# stochastic differential equations and diffusion processes

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## Stochastic Differential Equations And Diffusion Processes

**Author :**N. Ikeda

**ISBN :**9781483296159

**Genre :**Mathematics

**File Size :**52. 17 MB

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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.

## Stochastic Analysis And Diffusion Processes

**Author :**Gopinath Kallianpur

**ISBN :**9780199657070

**Genre :**Mathematics

**File Size :**69. 33 MB

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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.

## Stochastic Processes And Applications

**Author :**Grigorios A. Pavliotis

**ISBN :**9781493913237

**Genre :**Mathematics

**File Size :**71. 44 MB

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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.

## Statistical Methods For Stochastic Differential Equations

**Author :**Mathieu Kessler

**ISBN :**9781439849767

**Genre :**Mathematics

**File Size :**72. 30 MB

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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.

## Introduction To Stochastic Analysis

**Author :**Vigirdas Mackevicius

**ISBN :**9781118603246

**Genre :**Mathematics

**File Size :**89. 70 MB

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This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes. The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô’s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.

## Controlled Diffusion Processes

**Author :**N. V. Krylov

**ISBN :**9783540709145

**Genre :**Science

**File Size :**74. 74 MB

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Stochastic control theory is a relatively young branch of mathematics. The beginning of its intensive development falls in the late 1950s and early 1960s. ~urin~ that period an extensive literature appeared on optimal stochastic control using the quadratic performance criterion (see references in Wonham [76]). At the same time, Girsanov [25] and Howard [26] made the first steps in constructing a general theory, based on Bellman's technique of dynamic programming, developed by him somewhat earlier [4]. Two types of engineering problems engendered two different parts of stochastic control theory. Problems of the first type are associated with multistep decision making in discrete time, and are treated in the theory of discrete stochastic dynamic programming. For more on this theory, we note in addition to the work of Howard and Bellman, mentioned above, the books by Derman [8], Mine and Osaki [55], and Dynkin and Yushkevich [12]. Another class of engineering problems which encouraged the development of the theory of stochastic control involves time continuous control of a dynamic system in the presence of random noise. The case where the system is described by a differential equation and the noise is modeled as a time continuous random process is the core of the optimal control theory of diffusion processes. This book deals with this latter theory.

## Stochastic Differential Equations And Applications

**Author :**Avner Friedman

**ISBN :**9781483217888

**Genre :**Mathematics

**File Size :**67. 52 MB

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Stochastic Differential Equations and Applications, Volume 2 is an eight-chapter text that focuses on the practical aspects of stochastic differential equations. This volume begins with a presentation of the auxiliary results in partial differential equations that are needed in the sequel. The succeeding chapters describe the behavior of the sample paths of solutions of stochastic differential equations. These topics are followed by a consideration of an issue whether the paths can hit a given set with positive probability, as well as the stability of paths about a given manifold and with spiraling of paths about this manifold. Other chapters deal with the applications to partial equations, specifically with the Dirichlet problem for degenerate elliptic equations. These chapters also explore the questions of singular perturbations and the existence of fundamental solutions for degenerate parabolic equations. The final chapters discuss stopping time problems, stochastic games, and stochastic differential games. This book is intended primarily to undergraduate and graduate mathematics students.

## Stochastic Differential Equations Backward Sdes Partial Differential Equations

**Author :**Etienne Pardoux

**ISBN :**9783319057149

**Genre :**Mathematics

**File Size :**48. 79 MB

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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.

## Simulation And Inference For Stochastic Differential Equations

**Author :**Stefano M. Iacus

**ISBN :**0387758399

**Genre :**Computers

**File Size :**84. 92 MB

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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.

## Introduction To The Theory Of Diffusion Processes

**Author :**Nikolaĭ Vladimirovich Krylov

**ISBN :**0821846000

**Genre :**Mathematics

**File Size :**36. 55 MB

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Focusing on one of the major branches of probability theory, this book treats the large class of processes with continuous sample paths that possess the ``Markov property''. The exposition is based on the theory of stochastic analysis. The diffusion processes discussed are interpreted as solutions of Ito's stochastic integral equations. The book is designed as a self-contained introduction, requiring no background in the theory of probability or even in measure theory. In particular, the theory of local continuous martingales is covered without the introduction of the idea of conditional expectation. Krylov covers such subjects as the Wiener process and its properties, the theory of stochastic integrals, stochastic differential equations and their relation to elliptic and parabolic partial differential equations, Kolmogorov's equations, and methods for proving the smoothness of probabilistic solutions of partial differential equations. With many exercises and thought-provoking problems, this book would be an excellent text for a graduate course in diffusion processes and related subjects.