Theory of Stochastic Differential Equations with Jumps and Applications

Author: Rong SITU
Publisher: Springer Science & Business Media
ISBN: 0387251758
Format: PDF, ePub, Mobi
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Stochastic differential equations (SDEs) are a powerful tool in science, mathematics, economics and finance. This book will help the reader to master the basic theory and learn some applications of SDEs. In particular, the reader will be provided with the backward SDE technique for use in research when considering financial problems in the market, and with the reflecting SDE technique to enable study of optimal stochastic population control problems. These two techniques are powerful and efficient, and can also be applied to research in many other problems in nature, science and elsewhere.

Fluctuations of L vy Processes with Applications

Author: Andreas Kyprianou
Publisher: Springer Science & Business Media
ISBN: 3642376320
Format: PDF, Mobi
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Lévy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their application appears in the theory of many areas of classical and modern stochastic processes including storage models, renewal processes, insurance risk models, optimal stopping problems, mathematical finance, continuous-state branching processes and positive self-similar Markov processes. This textbook is based on a series of graduate courses concerning the theory and application of Lévy processes from the perspective of their path fluctuations. Central to the presentation is the decomposition of paths in terms of excursions from the running maximum as well as an understanding of short- and long-term behaviour. The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Lévy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical tractability. The second edition additionally addresses recent developments in the potential analysis of subordinators, Wiener-Hopf theory, the theory of scale functions and their application to ruin theory, as well as including an extensive overview of the classical and modern theory of positive self-similar Markov processes. Each chapter has a comprehensive set of exercises.

Numerical Solution of Stochastic Differential Equations

Author: Peter E. Kloeden
Publisher: Springer Science & Business Media
ISBN: 3662126168
Format: PDF, Kindle
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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

Beyond The Triangle Brownian Motion Ito Calculus And Fokker planck Equation Fractional Generalizations

Author: Umarov Sabir
Publisher: World Scientific
ISBN: 9813230991
Format: PDF, Kindle
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The book is devoted to the fundamental relationship between three objects: a stochastic process, stochastic differential equations driven by that process and their associated Fokker–Planck–Kolmogorov equations. This book discusses wide fractional generalizations of this fundamental triple relationship, where the driving process represents a time-changed stochastic process; the Fokker–Planck–Kolmogorov equation involves time-fractional order derivatives and spatial pseudo-differential operators; and the associated stochastic differential equation describes the stochastic behavior of the solution process. It contains recent results obtained in this direction. This book is important since the latest developments in the field, including the role of driving processes and their scaling limits, the forms of corresponding stochastic differential equations, and associated FPK equations, are systematically presented. Examples and important applications to various scientific, engineering, and economics problems make the book attractive for all interested researchers, educators, and graduate students. Contents: Introduction The Original Triangle: Brownian Motion, Itô Stochastic Calculus, and Fokker–Planck–Kolmogorov Equation Fractional Calculus Pseudo–Differential Operators Associated with Lévy Processes Stochastic Processes and Time-Changes Stochastic Calculus for Time-Changed Semimartingales and Its Applications to SDEs Fractional Fokker–Planck–Kolmogorov Equations Readership: Graduate students and researchers in science, engineering, economics. Keywords: Fractional Fokker-Planck Equations;Stochastic Differential Equations Driven by Time-changed Processes;Levy Processes;Fractional Brownian Motion;Inverse Stable Subordinators;Continuous Time Random Walk Approximations of Time-changed Processes;Pseudo-Differential Operators with Singular Symbols;Fractional Differential EquationsReview: Key Features: The novel theory of fractional Fokker–Planck–Kolmogorov equations and their connection with the associated stochastic differential equations driven by time-changed stochastic processes are discussed in detail The book is rich in new ideas and applications to various real world problems arising in natural science, engineering, and economics. Researchers may benefit from adapting the ideas to their own research and developing relevant theory The book contains discussions of some important open problems whose solutions make significant contributions

The Fast Solution of Boundary Integral Equations

Author: Sergej Rjasanow
Publisher: Springer Science & Business Media
ISBN: 0387340424
Format: PDF
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This book provides a detailed description of fast boundary element methods, all based on rigorous mathematical analysis. In particular, the authors use a symmetric formulation of boundary integral equations as well as discussing Galerkin discretisation. All the necessary related stability and error estimates are derived. The authors therefore describe the Adaptive Cross Approximation Algorithm, starting from the basic ideas and proceeding to their practical realization. Numerous examples representing standard problems are given.

Reflecting Stochastic Differential Equations with Jumps and Applications

Author: Situ Rong
Publisher: CRC Press
ISBN: 9781584881254
Format: PDF, ePub, Docs
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Many important physical variables satisfy certain dynamic evolution systems and can take only non-negative values. Therefore, one can study such variables by studying these dynamic systems. One can put some conditions on the coefficients to ensure non-negative values in deterministic cases. However, as a random process disturbs the system, the components of solutions to stochastic differential equations (SDE) can keep changing between arbitrary large positive and negative values-even in the simplest case. To overcome this difficulty, the author examines the reflecting stochastic differential equation (RSDE) with the coordinate planes as its boundary-or with a more general boundary. Reflecting Stochastic Differential Equations with Jumps and Applications systematically studies the general theory and applications of these equations. In particular, the author examines the existence, uniqueness, comparison, convergence, and stability of strong solutions to cases where the RSDE has discontinuous coefficients-with greater than linear growth-that may include jump reflection. He derives the nonlinear filtering and Zakai equations, the Maximum Principle for stochastic optimal control, and the necessary and sufficient conditions for the existence of optimal control. Most of the material presented in this book is new, including much new work by the author concerning SDEs both with and without reflection. Much of it appears here for the first time. With the application of RSDEs to various real-life problems, such as the stochastic population and neurophysiological control problems-both addressed in the text-scientists dealing with stochastic dynamic systems will find this an interesting and useful work.

Stochastic Partial Differential Equations

Author: Helge Holden
Publisher: Springer Science & Business Media
ISBN: 0387894888
Format: PDF, Mobi
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The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs. In this, the second edition, the authors build on the theory of SPDEs driven by space-time Brownian motion, or more generally, space-time Lévy process noise. Applications of the theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance. Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter.

Stochastic Calculus

Author: Mircea Grigoriu
Publisher: Springer Science & Business Media
ISBN: 9780817642426
Format: PDF, ePub
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"This self-contained text may be used for several graduate courses and as an important reference resource for applied scientists interested in analytical and numerical methods for solving stochastic problems."--BOOK JACKET.

Stochastic Analysis and Applications to Finance

Author: Tusheng Zhang
Publisher: World Scientific
ISBN: 9814383570
Format: PDF
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A collection of solicited and refereed articles from distinguished researchers across the field of stochastic analysis and its application to finance. It covers the topics ranging from Markov processes, backward stochastic differential equations, stochastic partial differential equations, and stochastic control, to risk measure and risk theory.

Statistical Methods for Stochastic Differential Equations

Author: Mathieu Kessler
Publisher: CRC Press
ISBN: 1439849765
Format: PDF, Kindle
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The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research. The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a spectrum of estimation methods, including nonparametric estimation as well as parametric estimation based on likelihood methods, estimating functions, and simulation techniques. Two chapters are devoted to high-frequency data. Multivariate models are also considered, including partially observed systems, asynchronous sampling, tests for simultaneous jumps, and multiscale diffusions. Statistical Methods for Stochastic Differential Equations is useful to the theoretical statistician and the probabilist who works in or intends to work in the field, as well as to the applied statistician or financial econometrician who needs the methods to analyze biological or financial time series.