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Author : H. Kunita
Publisher : Unknown
Page : 144 pages
File Size : 50,7 Mb
Release : 1986
Category : Flows (Differentiable dynamical systems).
ISBN : UCSD:31822003450087
Author : H. Kunita
Publisher : Springer
Page : 121 pages
File Size : 52,9 Mb
Release : 1987-03-09
Category : Science
ISBN : 3540177752
These are the notes of a lecture course given by the author at the T.I.F.R. Centre, Bangalore in late 1985. The contents are divided into three chapters concluding with an extensive bibliography. Chapters 1 and 2 deal with basic properties of stochastic flows and especially of Brownian flows and their relations with local characteristics and stochastic differential equations. An appendix on the generalized Ito#^ formula, Stratonovich integral and Stratonovich stochastic differential equations has been added to Chapter 2. By the way of applications of the foregoing, limit theorems for stochastic flows, along with a unifying general limit theorem, are then presented in Chapter 3 including: - Approximation theorems for stochastic differential equations and stochastic flows, due to Bismut, Ikeda-Watanabe, Malliavin, Dowell etc. - Limit theorems for driving processes, due to Papanicolaou-Stroock-Varadhan, and - Limit theorems for stochastic differential equations, due to Khasminkii, Papanicolaou-Kohler, Kesten-Papanicolaou etc.
Author : Fabrice Baudoin
Publisher : World Scientific
Page : 152 pages
File Size : 40,6 Mb
Release : 2004
Category : Mathematics
ISBN : 9781860944819
This book aims to provide a self-contained introduction to the local geometry of the stochastic flows associated with stochastic differential equations. It stresses the view that the local geometry of any stochastic flow is determined very precisely and explicitly by a universal formula referred to as the Chen-Strichartz formula. The natural geometry associated with the Chen-Strichartz formula is the sub-Riemannian geometry whose main tools are introduced throughout the text. By using the connection between stochastic flows and partial differential equations, we apply this point of view of the study of hypoelliptic operators written in Hormander's form.
Author : Philip Protter
Publisher : Springer
Page : 430 pages
File Size : 46,8 Mb
Release : 2013-12-21
Category : Mathematics
ISBN : 9783662100615
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.
Author : Ludwig Arnold,Hans Crauel,Jean-Pierre Eckmann
Publisher : Springer
Page : 372 pages
File Size : 55,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540464310
Since the predecessor to this volume (LNM 1186, Eds. L. Arnold, V. Wihstutz)appeared in 1986, significant progress has been made in the theory and applications of Lyapunov exponents - one of the key concepts of dynamical systems - and in particular, pronounced shifts towards nonlinear and infinite-dimensional systems and engineering applications are observable. This volume opens with an introductory survey article (Arnold/Crauel) followed by 26 original (fully refereed) research papers, some of which have in part survey character. From the Contents: L. Arnold, H. Crauel: Random Dynamical Systems.- I.Ya. Goldscheid: Lyapunov exponents and asymptotic behaviour of the product of random matrices.- Y. Peres: Analytic dependence of Lyapunov exponents on transition probabilities.- O. Knill: The upper Lyapunov exponent of Sl (2, R) cocycles:Discontinuity and the problem of positivity.- Yu.D. Latushkin, A.M. Stepin: Linear skew-product flows and semigroups of weighted composition operators.- P. Baxendale: Invariant measures for nonlinear stochastic differential equations.- Y. Kifer: Large deviationsfor random expanding maps.- P. Thieullen: Generalisation du theoreme de Pesin pour l' -entropie.- S.T. Ariaratnam, W.-C. Xie: Lyapunov exponents in stochastic structural mechanics.- F. Colonius, W. Kliemann: Lyapunov exponents of control flows.
Author : H. Kesten,R. Durrett
Publisher : Springer Science & Business Media
Page : 457 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461204596
This collection of articles is dedicated to Frank Spitzer on the occasion of his 65th birthday. The articles, written by a group of his friends, colleagues, former students and coauthors, are intended to demonstrate the major influence Frank has had on probability theory for the last 30 years and most likely will have for many years to come. Frank has always liked new phenomena, clean formulations and elegant proofs. He has created or opened up several research areas and it is not surprising that many people are still working out the consequences of his inventions. By way of introduction we have reprinted some of Frank's seminal articles so that the reader can easily see for himself the point of origin for much of the research presented here. These articles of Frank's deal with properties of Brownian motion, fluctuation theory and potential theory for random walks, and, of course, interacting particle systems. The last area was started by Frank as part of the general resurgence of treating problems of statistical mechanics with rigorous probabilistic tools.
Author : Robert C. Dalang,Marco Dozzi,Franco Flandoli,Francesco Russo
Publisher : Birkhäuser
Page : 393 pages
File Size : 52,6 Mb
Release : 2015-07-28
Category : Mathematics
ISBN : 9783034809092
This book presents in thirteen refereed survey articles an overview of modern activity in stochastic analysis, written by leading international experts. The topics addressed include stochastic fluid dynamics and regularization by noise of deterministic dynamical systems; stochastic partial differential equations driven by Gaussian or Lévy noise, including the relationship between parabolic equations and particle systems, and wave equations in a geometric framework; Malliavin calculus and applications to stochastic numerics; stochastic integration in Banach spaces; porous media-type equations; stochastic deformations of classical mechanics and Feynman integrals and stochastic differential equations with reflection. The articles are based on short courses given at the Centre Interfacultaire Bernoulli of the Ecole Polytechnique Fédérale de Lausanne, Switzerland, from January to June 2012. They offer a valuable resource not only for specialists, but also for other researchers and Ph.D. students in the fields of stochastic analysis and mathematical physics. Contributors: S. Albeverio M. Arnaudon V. Bally V. Barbu H. Bessaih Z. Brzeźniak K. Burdzy A.B. Cruzeiro F. Flandoli A. Kohatsu-Higa S. Mazzucchi C. Mueller J. van Neerven M. Ondreját S. Peszat M. Veraar L. Weis J.-C. Zambrini
Author : Francesco Russo,Pierre Vallois
Publisher : Springer Nature
Page : 656 pages
File Size : 50,7 Mb
Release : 2022-11-15
Category : Mathematics
ISBN : 9783031094460
The book constitutes an introduction to stochastic calculus, stochastic differential equations and related topics such as Malliavin calculus. On the other hand it focuses on the techniques of stochastic integration and calculus via regularization initiated by the authors. The definitions relies on a smoothing procedure of the integrator process, they generalize the usual Itô and Stratonovich integrals for Brownian motion but the integrator could also not be a semimartingale and the integrand is allowed to be anticipating. The resulting calculus requires a simple formalism: nevertheless it entails pathwise techniques even though it takes into account randomness. It allows connecting different types of pathwise and non pathwise integrals such as Young, fractional, Skorohod integrals, enlargement of filtration and rough paths. The covariation, but also high order variations, play a fundamental role in the calculus via regularization, which can also be applied for irregular integrators. A large class of Gaussian processes, various generalizations of semimartingales such that Dirichlet and weak Dirichlet processes are revisited. Stochastic calculus via regularization has been successfully used in applications, for instance in robust finance and on modeling vortex filaments in turbulence. The book is addressed to PhD students and researchers in stochastic analysis and applications to various fields.
Author : K.D. Elworthy,Y. Le Jan,Xue-Mei Li
Publisher : Springer
Page : 121 pages
File Size : 52,6 Mb
Release : 2007-01-05
Category : Mathematics
ISBN : 9783540470229
Stochastic differential equations, and Hoermander form representations of diffusion operators, can determine a linear connection associated to the underlying (sub)-Riemannian structure. This is systematically described, together with its invariants, and then exploited to discuss qualitative properties of stochastic flows, and analysis on path spaces of compact manifolds with diffusion measures. This should be useful to stochastic analysts, especially those with interests in stochastic flows, infinite dimensional analysis, or geometric analysis, and also to researchers in sub-Riemannian geometry. A basic background in differential geometry is assumed, but the construction of the connections is very direct and itself gives an intuitive and concrete introduction. Knowledge of stochastic analysis is also assumed for later chapters.
Author : Ian M Davies,Aubrey Truman
Publisher : World Scientific
Page : 326 pages
File Size : 40,9 Mb
Release : 1992-05-30
Category : Electronic
ISBN : 9789814554732
This volume contains papers which were presented at a series of short meetings collectively entitled “Stochastics and Quantum Mechanics” held in Swansea over the summer of 1990. Also included are some papers not presented at the meetings, but in the same subject area, authored by attendees or their co-workers. The topics covered include diffusion processes, stochastic mechanics, statistical mechanics, large deviations and Wiener-Hopf theory.The papers are in the main immediately accessible to workers in the field and provide a reasonable coverage of current areas of interest centering around uses of probabilistic methods in mathematical physics.
Author : Hiroshi Kunita,H. Kunita
Publisher : Cambridge University Press
Page : 364 pages
File Size : 52,5 Mb
Release : 1990
Category : Mathematics
ISBN : 0521599253
The main purpose of this book is to give a systematic treatment of the theory of stochastic differential equations and stochastic flow of diffeomorphisms, and through the former to study the properties of stochastic flows.The classical theory was initiated by K. Itô and since then has been much developed. Professor Kunita's approach here is to regard the stochastic differential equation as a dynamical system driven by a random vector field, including thereby Itô's theory as a special case. The book can be used with advanced courses on probability theory or for self-study.
Author : M. Emery,A. Nemirovski,D. Voiculescu
Publisher : Springer
Page : 359 pages
File Size : 45,8 Mb
Release : 2007-05-06
Category : Mathematics
ISBN : 9783540450290
This volume contains lectures given at the Saint-Flour Summer School of Probability Theory during 17th Aug. - 3rd Sept. 1998. The contents of the three courses are the following: - Continuous martingales on differential manifolds. - Topics in non-parametric statistics. - Free probability theory. The reader is expected to have a graduate level in probability theory and statistics. This book is of interest to PhD students in probability and statistics or operators theory as well as for researchers in all these fields. The series of lecture notes from the Saint-Flour Probability Summer School can be considered as an encyclopedia of probability theory and related fields.
Author : Kiyosi Ito,Takeyuki Hida
Publisher : Springer
Page : 228 pages
File Size : 45,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540398523
Author : Jialin Hong,Liying Sun
Publisher : Springer Nature
Page : 307 pages
File Size : 51,8 Mb
Release : 2023-02-21
Category : Mathematics
ISBN : 9789811976704
This book provides an accessible overview concerning the stochastic numerical methods inheriting long-time dynamical behaviours of finite and infinite-dimensional stochastic Hamiltonian systems. The long-time dynamical behaviours under study involve symplectic structure, invariants, ergodicity and invariant measure. The emphasis is placed on the systematic construction and the probabilistic superiority of stochastic symplectic methods, which preserve the geometric structure of the stochastic flow of stochastic Hamiltonian systems. The problems considered in this book are related to several fascinating research hotspots: numerical analysis, stochastic analysis, ergodic theory, stochastic ordinary and partial differential equations, and rough path theory. This book will appeal to researchers who are interested in these topics.
Author : Hiroshi Kunita
Publisher : Springer
Page : 352 pages
File Size : 55,6 Mb
Release : 2019-03-26
Category : Mathematics
ISBN : 9789811338014
This monograph presents a modern treatment of (1) stochastic differential equations and (2) diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and jump processes in this book is unique: Each chapter starts from continuous processes and then proceeds to processes with jumps.In the first part of the book, it is shown that solutions of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the relation between stochastic flows and heat equations is discussed. The latter part investigates fundamental solutions of these heat equations (heat kernels) through the study of the Malliavin calculus. The author obtains smooth densities for transition functions of various types of diffusions and jump-diffusions and shows that these density functions are fundamental solutions for various types of heat equations and backward heat equations. Thus, in this book fundamental solutions for heat equations and backward heat equations are constructed independently of the theory of partial differential equations.Researchers and graduate student in probability theory will find this book very useful.