DOCSLIB.ORG

Differentiable Measures and the Malliavin Calculus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Mathematical Surveys and Monographs Volume 164 Differentiable Measures and the Malliavin Calculus Vladimir I. Bogachev American Mathematical Society http://dx.doi.org/10.1090/surv/164 Differentiable Measures Differentiable Measures and the Malliavin Calculus and the Malliavin Calculus Vladimir I. Bogachev Mathematical Surveys and Monographs Volume 164 Differentiable Measures and the Malliavin Calculus Vladimir I. Bogachev American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 28Cxx, 46Gxx, 58Bxx, 60Bxx, 60Hxx. For additional information and updates on this book, visit www.ams.org/bookpages/surv-164 Library of Congress Cataloging-in-Publication Data Bogachev, V. I. (Vladimir Igorevich), 1961– Differentiable measures and the Malliavin calculus / Vladimir I. Bogachev p. cm. – (Mathematical surveys and monographs ; v. 164) Includes bibliographical references and index. ISBN 978-0-8218-4993-4 (alk. paper) 1. Measure theory. 2. Theory of distributions (Functional analysis) 3. Sobolev spaces. 4. Malliavin calculus. I. Title. QA312.B638 2010 515.42–dc22 2010005829 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 Contents Preface ix Chapter 1. Background material 1 1.1. Functional analysis 1 1.2. Measures on topological spaces 6 1.3. Conditional measures 19 1.4. Gaussian measures 23 1.5. Stochastic integrals 29 1.6. Comments and exercises 35 Chapter 2. Sobolev spaces on IRn 39 2.1. The Sobolev classes W p,k 39 2.2. Embedding theorems for Sobolev classes 45 2.3. The classes BV 51 2.4. Approximate differentiability and Jacobians 52 2.5. Restrictions and extensions 56 2.6. Weighted Sobolev classes 58 2.7. Fractional Sobolev classes 65 2.8. Comments and exercises 66 Chapter 3. Differentiable measures on linear spaces 69 3.1. Directional differentiability 69 3.2. Properties of continuous measures 73 3.3. Properties of differentiable measures 76 3.4. Differentiable measures on IRn 82 3.5. Characterization by conditional measures 88 3.6. Skorohod differentiability 91 3.7. Higher order differentiability 100 3.8. Convergence of differentiable measures 101 3.9. Comments and exercises 103 Chapter 4. Some classes of differentiable measures 105 4.1. Product measures 105 4.2. Gaussian and stable measures 107 4.3. Convex measures 112 4.4. Distributions of random processes 122 v vi Contents 4.5. Gibbs measures and mixtures of measures 127 4.6. Comments and exercises 131 Chapter 5. Subspaces of differentiability of measures 133 5.1. Geometry of subspaces of differentiability 133 5.2. Examples 136 5.3. Disposition of subspaces of differentiability 141 5.4. Differentiability along subspaces 149 5.5. Comments and exercises 155 Chapter 6. Integration by parts and logarithmic derivatives 157 6.1. Integration by parts formulae 157 6.2. Integrability of logarithmic derivatives 161 6.3. Differentiability of logarithmic derivatives 168 6.4. Quasi-invariance and differentiability 170 6.5. Convex functions 173 6.6. Derivatives along vector fields 180 6.7. Local logarithmic derivatives 183 6.8. Comments and exercises 187 Chapter 7. Logarithmic gradients 189 7.1. Rigged Hilbert spaces 189 7.2. Definition of logarithmic gradient 190 7.3. Connections with vector measures 194 7.4. Existence of logarithmic gradients 198 7.5. Measures with given logarithmic gradients 204 7.6. Uniqueness problems 209 7.7. Symmetries of measures and logarithmic gradients 217 7.8. Mappings and equations connected with logarithmic gradients 222 7.9. Comments and exercises 223 Chapter 8. Sobolev classes on infinite dimensional spaces 227 8.1. The classes W p,r 227 8.2. The classes Dp,r 230 8.3. Generalized derivatives and the classes Gp,r 233 8.4. The semigroup approach 235 8.5. The Gaussian case 239 8.6. The interpolation approach 242 8.7. Connections between different definitions 246 8.8. The logarithmic Sobolev inequality 250 8.9. Compactness in Sobolev classes 253 8.10. Divergence 254 8.11. An approach via stochastic integrals 257 8.12. Some identities of the Malliavin calculus 263 8.13. Sobolev capacities 265 8.14. Comments and exercises 274 Contents vii Chapter 9. The Malliavin calculus 279 9.1. General scheme 279 9.2. Absolute continuity of images of measures 282 9.3. Smoothness of induced measures 288 9.4. Infinite dimensional oscillatory integrals 297 9.5. Surface measures 299 9.6. Convergence of nonlinear images of measures 307 9.7. Supports of induced measures 319 9.8. Comments and exercises 323 Chapter 10. Infinite dimensional transformations 329 10.1. Linear transformations of Gaussian measures 329 10.2. Nonlinear transformations of Gaussian measures 334 10.3. Transformations of smooth measures 338 10.4. Absolutely continuous flows 340 10.5. Negligible sets 342 10.6. Infinite dimensional Rademacher’s theorem 350 10.7. Triangular and optimal transformations 358 10.8. Comments and exercises 365 Chapter 11. Measures on manifolds 369 11.1. Measurable manifolds and Malliavin’s method 370 11.2. Differentiable families of measures 379 11.3. Current and loop groups 390 11.4. Poisson spaces 393 11.5. Diffeomorphism groups 394 11.6. Comments and exercises 398 Chapter 12. Applications 401 12.1. A probabilistic approach to hypoellipticity 401 12.2. Equations for measures 407 12.3. Logarithmic gradients and symmetric diffusions 414 12.4. Dirichlet forms and differentiable measures 416 12.5. The uniqueness problem for invariant measures 420 12.6. Existence of Gibbs measures 422 12.7. Comments and exercises 424 References 427 Subject Index 483 Preface Among the most notable events in nonlinear functional analysis for the last three decades one should mention the development of the theory of differentiable measures and the Malliavin calculus. These two closely related theories can be regarded at the same time as infinite dimensional analogues of such classical fields as geometric measure theory, the theory of Sobolev spaces, and the theory of generalized functions. The theory of differentiable measures was suggested by S.V. Fomin in his address at the International Congress of Mathematicians in Moscow in 1966 as a candidate for an infinite dimensional substitute of the Sobolev– Schwartz theory of distributions (see [440]–[443], [82], [83]). It was Fomin who realized that in the infinite dimensional case instead of a pair of spaces (test functions – generalized functions) it is natural to consider four spaces: a certain space of functions S, its dual S0, a certain space of measures M, and its dual M 0. In the finite dimensional case, M is identified with S by means of Lebesgue measure which enables us to represent measures via their densities. The absence of translationally invariant nonzero measures destroys this identification in the infinite dimensional case. Fomin found a convenient definition of a smooth measure on an infinite dimensional space corresponding to a measure with a smooth density on IRn. By means of this definition one introduces the space M of smooth measures. The Fourier transform then acts between S and M and between S0 and M 0. In this way a theory of pseudo-differential operators can be developed: the initial objective of Fomin was a theory of infinite dimensional partial differential equations. However, as it often happens with fruitful ideas, the theory of differentiable measures overgrew the initial framework. This theory has be- come an efficient tool in a wide variety of the most diverse applications such as stochastic analysis, quantum field theory, and nonlinear analysis. At present it is a rapidly developing field abundant with many challeng- ing problems of great importance for better understanding of the nature of infinite dimensional phenomena. It is worth mentioning that before the pioneering works of Fomin re- lated ideas appeared in several papers by T. Pitcher on the distributions of random processes in functional spaces. T. Pitcher investigated an even more general situation, namely he studied the differentiability of a family ix x Preface −1 of measures µt = µ◦Tt generated by a family of transformations Tt of a fixed measure µ. Fomin’s differentiability corresponds to the case where the measures µt are the shifts µth of a measure µ on a linear space. However, the theory constructed for this more special case is much richer. Let us also mention similar ideas of L. Gross [322] developed in the analysis on Wiener space. In the mid 1970s P. Malliavin [751] suggested an elegant method for proving the smoothness of the transition probabilities of finite dimensional diffusions. The essence of the method is to consider the transition proba- bilities Pt as images of the Wiener measure under some nonlinear transfor- mations (generated by stochastic differential equations) and then apply an integration by parts formula on the Wiener space leading to certain estimates ∞ of the generalized derivatives of Pt which ensure the membership in C .
Recommended publications
  • Math 346 Lecture #3 6.3 the General Fréchet Derivative

    Math 346 Lecture #3 6.3 the General Fréchet Derivative

    Math 346 Lecture #3 6.3 The General Fr´echet Derivative We now extend the notion of the Fr´echet derivative to the Banach space setting. Throughout let (X; k · kX ) and (Y; k · kY ) be Banach spaces, and U an open set in X. 6.3.1 The Fr´echet Derivative Definition 6.3.1. A function f : U ! Y is Fr´echet differentiable at x 2 U if there exists A 2 B(X; Y ) such that kf(x + h) − f(x) − A(h)k lim Y = 0; h!0 khkX and we write Df(x) = A. We say f is Fr´echet differentiable on U if f is Fr´echet differ- entiable at each x 2 U. We often refer to Fr´echet differentiable simply as differentiable. Note. When f is Fr´echet differentiable at x 2 U, the derivative Df(x) is unique. This follows from Proposition 6.2.10 (uniqueness of the derivative for finite dimensional X and Y ) whose proof carries over without change to the general Banach space setting (see Remark 6.3.9). Remark 6.3.2. The existence of the Fr´echet derivative does not change when the norm on X is replace by a topologically equivalent one and/or the norm on Y is replaced by a topologically equivalent one. Example 6.3.3. Any L 2 B(X; Y ) is Fr´echet differentiable with DL(x)(v) = L(v) for all x 2 X and all v 2 X. The proof of this is exactly the same as that given in Example 6.2.5 for finite-dimensional Banach spaces.
  • Application of Malliavin Calculus and Wiener Chaos to Option Pricing Theory

    Application of Malliavin Calculus and Wiener Chaos to Option Pricing Theory

    Application of Malliavin Calculus and Wiener Chaos to Option Pricing Theory by Eric Ben-Hamou The London School of Economics and Political Science Thesis submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy at the University of London UMI Number: U615447 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U615447 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 weseS F 7855 % * A bstract This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al.
  • Which Moments of a Logarithmic Derivative Imply Quasiinvariance?

    Which Moments of a Logarithmic Derivative Imply Quasiinvariance?

    Doc Math J DMV Which Moments of a Logarithmic Derivative Imply Quasi invariance Michael Scheutzow Heinrich v Weizsacker Received June Communicated by Friedrich Gotze Abstract In many sp ecial contexts quasiinvariance of a measure under a oneparameter group of transformations has b een established A remarkable classical general result of AV Skorokhod states that a measure on a Hilb ert space is quasiinvariant in a given direction if it has a logarithmic aj j derivative in this direction such that e is integrable for some a In this note we use the techniques of to extend this result to general oneparameter families of measures and moreover we give a complete char acterization of all functions for which the integrability of j j implies quasiinvariance of If is convex then a necessary and sucient condition is that log xx is not integrable at Mathematics Sub ject Classication A C G Overview The pap er is divided into two parts The rst part do es not mention quasiinvariance at all It treats only onedimensional functions and implicitly onedimensional measures The reason is as follows A measure on R has a logarithmic derivative if and only if has an absolutely continuous Leb esgue density f and is given by 0 f x ae Then the integrability of jj is equivalent to the Leb esgue x f 0 f jf The quasiinvariance of is equivalent to the statement integrability of j f that f x Leb esgueae Therefore in the case of onedimensional measures a function allows a quasiinvariance criterion as indicated in the abstract
  • Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications

    Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications

    Purdue University Purdue e-Pubs Open Access Dissertations Theses and Dissertations January 2015 Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications. Rolando Dangnanan Navarro Purdue University Follow this and additional works at: https://docs.lib.purdue.edu/open_access_dissertations Recommended Citation Navarro, Rolando Dangnanan, "Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications." (2015). Open Access Dissertations. 1422. https://docs.lib.purdue.edu/open_access_dissertations/1422 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Graduate School Form 30 Updated 1/15/2015 PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Rolando D. Navarro, Jr. Entitled Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications For the degree of Doctor of Philosophy Is approved by the final examining committee: Frederi Viens Chair Jonathon Peterson Michael Levine Jose Figueroa-Lopez To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material. Approved by Major Professor(s): Frederi Viens Approved by: Jun Xie 11/24/2015 Head of the Departmental Graduate Program Date MALLIAVIN CALCULUS IN THE CANONICAL LEVY´ PROCESS: WHITE NOISE THEORY AND FINANCIAL APPLICATIONS A Dissertation Submitted to the Faculty of Purdue University by Rolando D. Navarro, Jr.
  • Weak Approximations for Arithmetic Means of Geometric Brownian Motions and Applications to Basket Options Romain Bompis

    Weak Approximations for Arithmetic Means of Geometric Brownian Motions and Applications to Basket Options Romain Bompis

    Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options Romain Bompis To cite this version: Romain Bompis. Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options. 2017. hal-01502886 HAL Id: hal-01502886 https://hal.archives-ouvertes.fr/hal-01502886 Preprint submitted on 6 Apr 2017 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. Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options ∗ Romain Bompis y Abstract. In this work we derive new analytical weak approximations for arithmetic means of geometric Brownian mo- tions using a scalar log-normal Proxy with an averaged volatility. The key features of the approach are to keep the martingale property for the approximations and to provide new integration by parts formulas for geometric Brownian motions. Besides, we also provide tight error bounds using Malliavin calculus, estimates depending on a suitable dispersion measure for the volatilities and on the maturity. As applications we give new price and im- plied volatility approximation formulas for basket call options. The numerical tests reveal the excellent accuracy of our results and comparison with the other known formulas of the literature show a valuable improvement.
  • A Logarithmic Derivative Lemma in Several Complex Variables and Its Applications

    A Logarithmic Derivative Lemma in Several Complex Variables and Its Applications

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 12, December 2011, Pages 6257–6267 S 0002-9947(2011)05226-8 Article electronically published on June 27, 2011 A LOGARITHMIC DERIVATIVE LEMMA IN SEVERAL COMPLEX VARIABLES AND ITS APPLICATIONS BAO QIN LI Abstract. We give a logarithmic derivative lemma in several complex vari- ables and its applications to meromorphic solutions of partial differential equa- tions. 1. Introduction The logarithmic derivative lemma of Nevanlinna is an important tool in the value distribution theory of meromorphic functions and its applications. It has two main forms (see [13, 1.3.3 and 4.2.1], [9, p. 115], [4, p. 36], etc.): Estimate (1) and its consequence, Estimate (2) below. Theorem A. Let f be a non-zero meromorphic function in |z| <R≤ +∞ in the complex plane with f(0) =0 , ∞. Then for 0 <r<ρ<R, 1 2π f (k)(reiθ) log+ | |dθ 2π f(reiθ) (1) 0 1 1 1 ≤ c{log+ T (ρ, f) + log+ log+ +log+ ρ +log+ +log+ +1}, |f(0)| ρ − r r where k is a positive integer and c is a positive constant depending only on k. Estimate (1) with k = 1 was originally due to Nevanlinna, which easily implies the following version of the lemma with exceptional intervals of r for meromorphic functions in the plane: 1 2π f (reiθ) (2) + | | { } log iθ dθ = O log(rT(r, f )) , 2π 0 f(re ) for all r outside a countable union of intervals of finite Lebesgue measure, by using the Borel lemma in a standard way (see e.g.
  • FROM CLASSICAL MECHANICS to QUANTUM FIELD THEORY, a TUTORIAL Copyright © 2020 by World Scientific Publishing Co

    FROM CLASSICAL MECHANICS to QUANTUM FIELD THEORY, a TUTORIAL Copyright © 2020 by World Scientific Publishing Co

    FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL 11556_9789811210488_TP.indd 1 29/11/19 2:30 PM This page intentionally left blank FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL Manuel Asorey Universidad de Zaragoza, Spain Elisa Ercolessi University of Bologna & INFN-Sezione di Bologna, Italy Valter Moretti University of Trento & INFN-TIFPA, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 11556_9789811210488_TP.indd 2 29/11/19 2:30 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY, A TUTORIAL Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-121-048-8 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11556#t=suppl Desk Editor: Nur Syarfeena Binte Mohd Fauzi Typeset by Stallion Press Email: [email protected] Printed in Singapore Syarfeena - 11556 - From Classical Mechanics.indd 1 02-12-19 3:03:23 PM January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page v Preface This book grew out of the mini courses delivered at the Fall Workshop on Geometry and Physics, in Granada, Zaragoza and Madrid.
  • Differentiability Problems in Banach Spaces

    Differentiability Problems in Banach Spaces

    Derivatives Differentiability problems in Banach spaces For vector valued functions there are two main version of derivatives: Gâteaux (or weak) derivatives and Fréchet (or strong) derivatives. For a function f from a Banach space X 1 David Preiss into a Banach space Y the Gâteaux derivative at a point x0 2 X is by definition a bounded linear operator T : X −! Y so that Expanded notes of a talk based on a nearly finished research monograph “Fréchet differentiability of Lipschitz functions and porous sets in Banach for every u 2 X, spaces” f (x + tu) − f (x ) written jointly with lim 0 0 = Tu t!0 t Joram Lindenstrauss2 and Jaroslav Tišer3 The operator T is called the Fréchet derivative of f at x if it is a with some results coming from a joint work with 0 4 5 Gâteaux derivative of f at x0 and the limit above holds uniformly Giovanni Alberti and Marianna Csörnyei in u in the unit ball (or unit sphere) in X. So T is the Fréchet derivative of f at x0 if 1Warwick 2Jerusalem 3Prague 4Pisa 5London f (x0 + u) = f (x0) + Tu + o(kuk) as kuk ! 0: 1 / 24 2 / 24 Existence of derivatives Sharpness of Lebesgue’s result The first continuous nowhere differentiable f : R ! R was constructed by Bolzano about 1820 (unpublished), who Lebesgue’s result is sharp in the sense that for every A ⊂ of however did not give a full proof. Around 1850, Riemann R measure zero there is a Lipschitz (and monotone) function mentioned such an example, which was later found slightly f : ! which fails to have a derivative at any point of A.
  • Value Functions and Optimality Conditions for Nonconvex

    Value Functions and Optimality Conditions for Nonconvex

    Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces∗ H´el`ene Frankowska† CNRS Institut de Math´ematiques de Jussieu – Paris Rive Gauche Sorbonne Universit´e, Campus Pierre et Marie Curie Case 247, 4 Place Jussieu, 75252 Paris, France e-mail: [email protected] Nobusumi Sagara‡ Department of Economics, Hosei University 4342, Aihara, Machida, Tokyo, 194–0298, Japan e-mail: [email protected] February 11, 2020 arXiv:1801.00400v3 [math.OC] 10 Feb 2020 ∗The authors are grateful to the two anonymous referees for their helpful comments and suggestions on the earlier version of this manuscript. †The research of this author was supported by the Air Force Office of Scientific Re- search, USA under award number FA9550-18-1-0254. It also partially benefited from the FJMH Program PGMO and from the support to this program from EDF-THALES- ORANGE-CRITEO under grant PGMO 2018-0047H. ‡The research of this author benefited from the support of JSPS KAKENHI Grant Number JP18K01518 from the Ministry of Education, Culture, Sports, Science and Tech- nology, Japan. Abstract We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini–Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler–Lagrange condition and the maximum principle.
  • On Fréchet Differentiability of Lipschitz Maps

    On Fréchet Differentiability of Lipschitz Maps

    Annals of Mathematics, 157 (2003), 257–288 On Fr´echet differentiability of Lipschitz maps between Banach spaces By Joram Lindenstrauss and David Preiss Abstract Awell-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point ofFr´echet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called Γ-null sets), whose definition involves both the notions of category and mea- sure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Fr´echet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces; no such results have been known previously. Our main result states that a Lipschitz map between separable Banach spaces is Fr´echet differentiable Γ-almost everywhere provided that it is reg- ularly Gˆateaux differentiable Γ-almost everywhere and the Gˆateaux deriva- tives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikod´ym property are Gˆateaux differentiable Γ-almost everywhere. Moreover, Gˆateaux differentiability im- plies regular Gˆateaux differentiability with exception of another kind of neg- ligible sets, so-called σ-porous sets. The answer to the question is therefore positive in every space in which every σ-porous set is Γ-null.
  • Stochastic Differential Equations and Hypoelliptic

    Stochastic Differential Equations and Hypoelliptic

    . STOCHASTIC DIFFERENTIAL EQUATIONS AND HYPOELLIPTIC OPERATORS Denis R. Bell Department of Mathematics University of North Florida Jacksonville, FL 32224 U. S. A. In Real and Stochastic Analysis, 9-42, Trends Math. ed. M. M. Rao, Birkhauser, Boston. MA, 2004. 1 1. Introduction The first half of the twentieth century saw some remarkable developments in analytic probability theory. Wiener constructed a rigorous mathematical model of Brownian motion. Kolmogorov discovered that the transition probabilities of a diffusion process define a fundamental solution to the associated heat equation. Itˆo developed a stochas- tic calculus which made it possible to represent a diffusion with a given (infinitesimal) generator as the solution of a stochastic differential equation. These developments cre- ated a link between the fields of partial differential equations and stochastic analysis whereby results in the former area could be used to prove results in the latter. d More specifically, let X0,...,Xn denote a collection of smooth vector fields on R , regarded also as first-order differential operators, and define the second-order differen- tial operator n 1 L ≡ X2 + X . (1.1) 2 i 0 i=1 Consider also the Stratonovich stochastic differential equation (sde) n dξt = Xi(ξt) ◦ dwi + X0(ξt)dt (1.2) i=1 where w =(w1,...,wn)isann−dimensional standard Wiener process. Then the solution ξ to (1.2) is a time-homogeneous Markov process with generator L, whose transition probabilities p(t, x, dy) satisfy the following PDE (known as the Kolmogorov forward equation) in the weak sense ∂p = L∗p. ∂t y A differential operator G is said to be hypoelliptic if, whenever Gu is smooth for a distribution u defined on some open subset of the domain of G, then u is smooth.
  • An Intuitive Introduction to Fractional and Rough Volatilities

    An Intuitive Introduction to Fractional and Rough Volatilities

    mathematics Review An Intuitive Introduction to Fractional and Rough Volatilities Elisa Alòs 1,* and Jorge A. León 2 1 Department d’Economia i Empresa, Universitat Pompeu Fabra, and Barcelona GSE c/Ramon Trias Fargas, 25-27, 08005 Barcelona, Spain 2 Departamento de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 07000 México City, Mexico; [email protected] * Correspondence: [email protected] Abstract: Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments. Keywords: derivative operator in the Malliavin calculus sense; fractional Brownian motion; future average volatility; Hull and White formula; Itô’s formula; Skorohod integral; stochastic volatility models; implied volatility; skews and smiles; rough volatility Citation: Alòs, E.; León, J.A.