DOCSLIB.ORG

On Stochastic Integration for White Noise Distribution Theory. Said Kalema Ngobi Louisiana State University and Agricultural & Mechanical College

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Stochastic Integration for White Noise Distribution Theory. Said Kalema Ngobi Louisiana State University and Agricultural & Mechanical College Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 2000 On Stochastic Integration for White Noise Distribution Theory. Said Kalema Ngobi Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Ngobi, Said Kalema, "On Stochastic Integration for White Noise Distribution Theory." (2000). LSU Historical Dissertations and Theses. 7381. https://digitalcommons.lsu.edu/gradschool_disstheses/7381 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ON STOCHASTIC INTEGRATION FOR WHITE NOISE DISTRIBUTION THEORY A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Said K. Ngobi B. Statistics, Makerere University, Kampala, Uganda, 1986 M.S., Louisiana State University, 1990 December 2000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9998700 _ ___ __ (§) UMI UMI Microform 9998700 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments This dissertation would not see the light of the day without several significant contributions. The most important of them, without any doubt, was given by my advisor, Professor Hui-Hsiung Kuo, who continuously dedicated his time and attention to ensure that I finish up my work. I am grateful for his advice, guidance, patience and mentorship. I thank Professor Padmanabhan Sundar for introducing me to the field of stochastic integration and Dr. George Cochran for proof-reading my dissertation. I give special thanks to my committee members for being patient. I thank all the people in the Department of Mathematics for providing me with a pleasant working environment. A special thanks to Professor Leonard Richardson who granted me an assistantship to come from Uganda and study at LSU. To my fellow graduate students, I am very grateful for all your friendship and support. I wish to extend a very special thank you to my family in Uganda, for your encouragement and support which were strong enough to overcome the tremendous distance of separation. To Nasser, Zaharah and Faisal, I will always remember you and I dedicate this work to my late Father. Finally, a heartfelt thank you to Monica, Ishmael and Shereef, who gave me a lot of emotional support. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Acknowledgments ................................................................................................... ii A bstract ........................................................................................................................ iv Chapter 1. Introduction ..................................................................................... 1 Chapter 2. Background ....................................................................................... 3 2.1 Concept and Notations .............................................................................. 3 2.2 Hermite Polynomials, Wick Tensors, and Multiple Wiener Integrals 7 2.3 The White Noise Differential O perator ................................................ 9 2.4 Characterization Theorems and Convergence of Generalized Functions 10 2.5 Pettis and Bochner Integrals .................................................................... 13 2.6 White Noise Integrals ................................................................................. 15 2.7 Donsker’s Delta Function ........................................................................... 19 Chapter 3. A Generalization of the Ito Formula ...................................... 21 3.1 The Ito In te g ra l ........................................................................................... 21 3.2 The Ito Form ula ........................................................................................... 23 3.3 An Extension of the Ito Integral ............................................................... 25 3.4 An Anticipating Ito Form ula...................................................................... 30 Chapter 4. A Generalization of the Clark-Ocone Formula ................ 45 4.1 Representation of Brownian Functionals by Stochastic Integrals . 45 4.2 A Generalized Clark-Ocone Formula ................................................... 54 4.3 The Formula for the Hermite Brownian Functional ............................ 60 4.4 The Formula for Donsker’s Delta Function ............................................ 63 4.5 Generalization to Compositions with Tempered Distributions .... 66 References .................................................................................................................. 68 V ita ............................................................................................................................... 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract This thesis consists of two parts, each part concentrating on a different problem from the theory of Stochastic Integration. Chapter 1 contains the introduction explaining the results in this dissertation in general terms. We use the infinite di­ mensional space <S'(K), which is the dual of the Schwartz space S(M), endowed with the gaussian measure [i. The Hilbert space (L2) is defined as (L2) = L2(S'(R), (i), and our results are based on the Gel’fand triple (S)p c (L2) C (S)£- The necessary preliminary background in white noise analysis are well elaborated in Chapter 2. In the first part of the dissertation we present a generalization of the Ito Formula to anticipating processes in the white noise framework. This is contained in Chapter 3. We first introduce an extension of the Ito integral to anticipating processes called the Hitsuda Skorokhod integral. We then use the anticipating Ito formula for processes of the type 9(X(t), B(c)), where X (t) is a Wiener integral and B(t) = (•> l[o,t)) is Brownian motion for a < c < b, which was obtained in [21] as a motivation to obtain our first main result. In this result, we generalize the formula in [21] to processes of the form 0(X (t),F ) where X(t) is a Wiener integral and F E W 1/2. The space W 1/2 is a special subspace of (L2). Chapter 4 contains the second part of our work. We first state the Clark-Ocone (C-O) formula in the white noise framework. Before the main result we verify the formula for Brownian functionals using the very important white noise tool, the S- transform. Previous proofs of this formula have not used this method. As our second main result, we extend the formula to generalized Brownian functionals of the form $ = £ ~ =0(: '®n: ifn)i where f n E L2(R)®n. The formula for such $ takes on the same form as that for Brownian functionals. As examples we verify the formula iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for the Hermite Brownian functional : B(t)n : t using the Ito formula. We also compute the C-O formula for the Donsker’s delta function using the result for the Hermite Brownian functional. Finally, the formula is generalized to compositions of tempered distributions with
Load more

Recommended publications
  • A Generalized White Noise Space Approach to Stochastic Integration for a Class of Gaussian Stationary Increment Processes
    Chapman University Chapman University Digital Commons Mathematics, Physics, and Computer Science Science and Technology Faculty Articles and Faculty Articles and Research Research 2013 A Generalized White Noise Space Approach To Stochastic Integration for a Class of Gaussian Stationary Increment Processes Daniel Alpay Chapman University, [email protected] Alon Kipnis Ben Gurion University of the Negev Follow this and additional works at: http://digitalcommons.chapman.edu/scs_articles Part of the Algebra Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation D. Alpay and A. Kipnis. A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes. Opuscula Mathematica, vol. 33 (2013), pp. 395-417. This Article is brought to you for free and open access by the Science and Technology Faculty Articles and Research at Chapman University Digital Commons. It has been accepted for inclusion in Mathematics, Physics, and Computer Science Faculty Articles and Research by an authorized administrator of Chapman University Digital Commons. For more information, please contact [email protected]. A Generalized White Noise Space Approach To Stochastic Integration for a Class of Gaussian Stationary Increment Processes Comments This article was originally published in Opuscula Mathematica, volume 33, issue 3, in 2013 DOI: 10.7494/ OpMath.2013.33.3.395 Copyright Wydawnictwa AGH This article is available at Chapman University Digital Commons: http://digitalcommons.chapman.edu/scs_articles/396 Opuscula Math. 33, no. 3 (2013), 395–417 http://dx.doi.org/10.7494/OpMath.2013.33.3.395 Opuscula Mathematica A GENERALIZED WHITE NOISE SPACE APPROACH TO STOCHASTIC INTEGRATION FOR A CLASS OF GAUSSIAN STATIONARY INCREMENT PROCESSES Daniel Alpay and Alon Kipnis Communicated by Palle E.T.
    [Show full text]
  • A Short Introduction to the Quantum Formalism[S]
    A short introduction to the quantum formalism[s] François David Institut de Physique Théorique CNRS, URA 2306, F-91191 Gif-sur-Yvette, France CEA, IPhT, F-91191 Gif-sur-Yvette, France [email protected] These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May- June 2012; (ii) a previous letter [Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics. arXiv:1211.5627v1 [math-ph] 24 Nov 2012 Preprint IPhT t12/042 ii CONTENTS Contents 1 Introduction 1-1 1.1 Motivation . 1-1 1.2 Organization . 1-2 1.3 What this course is not! . 1-3 1.4 Acknowledgements . 1-3 2 Reminders 2-1 2.1 Classical mechanics .
    [Show full text]
  • Functional Integration on Paracompact Manifods Pierre Grange, E
    Functional Integration on Paracompact Manifods Pierre Grange, E. Werner To cite this version: Pierre Grange, E. Werner. Functional Integration on Paracompact Manifods: Functional Integra- tion on Manifold. Theoretical and Mathematical Physics, Consultants bureau, 2018, pp.1-29. hal- 01942764 HAL Id: hal-01942764 https://hal.archives-ouvertes.fr/hal-01942764 Submitted on 3 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Functional Integration on Paracompact Manifolds Pierre Grangé Laboratoire Univers et Particules, Université Montpellier II, CNRS/IN2P3, Place E. Bataillon F-34095 Montpellier Cedex 05, France E-mail: [email protected] Ernst Werner Institut fu¨r Theoretische Physik, Universita¨t Regensburg, Universita¨tstrasse 31, D-93053 Regensburg, Germany E-mail: [email protected] ........................................................................ Abstract. In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette (CDWM) focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. CDWM’s central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures.
    [Show full text]
  • Well-Defined Lagrangian Flows for Absolutely Continuous Curves of Probabilities on the Line
    Graduate Theses, Dissertations, and Problem Reports 2016 Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Amsaad, Mohamed H., "Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line" (2016). Graduate Theses, Dissertations, and Problem Reports. 5098. https://researchrepository.wvu.edu/etd/5098 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Adrian Tudorascu, Ph.D., Chair Harumi Hattori, Ph.D. Harry Gingold, Ph.D. Tudor Stanescu, Ph.D. Charis Tsikkou, Ph.D. Department Of Mathematics Morgantown, West Virginia 2016 Keywords: Continuity Equation, Lagrangian Flow, Optimal Transport, Wasserstein metric, Wasserstein space.
    [Show full text]
  • Functional Derivative
    Appendix Functional Derivative Appendix A: Calculus of Variation Consider a mapping from a vector space of functions to real number. Such mapping is called functional. One of the most representative examples of such a mapping is the action functional of analytical mechanics, A ,which is defined by Zt2 A½xtðÞ Lxt½ðÞ; vtðÞdt ðA:1Þ t1 where t is the independent variable, xtðÞis the trajectory of a particle, vtðÞ¼dx=dt is the velocity of the particle, and L is the Lagrangian which is the difference of kinetic energy and the potential: m L ¼ v2 À uxðÞ ðA:2Þ 2 Calculus of variation is the mathematical theory to find the extremal function xtðÞwhich maximizes or minimizes the functional such as Eq. (A.1). We are interested in the set of functions which satisfy the fixed boundary con- ditions such as xtðÞ¼1 x1 and xtðÞ¼2 x2. An arbitrary element of the function space may be defined from xtðÞby xtðÞ¼xtðÞþegðÞt ðA:3Þ where ε is a real number and gðÞt is any function satisfying gðÞ¼t1 gðÞ¼t2 0. Of course, it is obvious that xtðÞ¼1 x1 and xtðÞ¼2 x2. Varying ε and gðÞt generates any function of the vector space. Substitution of Eq. (A.3) to Eq. (A.1) gives Zt2 dx dg aðÞe A½¼xtðÞþegðÞt L xtðÞþegðÞt ; þ e dt ðA:4Þ dt dt t1 © Springer Science+Business Media Dordrecht 2016 601 K.S. Cho, Viscoelasticity of Polymers, Springer Series in Materials Science 241, DOI 10.1007/978-94-017-7564-9 602 Appendix: Functional Derivative From Eq.
    [Show full text]
  • 35 Nopw It Is Time to Face the Menace
    35 XIV. CORRELATIONS AND SUSCEPTIBILITY A. Correlations - saddle point approximation: the easy way out Nopw it is time to face the menace - path integrals. We were tithering on the top of this ravine for long enough. We need to take the plunge. The meaning of the path integral is very simple - tofind the partition function of a system locallyfluctuating, we need to consider the all the possible patterns offlucutations, and the only way to do this is through a path integral. But is the path integral so bad? Well, if you believe that anyfluctuation is still quite costly, then the path integral, like the integral over the exponent of a quadratic function, is pretty much determined by where the argument of the exponent is a minimum, plus quadraticfluctuations. These belief is translated into equations via the saddle point approximation. Wefind the minimum of the free energy functional, and then expand upto second order in the exponent. 1 1 F [m(x),h] = ddx γ( m (x))2 +a(T T )m (x)2 + um (x)4 hm + ( m (x))2 ∂2F (219) L ∇ min − c min 2 min − min 2 − min � � � I write here deliberately the vague partial symbol, in fact, this should be the functional derivative. Does this look familiar? This should bring you back to the good old classical mechanics days. Where the Lagrangian is the only thing you wanted to know, and the Euler Lagrange equations were the only guidance necessary. Non of this non-commuting business, or partition function annoyance. Well, these times are back for a short time! In order tofind this, we do a variation: m(x) =m min +δm (220) Upon assignment wefind: u F [m(x),h] = ddx γ( (m +δm)) 2 +a(T T )(m +δm) 2 + (m +δm) 4 h(m +δm) (221) L ∇ min − c min 2 min − min � � � Let’s expand this to second order: d 2 2 d x γ (mmin) + δm 2γ m+γ( δm) → ∇ 2 ∇ · ∇ ∇ 2 +a (mmin) +δm 2am+aδm � � ∇ u 4 ·3 2 2 (222) + 2 mmin +2umminδm +3umminδm hm hδm] − min − Collecting terms in power ofδm, thefirst term is justF L[mmin].
    [Show full text]
  • Noncausal Stochastic Calculus Noncausal Stochastic Calculus
    Shigeyoshi Ogawa Noncausal Stochastic Calculus Noncausal Stochastic Calculus Shigeyoshi Ogawa Noncausal Stochastic Calculus 123 Shigeyoshi Ogawa Department of Mathematical Sciences Ritsumeikan University Kusatsu, Shiga Japan ISBN 978-4-431-56574-1 ISBN 978-4-431-56576-5 (eBook) DOI 10.1007/978-4-431-56576-5 Library of Congress Control Number: 2017945261 © Springer Japan KK 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Japan KK The registered company address is: Chiyoda First Bldg.
    [Show full text]
  • On Malliavin's Proof of H\" Ormander's Theorem
    On Malliavin’s proof of Hormander’s¨ theorem May 29, 2018 Martin Hairer Mathematics Department, University of Warwick Email: [email protected] Abstract The aim of this note is to provide a short and self-contained proof of H¨ormander’s theorem about the smoothness of transition probabilities for a diffusion under H¨ormander’s “brackets condition”. While both the result and the technique of proof are well-known, the exposition given here is novel in two aspects. First, we introduce Malliavin calculus in an “intuitive” way, without using Wiener’s chaos decomposition. While this may make it difficult to prove some of the standard results in Malliavin calculus (boundedness of the derivative operator in Lp spaces for example), we are able to bypass these and to replace them by weaker results that are still sufficient for our purpose. Second, we introduce a notion of “almost implication” and “almost truth” (somewhat similar to what is done in fuzzy logic) which allows, once the foundations of Malliavin calculus are laid out, to give a very short and streamlined proof of H¨ormader’s theorem that focuses on the main ideas without clouding it by technical details. Dedicated to the memory of Paul Malliavin. 1 Introduction arXiv:1103.1998v1 [math.PR] 10 Mar 2011 One of the main tools in many results on the convergence to equilibrium of Markov processes is the presence of some form of “smoothing” for the semigroup. For example, if a Markov operator over a Polish space possesses the strong Feller P X property (namely it maps b( ), the space of bounded measurable functions into ( ), the space of boundedB X continuous functions), then one can conclude that Cb X any two ergodic invariant measures for must either coincide or have disjoint topological supports.
    [Show full text]
  • On the Solution and Application of Rational Expectations Models With
    On the Solution and Application of Rational Expectations Models with Function-Valued States⇤ David Childers† November 16, 2015 Abstract Many variables of interest to economists take the form of time varying distribu- tions or functions. This high-dimensional ‘functional’ data can be interpreted in the context of economic models with function valued endogenous variables, but deriving the implications of these models requires solving a nonlinear system for a potentially infinite-dimensional function of infinite-dimensional objects. To overcome this difficulty, I provide methods for characterizing and numerically approximating the equilibria of dynamic, stochastic, general equilibrium models with function-valued state variables by linearization in function space and representation using basis functions. These meth- ods permit arbitrary infinite-dimensional variation in the state variables, do not impose exclusion restrictions on the relationship between variables or limit their impact to a finite-dimensional sufficient statistic, and, most importantly, come with demonstrable guarantees of consistency and polynomial time computational complexity. I demon- strate the applicability of the theory by providing an analytical characterization and computing the solution to a dynamic model of trade, migration, and economic geogra- phy. ⇤Preliminary, subject to revisions. For the latest version, please obtain from the author’s website at https://sites.google.com/site/davidbchilders/DavidChildersFunctionValuedStates.pdf †Yale University, Department of Economics. The author gratefully acknowledges financial support from the Cowles Foundation. This research has benefited from discussion and feedback from Peter Phillips, Tony Smith, Costas Arkolakis, John Geanakoplos, Xiaohong Chen, Tim Christensen, Kieran Walsh, and seminar participants at the Yale University Econometrics Seminar, Macroeconomics Lunch, and Financial Markets Reading Group.
    [Show full text]
  • Characteristic Hypersurfaces and Constraint Theory
    Characteristic Hypersurfaces and Constraint Theory Thesis submitted for the Degree of MSc by Patrik Omland Under the supervision of Prof. Stefan Hofmann Arnold Sommerfeld Center for Theoretical Physics Chair on Cosmology Abstract In this thesis I investigate the occurrence of additional constraints in a field theory, when formulated in characteristic coordinates. More specifically, the following setup is considered: Given the Lagrangian of a field theory, I formulate the associated (instantaneous) Hamiltonian problem on a characteristic hypersurface (w.r.t. the Euler-Lagrange equations) and find that there exist new constraints. I then present conditions under which these constraints lead to symplectic submanifolds of phase space. Symplecticity is desirable, because it renders Hamiltonian vector fields well-defined. The upshot is that symplecticity comes down to analytic rather than algebraic conditions. Acknowledgements After five years of study, there are many people I feel very much indebted to. Foremost, without the continuous support of my parents and grandparents, sister and aunt, for what is by now a quarter of a century I would not be writing these lines at all. Had Anja not been there to show me how to get to my first lecture (and in fact all subsequent ones), lord knows where I would have ended up. It was through Prof. Cieliebaks inspiring lectures and help that I did end up in the TMP program. Thank you, Robert, for providing peculiar students with an environment, where they could forget about life for a while and collectively find their limits, respectively. In particular, I would like to thank the TMP lonely island faction.
    [Show full text]
  • Variational Theory in the Context of General Relativity Abstract
    Variational theory in the context of general relativity Hiroki Takeda1, ∗ 1Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan (Dated: October 25, 2018) Abstract This is a brief note to summarize basic concepts of variational theory in general relativity for myself. This note is mainly based on Hawking(2014)"Singularities and the geometry of spacetime" and Wald(1984)"General Relativity". ∗ [email protected] 1 I. BRIEF REVIEW OF GENERAL RELATIVITY AND VARIATIONAL THEORY A. Variation defnition I.1 (Action) Let M be a differentiable manifold and Ψ be some tensor fields on M . Ψ are simply called the fields. Consider the function of Ψ, S[Ψ]. This S is a map from the set of the fields on M to the set of real numbers and called the action. a···b Ψ actually represents some fields Ψ(i) c···d(i = 1; 2; ··· ; n). I left out the indices i that indicate the number of fields and the abstract indices for the tensors. The tensor of the a···b a···b tensor fields Ψ(i) c···d(i = 1; 2; ··· ; n) at x 2 M is written by Ψ(i)jx c···d(i = 1; 2; ··· ; n). Similarly I denote the tensor of the tensor fields Ψ at x 2 M by Ψjx. defnition I.2 (Variation) Let D ⊂ M be a submanifold of M and Ψjx(u), u 2 (−ϵ, ϵ), x 2 M be the tensors of one-parameter family of the tensor fields Ψ(u), u 2 (−ϵ, ϵ) at x 2 M such that (1) Ψjx(0) = Ψ in M ; (1) (2) Ψjx(u) = Ψ in M − D: (2) Then I define the variation of the fields as follows, dΨ(u) δΨ := : (3) du u=0 In this note, I suppose the existence of the derivatives dS=duju=0 for all one-parameter family Ψ(u).
    [Show full text]
  • Full Version
    Theory of Stochastic Processes Vol. 14 (30), no. 2, 2008, pp. 10–18 UDC 519.21 KHALIFA ES-SEBAIY AND CIPRIAN A. TUDOR LEVY´ PROCESSES AND ITO–SKOROKHODˆ INTEGRALS We study Skorokhod integral processes on L´evy spaces and prove an equivalence between this class of processes and the class of Itˆo–Skorokhod processes (in the sense of [14]). Using this equivalence, we introduce a stochastic analysis of the Itˆotypefor anticipating integrals on L´evy spaces. Introduction We study in this work the anticipating integrals with respect to a L´evy process. The anticipating integral on the Wiener space, known in general as the Skorokhod integral (and sometimes as the Hitsuda integral), constitutes an extension of the standard Itˆo integral to non-adapted integrands. It is nothing else than the classical Itˆointegralifthe integrand is adapted. The Skorokhod integral has been extended to the Poisson process, and next it has been defined with respect to a normal martingale (see [3]) due to the Fock space structure generated by such processes. Recently, an anticipating calculus of the Malliavin-type has been defined on L´evy spaces again by using some multiple stochastic integrals with respect to a L´evy process which have been defined, in essence, in the old paper by K. Itˆo (see [4]). We refer to [8], [9], or [13] for the Malliavin calculus on L´evy spaces and possible applications to mathematical finance. The purpose of this paper is to understand the relation between anticipating Sko- rokhod integral processes and Itˆo-Skorokhod integral processes (in the sense of [14] or [11]) in the L´evy case.
    [Show full text]