Stochastic Differential Equations and Diffusion Processes

Author: N. Ikeda,S. Watanabe

Publisher: Elsevier

ISBN: 1483296156

Category: Mathematics

Page: 572

View: 7085


Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis. A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sections discussing complex (conformal) martingales and Kahler diffusions have been added.

Numerical Solution of Stochastic Differential Equations

Author: Peter E. Kloeden,Eckhard Platen

Publisher: Springer Science & Business Media

ISBN: 3662126168

Category: Mathematics

Page: 636

View: 9452


The numerical analysis of stochastic differential equations (SDEs) differs significantly from that of ordinary differential equations. This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations. From the reviews: "The authors draw upon their own research and experiences in obviously many disciplines... considerable time has obviously been spent writing this in the simplest language possible." --ZAMP

Stochastic Differential Equations, Backward SDEs, Partial Differential Equations

Author: Etienne Pardoux,Aurel Rӑşcanu

Publisher: Springer

ISBN: 3319057146

Category: Mathematics

Page: 667

View: 556


This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has become an important subject of Mathematics and Applied Mathematics, because of its mathematical richness and its importance for applications in many areas of Physics, Biology, Economics and Finance, where random processes play an increasingly important role. One important aspect is the connection between diffusion processes and linear partial differential equations of second order, which is in particular the basis for Monte Carlo numerical methods for linear PDEs. Since the pioneering work of Peng and Pardoux in the early 1990s, a new type of SDEs called backward stochastic differential equations (BSDEs) has emerged. The two main reasons why this new class of equations is important are the connection between BSDEs and semilinear PDEs, and the fact that BSDEs constitute a natural generalization of the famous Black and Scholes model from Mathematical Finance, and thus offer a natural mathematical framework for the formulation of many new models in Finance.

Stochastic Processes and Applications

Diffusion Processes, the Fokker-Planck and Langevin Equations

Author: Grigorios A. Pavliotis

Publisher: Springer

ISBN: 1493913239

Category: Mathematics

Page: 339

View: 2700


This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated. The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to equilibrium for diffusion processes, inference methods for stochastic differential equations, derivation of the generalized Langevin equation, exit time problems) cannot be easily found in textbook form and will be useful to both researchers and students interested in the applications of stochastic processes.

Stochastic Differential Equations

With Applications to Physics and Engineering

Author: K. Sobczyk

Publisher: Springer Science & Business Media

ISBN: 9401137129

Category: Mathematics

Page: 400

View: 6580


'Et moi, ..~ si lavait su CO.llUlJalt en revc:nir, One acMcc matbcmatica bu JaIdcred the human rac:c. It bu put COIDIDOD _ beet je n'y serais point aBe.' Jules Verne wbac it bdoup, 0Jl!be~ IbcII _t to!be dusty cauialcr Iabc & d 'diMardod__ The series is divergent; thc:reforc we may be -'. I!.ticT. Bc:I1 able to do something with it. O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world when: 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 statcmalts as: 'One service topology has rendered mathematical physics ...-; 'One service logic has rendered c0m 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 19n. 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 branc:hes. It also happens, quite often in fact, that branches which were thought to be completely.

Stochastic Analysis and Diffusion Processes

Author: Gopinath Kallianpur,P Sundar

Publisher: Oxford University Press

ISBN: 0199657068

Category: Mathematics

Page: 352

View: 347


Beginning with the concept of random processes and Brownian motion and building on the theory and research directions in a self-contained manner, this book provides an introduction to stochastic analysis for graduate students, researchers and applied scientists interested in stochastic processes and their applications.

Probability Theory III

Stochastic Calculus

Author: Yurij V. Prokhorov,Albert N. Shiryaev

Publisher: Springer Science & Business Media

ISBN: 3662036401

Category: Mathematics

Page: 256

View: 9275


This volume of the Encyclopaedia is a survey of stochastic calculus, an increasingly important part of probability, authored by well-known experts in the field. The book addresses graduate students and researchers in probability theory and mathematical statistics, as well as physicists and engineers who need to apply stochastic methods.

Wahrscheinlichkeitstheorie und Stochastische Prozesse

Author: Michael Mürmann

Publisher: Springer-Verlag

ISBN: 364238160X

Category: Mathematics

Page: 428

View: 8764


Dieses Lehrbuch beschäftigt sich mit den zentralen Gebieten einer maßtheoretisch orientierten Wahrscheinlichkeitstheorie im Umfang einer zweisemestrigen Vorlesung. Nach den Grundlagen werden Grenzwertsätze und schwache Konvergenz behandelt. Es folgt die Darstellung und Betrachtung der stochastischen Abhängigkeit durch die bedingte Erwartung, die mit der Radon-Nikodym-Ableitung realisiert wird. Sie wird angewandt auf die Theorie der stochastischen Prozesse, die nach der allgemeinen Konstruktion aus der Untersuchung von Martingalen und Markov-Prozessen besteht. Neu in einem Lehrbuch über allgemeine Wahrscheinlichkeitstheorie ist eine Einführung in die stochastische Analysis von Semimartingalen auf der Grundlage einer geeigneten Stetigkeitsbedingung mit Anwendungen auf die Theorie der Finanzmärkte. Das Buch enthält zahlreiche Übungen, teilweise mit Lösungen. Neben der Theorie vertiefen Anmerkungen, besonders zu mathematischen Modellen für Phänomene der Realität, das Verständnis.​

Stochastic Differential Equations

Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cortona (Arezzo), Italy, May 29-June 10, 1978

Author: Jaures Cecconi

Publisher: Springer Science & Business Media

ISBN: 9783642110795

Category: Mathematics

Page: 249

View: 7827


C. Doleans-Dade: Stochastic processes and stochastic differential equations.- A. Friedman: Stochastic differential equations and applications.- D.W. Stroock, S.R.S. Varadhan: Theory of diffusion processes.- G.C. Papanicolaou: Wave propagation and heat conduction in a random medium.- C. Dewitt Morette: A stochastic problem in Physics.- G.S. Goodman: The embedding problem for stochastic matrices.

Diffusion Processes and Related Problems in Analysis, Volume II

Stochastic Flows

Author: V. Wihstutz,M.A. Pinsky

Publisher: Springer Science & Business Media

ISBN: 1461203899

Category: Mathematics

Page: 346

View: 2873


During the weekend of March 16-18, 1990 the University of North Carolina at Charlotte hosted a conference on the subject of stochastic flows, as part of a Special Activity Month in the Department of Mathematics. This conference was supported jointly by a National Science Foundation grant and by the University of North Carolina at Charlotte. Originally conceived as a regional conference for researchers in the Southeastern United States, the conference eventually drew participation from both coasts of the U. S. and from abroad. This broad-based par ticipation reflects a growing interest in the viewpoint of stochastic flows, particularly in probability theory and more generally in mathematics as a whole. While the theory of deterministic flows can be considered classical, the stochastic counterpart has only been developed in the past decade, through the efforts of Harris, Kunita, Elworthy, Baxendale and others. Much of this work was done in close connection with the theory of diffusion processes, where dynamical systems implicitly enter probability theory by means of stochastic differential equations. In this regard, the Charlotte conference served as a natural outgrowth of the Conference on Diffusion Processes, held at Northwestern University, Evanston Illinois in October 1989, the proceedings of which has now been published as Volume I of the current series. Due to this natural flow of ideas, and with the assistance and support of the Editorial Board, it was decided to organize the present two-volume effort.

Simulation and Inference for Stochastic Differential Equations

With R Examples

Author: Stefano M. Iacus

Publisher: Springer Science & Business Media

ISBN: 9780387758398

Category: Computers

Page: 286

View: 1544


This book covers a highly relevant and timely topic that is of wide interest, especially in finance, engineering and computational biology. The introductory material on simulation and stochastic differential equation is very accessible and will prove popular with many readers. While there are several recent texts available that cover stochastic differential equations, the concentration here on inference makes this book stand out. No other direct competitors are known to date. With an emphasis on the practical implementation of the simulation and estimation methods presented, the text will be useful to practitioners and students with minimal mathematical background. What’s more, because of the many R programs, the information here is appropriate for many mathematically well educated practitioners, too.

Stochastic Differential Equations

Theory and Applications

Author: Ludwig Arnold

Publisher: Severn House Paperbacks

ISBN: 9780486482361

Category: Stochastic differential equations

Page: 256

View: 7581


Practical and not too rigorous, this highly readable text on stochastic calculus provides an excellent introduction to stochastic partial differential equations. Written at a moderately advanced level, it covers important topics often ignored by other texts on the subject—including Fokker-Planck equations—and it functions as both a classroom text and a reference for professionals and students. The only prerequisite is the mathematical preparation usual for students of physical and engineering sciences. An introductory chapter, intended for reference and review, covers the basics of probability theory. Subsequent chapters focus on Markov and diffusion processes, Wiener process and white noise, and stochastic integrals and differential equations. Additional topics include questions of modeling and approximation, stability of stochastic dynamic systems, optimal filtering of a disturbed signal, and optimal control of stochastic dynamic systems.

Stochastic versus Deterministic Systems of Differential Equations

Author: G. S. Ladde,M. Sambandham

Publisher: CRC Press

ISBN: 0824758757

Category: Mathematics

Page: 300

View: 925


This peerless reference/text unfurls a unified and systematic study of the two types of mathematical models of dynamic processes-stochastic and deterministic-as placed in the context of systems of stochastic differential equations. Using the tools of variational comparison, generalized variation of constants, and probability distribution as its methodological backbone, Stochastic Versus Deterministic Systems of Differential Equations addresses questions relating to the need for a stochastic mathematical model and the between-model contrast that arises in the absence of random disturbances/fluctuations and parameter uncertainties both deterministic and stochastic.

Multidimensional Diffusion Processes

Author: Daniel W. Stroock,S.R.S. Varadhan

Publisher: Springer

ISBN: 3540289992

Category: Mathematics

Page: 338

View: 3588


From the reviews: "This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. [...] 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

Generalized Diffusion Processes

Author: Nikola_ Ivanovich Portenko

Publisher: American Mathematical Soc.

ISBN: 9780821898260

Category: Mathematics

Page: 180

View: 2020


Diffusion processes serve as a mathematical model for the physical phenomenon of diffusion. One of the most important problems in the theory of diffusion processes is the development of methods for constructing these processes from a given diffusion matrix and a given drift vector. Focusing on the investigation of this problem, this book is intended for specialists in the theory of random processes and its applications. A generalized diffusion process (that is, a continuous Markov process for which the Kolmogorov local characteristics exist in the generalized sense) can serve as a model for diffusion in a medium moving in a nonregular way. The author constructs generalized diffusion processes under two assumptions: first, that the diffusion matrix is sufficiently regular; and second, that the drift vector is a function integrable to some power, or is a generalized function of the type of the derivative of a measure.

Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Author: Eckhard Platen,Nicola Bruti-Liberati

Publisher: Springer Science & Business Media

ISBN: 364213694X

Category: Mathematics

Page: 856

View: 4543


In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitative methods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics.

The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions

Author: Christian Soize

Publisher: World Scientific

ISBN: 9789810217556

Category: Science

Page: 321

View: 2770


This is an analysis of multidimensional nonlinear dissipative Hamiltonian dynamical systems subjected to parametric and external stochastic excitations by the Fokker-Planck equation method.The author answers three types of questions concerning this area. First, what probabilistic tools are necessary for constructing a stochastic model and deriving the FKP equation for nonlinear stochastic dynamical systems? Secondly, what are the main results concerning the existence and uniqueness of an invariant measure and its associated stationary response? Finally, what is the class of multidimensional dynamical systems that have an explicit invariant measure and what are the fundamental examples for applications?

Harnack Inequalities for Stochastic Partial Differential Equations

Author: Feng-Yu Wang

Publisher: Springer Science & Business Media

ISBN: 1461479347

Category: Mathematics

Page: 125

View: 3423


​In this book the author presents a self-contained account of Harnack inequalities and applications for the semigroup of solutions to stochastic partial and delayed differential equations. Since the semigroup refers to Fokker-Planck equations on infinite-dimensional spaces, the Harnack inequalities the author investigates are dimension-free. This is an essentially different point from the above mentioned classical Harnack inequalities. Moreover, the main tool in the study is a new coupling method (called coupling by change of measures) rather than the usual maximum principle in the current literature.

Stochastic Differential Equations

Convection-Diffusion Equation, Doléans-Dade Exponential, Dynkin's Formula, Euler-Maruyama Method, Filtering Problem

Author: Source Wikipedia


ISBN: 9781230498966


Page: 90

View: 5931


Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 28. Chapters: Convection-diffusion equation, Doleans-Dade exponential, Dynkin's formula, Euler-Maruyama method, Filtering problem (stochastic processes), Freidlin-Wentzell theorem, Green measure, Hormander's condition, Infinitesimal generator (stochastic processes), It diffusion, Kalman filter, Kolmogorov backward equations (diffusion), Milstein method, Ornstein-Uhlenbeck process, Random dynamical system, Reversible diffusion, Runge-Kutta method (SDE), Stochastic differential equation, Stochastic partial differential equation, Stochastic processes and boundary value problems, Tanaka equation, Telegraph process, Wiener equation, Zakai equation.