11-29
Category : Mathematics
ISBN : 9783642161940

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Stochastic Differential Equations in Infinite Dimensions by Leszek Gawarecki,Vidyadhar Mandrekar Pdf

The systematic study of existence, uniqueness, and properties of solutions to stochastic differential equations in infinite dimensions arising from practical problems characterizes this volume that is intended for graduate students and for pure and applied mathematicians, physicists, engineers, professionals working with mathematical models of finance. Major methods include compactness, coercivity, monotonicity, in a variety of set-ups. The authors emphasize the fundamental work of Gikhman and Skorokhod on the existence and uniqueness of solutions to stochastic differential equations and present its extension to infinite dimension. They also generalize the work of Khasminskii on stability and stationary distributions of solutions. New results, applications, and examples of stochastic partial differential equations are included. This clear and detailed presentation gives the basics of the infinite dimensional version of the classic books of Gikhman and Skorokhod and of Khasminskii in one concise volume that covers the main topics in infinite dimensional stochastic PDE’s. By appropriate selection of material, the volume can be adapted for a 1- or 2-semester course, and can prepare the reader for research in this rapidly expanding area.