Probability Theory and Stochastic Modelling 79 T E. Govindan Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications Probability Theory and Stochastic Modelling Volume 79 Editors-in-Chief Søren Asmussen, Aarhus, Denmark Peter W Glynn, Stanford, USA Thomas G Kurtz, Madison, WI, USA Yves Le Jan, Orsay, France Advisory Board Martin Hairer, Coventry, UK Peter Jagers, Gothenburg, Sweden Ioannis Karatzas, New York, NY, USA Frank P Kelly, Cambridge, UK Andreas E Kyprianou, Bath, UK Bernt Øksendal, Oslo, Norway George Papanicolaou, Stanford, CA, USA Etienne Pardoux, Marseille, France Edwin Perkins, Vancouver, BC, Canada Halil Mete Soner, Zürich, Switzerland The Probability Theory and Stochastic Modelling series is a merger and continuation of Springer’s two well established series Stochastic Modelling and Applied Probability and Probability and Its Applications series It publishes research monographs that make a significant contribution to probability theory or an applications domain in which advanced probability methods are fundamental Books in this series are expected to follow rigorous mathematical standards, while also displaying the expository quality necessary to make them useful and accessible to advanced students as well as researchers The series covers all aspects of modern probability theory including • • • • • • Gaussian processes Markov processes Random fields, point processes and random sets Random matrices Statistical mechanics and random media Stochastic analysis as well as applications that include (but are not restricted to): • Branching processes and other models of population growth • Communications and processing networks • Computational methods in probability and stochastic processes, including simulation • Genetics and other stochastic models in biology and the life sciences • Information theory, signal processing, and image synthesis • Mathematical economics and finance • Statistical methods (e.g empirical processes, MCMC) • Statistics for stochastic processes • Stochastic control • Stochastic models in operations research and stochastic optimization • Stochastic models in the physical sciences More information about this series at http://www.springer.com/series/13205 T E Govindan Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications 123 T E Govindan National Polytechnic Institute Mexico City, Mexico tegovindan@yahoo.com ISSN 2199-3130 ISSN 2199-3149 (electronic) Probability Theory and Stochastic Modelling ISBN 978-3-319-45682-9 ISBN 978-3-319-45684-3 (eBook) DOI 10.1007/978-3-319-45684-3 Library of Congress Control Number: 2016950521 Mathematics Subject Classification (2010): 60H05, 60H10, 60H15, 60H20, 60H30, 60H25, 65C30, 93E03, 93D09, 93D20, 93E15, 93E20, 37L55, 35R60 © Springer International Publishing Switzerland 2016 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 Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland In fond memory of my maternal great grandmother and my maternal grandmother To my mother Mrs G Suseela and to my father Mr T E Sarangan In fond memory of my Kutty Preface It is well known that the celebrated Hille-Yosida theorem, discovered independently by Hille [1] and Yosida [1], gave the first characterization of the infinitesimal generator of a strongly continuous semigroup of contractions This was the beginning of a systematic development of the theory of semigroups of bounded linear operators The bounded linear operator Aλ appearing in the sufficiency part of Yosida’s proof of this theorem is called the Yosida approximation of A; see Pazy [1] The objective of this research monograph is to present a systematic study on Yosida approximations of stochastic differential equations in infinite dimensions and applications On the other hand, a study on stochastic differential equations (SDEs) in infinite dimensions was initiated in the mid-1960s; see, for instance, Curtain and Falb [1, 2], Chojnowska-Michalik [1], Ichikawa [1–4], and Metivier and Pistone [1] using the semigroup theoretic approach and Pardoux [1] using the variational approach of Lions [1] from the deterministic case Note, however, that a strong foundation of SDEs, in infinite dimensions in the semilinear case was first laid by Ichikawa [1–4] It is also worth mentioning here the earlier works of Haussman [1] and Zabczyk [1] All these aforementioned attempts in infinite dimensions were generalizations of stochastic ordinary differential equations introduced by K Itô in the 1940s and independently by Gikhman [1] in a different form, perhaps motivated by applications to stochastic partial differential equations in one dimension, like heat equations Today, SDEs in the sense of Itô, in infinite dimensions are a wellestablished area of research; see the excellent monographs by Curtain and Pritchard [1], Itô [1], Rozovskii [1], Ahmed [1], Da Prato and Zabczyk [1], Kallianpur and Xiong [1], and Gawarecki and Mandrekar [1] Throughout this book, we shall use mainly the semigroup theoretic approach as it is our interest to study mild solutions of SDEs in infinite dimensions However, we shall also use the variational approach to study stochastic evolution equations with delay and multivalued stochastic partial differential equations To the best of our knowledge, Ichikawa [2] was the first to use Yosida approximations to study control problems for SDEs It is a well-known fact that Itô’s formula is not applicable to mild solutions; see Curtain [1] This motivates the vii viii Preface need to look for a way out, and Yosida approximations come in handy as these Yosida approximating SDEs have the so-called strong solutions and Itô’s formula is applicable only to strong solutions Yosida approximations, since then, have been used widely for various classes of SDEs; see Chapters and below, to study many diverse problems considered in Chapters and The book begins in Chapter with a brief introduction mentioning motivating problems like heat equations, an electric circuit, an interacting particle system, a lumped control system, and the option and stock price dynamics to study the corresponding abstract stochastic equations in infinite dimensions like stochastic evolution equations including such equations with delay, McKean-Vlasov stochastic evolution equations, neutral stochastic partial differential equations, and stochastic evolution equations with Poisson jumps The book also deals with stochastic integrodifferential equations, multivalued stochastic differential equations, stochastic evolution equations with Markovian switchings driven by Lévy martingales, and time-varying stochastic evolution equations In Chapter 2, to make the book as self-contained as possible and reader friendly, some important mathematical machinery, namely, concepts and definitions, lemmas, and theorems, that will be needed later on in the book will be provided As the book studies SDEs using mainly the semigroup theory, it is first intended to provide this theory starting with the fundamental Hille-Yosida theorem and then define precisely the Yosida approximations as well as such approximations for multivalued monotone maps There is an interesting connection between the semigroup theory and the probability theory Using this, we shall also delve into some recent results on asymptotic expansions and optimal convergence rate of Yosida approximations Next, some basics from probability and analysis in Banach spaces are considered like those of the concepts of probability and random variables, Wiener process, Poisson process, and Lévy process, among others With this preparation, stochastic calculus in infinite dimensions is dealt with next, namely, the concepts of Itô stochastic integral with respect to Q-Wiener and cylindrical Wiener processes, stochastic integral with respect to a compensated Poisson random measure, and Itô’s formulas in various settings In some parts of the book, the theory of stochastic convolution integrals is needed So, we then state some results from this theory without proofs This chapter coupled with Appendices dealing with multivalued maps, maximal monotone operators, duality maps, random multivalued maps, and operators on Hilbert spaces, more precisely, notions of trace class operators, nuclear and Hilbert-Schmidt operators, etc., should give a sound background Since there are many excellent references on this subject matter like Curtain and Pritchard [1], Ahmed [1], Altman [1], Bharucha-Reid [1], Bichteler [1], Da Prato and Zabczyk [1, 2], Dunford and Schwartz [1], Ichikawa [3], Gawarecki and Mandrekar [1], Joshi and Bose [1], Pazy [1], Barbu [1, 2], Knoche [1], Peszat and Zabczyk [1], Prévơt and Rưckner [1], Padgett [1], Padgett and Rao [1], Stephan [1], Tudor [1], Yosida [1], and Vilkiene [1–3], among others, the objective here is to keep this chapter brief Chapter addresses the main results on Yosida approximations of stochastic differential equations in infinite dimensions in the sense of Itô The chapter begins by motivating this study from linear stochastic evolution equations After Preface ix a brief discussion on linear equations, the pioneering work by Ichikawa (1982) on semilinear stochastic evolution equations is considered in detail next We introduce Yosida approximating system as it has strong solutions so that Itô’s formula can be applied It will be interesting to show that these approximating strong solutions converge to mild solutions of the original system in mean square This result is then generalized to stochastic evolution equations with delay We next consider a special form of a stochastic evolution equation that is related to the so-called McKean-Vlasov measure-valued stochastic evolution equation We introduce Yosida approximations to this class of equations, showing their existence and uniqueness of strong solutions and also the mean-square convergence of these strong solutions to the mild solutions of the original system We next generalize this theory to McKean-Vlasov-type stochastic evolution equations with a multiplicative diffusion In the rest of the chapter, we consider Yosida approximation problems of many more general stochastic models including neutral stochastic partial functional differential equations, stochastic integrodifferential equations, multivalued-valued stochastic differential equations, and time-varying stochastic evolution equations The chapter concludes with some interesting Yosida approximations of controlled stochastic differential equations, notably, stochastic evolution equations driven by stochastic vector measures, McKean-Vlasov measure-valued evolution equations, and also stochastic equations with partially observed relaxed controls In Chapter 4, we consider Yosida approximations of stochastic differential equations with Poisson jumps More precisely, we introduce Yosida approximations to stochastic delay evolution equations with Poisson jumps, stochastic evolution equations with Markovian switching driven by Lévy martingales, multivaluedvalued stochastic differential equations driven by Poisson noise, and also such equations with a general drift term with respect to a general measure As before, we shall also obtain mean-square convergence results of strong solutions of such Yosida approximate systems to mild solutions of the original equations In Chapter 5, many consequences and applications of Yosida approximations to stochastic stability theory are given First, we consider the pioneering work of Ichikawa (1982) on exponential stability of semilinear stochastic evolution equation in detail and also the stability in distribution of mild solutions of such semilinear equations As an interesting consequence, exponential stabilizability for mild solutions of semilinear stochastic evolution equations is considered next Since an uncertainty is present in the system, we obtain robustness in stability of such systems with constant and general decays This study is then generalized to stochastic equations with delay; that is, polynomial stability with a general decay is established for such delay systems Consequently, robust exponential stabilization of such delay equations is obtained Subsequently, stability in distribution is considered for stochastic evolution equations with delays driven by Poisson jumps Moreover, moment exponential stability and also almost sure exponential stability of sample paths of mild solutions of stochastic evolution equations with Markovian switching with Poisson jumps are dealt with We also study the weak convergence of induced probability measures of mild solutions of McKean-Vlasov stochastic evolution equations, neutral stochastic partial functional differential equations, x Preface and stochastic integrodifferential equations Furthermore, the exponential stability of mild solutions of McKean-Vlasov-type stochastic evolution equations with a multiplicative diffusion, stochastic integrodifferential evolution equations, and timevarying stochastic evolution equations are considered Finally, in Chapter 6, it will be interesting to consider some applications of Yosida approximations to stochastic optimal control problems like optimal control over finite time horizon, a periodic control problem of stochastic evolution equations, and an optimal control problem of McKean-Vlasov measure-valued evolution equations Moreover, we also consider some necessary conditions of optimality of relaxed controls of stochastic evolution equations The chapter as well as the book concludes with optimal feedback control problems of stochastic evolution equations driven by stochastic vector measures I have tried to keep the work of various authors drawn from all over the literature as original as possible I thank sincerely all of them whose work have been included in the book with due citations they deserve in the bibliographical notes and remarks and elsewhere I believe to the best of my knowledge that I have covered in this monograph all the work that I have known There may be more interesting materials, but it is impossible to include all in one book I apologize to those authors in case I have missed out their work This is certainly not deliberate Mexico City, Mexico July 22, 2016 T E Govindan Bibliography 393 [2] Almost sure asymptotic stability of stochastic partial differential equations with jumps, SIAM J Control Optimiz 49, No 2, 771–787, 2011 V Barbu [1] Analysis and Control of nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, Vol 190, Academic Press, Boston, 1993 [2] Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, NY, 2010 H Becker and V Mandrekar [1] On the existence of optimal random controls, J Math Mechan 18, 1151–1166, 1969 V Bentkus [1] A new method for approximations in probability and operator theories, Lithuanian Math J 43, 367–388, 2003 V Bentkus and V Paulauskas [1] Optimal error estimates in operator-norm approximations of semigroups, Letters Math Physics 68, 131–138, 2004 A T Bharucha-Reid [1] Random Integral Equations, Academic Press, NY, 1972 F Biagini, Y Hu, B Oksendal and T Zhang [1] Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, 2008 K Bichteler [1] Stochastic Integration with Jumps, Encyclopedia of Mathematics and its Applications, Vol 89, Cambridge Univ Press, Cambridge, 2002 F Black and M Scholes [1] The pricing of options and corporate liabilities, J Political Economy 81, 637–659, 1973 V S Borkar and T E Govindan [1] Optimal control of semilinear stochastic evolution equations, Nonlinear Anal.: Theory, Methods Appl 23, 15–35, 1994 Z Brezniak, W Liu and J Zhu [1] Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, arXiv.11080343v4.2011 Z Brezniak, E Hausenblas and J Zhu [1] Maximal inequality of stochastic convolution driven by compensated random measures in Banach spaces, arXiv:1005.1600 T Caraballo and J Real [1] Partial differential equations with delayed random perturbations: existence, uniqueness and stability of solutions, Stochastic Anal Appl 11, 497–511, 1993 T Caraballo, M J Garrido-Atienza and T Taniguchi [1] The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal 74, 3671–3684, 2011 394 Bibliography C Castaing and M Valadier [1] Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol 580, Springer-Verlag, Berlin, 1977 M F Chen [1] From Markov Chains to Non-Equilibrium Particle Systems, World Scientific, Singapore, 1992 G Chen and R Grimmer [1] Semigroups and integral equations, J Integral Eqns 2, 133–154, 1980 A Chojnowska-Michalik [1] Stochastic Differential Equations in Hilbert Spaces and Their Applications, Thesis, Institute of Mathematics, Polish Academy of Sciences, 1976 P L Chow [1] Stochastic Partial Differential Equations, CRC Press, 2007 E N Chukwu [1] Stability and Time-Optimal Control of Hereditary Systems, Academic Press, NY, 1992 R Cont and P Tankov [1] Non-parametric calibration of jump-diffusion option pricing models, J Computational Finance 7, No 3, 2004 S G Cox [1] Stochastic Differential Equations in Banach Spaces: Decoupling, Delay Equations and Approximations in Space and Time, Doctoral thesis, Delft University of Technology, Vancouver, Canada, 2012 R F Curtain [1] Stability of stochastic partial differential equation, J Math Anal Appl 79, 352–369, 1981 R F Curtain and P L Falb [1] Stochastic differential equations in Hilbert spaces, J Differential Eqns 10, 412–430, 1971 [2] Itô’s lemma in infinite dimensions, J Math Anal Appl 31, 434–448, 1970 R F Curtain and A J Pritchard [1] Infinite Dimensional Linear System Theory, Lecture Notes in Control and Information Sciences No 8, Springer-Verlag, NY, Berlin, 1978 [2] The infinite dimensional Riccati equation for systems defined by evolution operators, SIAM J Control Optimization 14, 951–983, 1976 D A Dawson and J Gärtner [1] Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics 20, 247–308, 1987 G Da Prato [1] Synthesis of optimal control for an infinite dimensional periodic problem, SIAM J Control Optimiz 25, 706–714, 1987 [2] An Introduction to Infinite-Dimensional Analysis, Springer, Berlin, 2006 G Da Prato, A Jentzen and M Röckner [1] A mild Itô formula for SPDEs, Research Report No 2013–46, ETH, Switzerland, 2013 Bibliography 395 G Da Prato and J Zabczyk [1] Stochastic Equations in Infinite Dimensions, Cambridge Univ Press, 1992 [2] A note on stochastic convolution, Stochastic Anal Appl 10, 143–153, 1992 [3] Second Order Partial Differential Equations in Hilbert spaces, Cambridge Univ Press, 2004 [4] Ergodicity for Infinite Dimensional Systems, Cambridge Univ Press, 1996 T Datko [1] Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J Math Anal 3, 428–445, 1972 D Duffie, J Pan and K Singleton [1] Transform analysis and asset pricing for affine jump-diffusions, Econometrica 68, No 6, 1343–1376, 2000 N Dunford and J T Schwartz [1] Linear Operators, Part 1, Interscience Publishers, NY, 1958 L C Evans [1] Partial Differential Equations, 2nd Edn., Graduate Studies in Mathematics, Vol 19, AMS, Providence, RI, 2010 E Fama [1] Efficient capital markets: A review of theory and empirical work, J of Finance 25, 383–417, 1970 T Funaki [1] Random motions of string and related stochastic evolution equations, Nagoya Math J 89, 129–193, 1983 J Gärtner [1] On the McKean-Vlasov limit for interacting diffusions I, II, Akademie Der Wissenschaften Der DDR, Karl-Weierstrass-Institute Für Mathematik, Berlin, 1986 L Gawarecki and V Mandrekar [1] Stochastic Differential Equations in Infinite Dimensions: with applications to stochastic partial differential equations, Springer, Berlin, 2011 H Gemam and A Roncoroni [1] Understanding the fine structure of electricity prices, J of Business 79, No 3, 1225–1261, 2006 I I Gikhman [1] A method of constructing random processes, Dokl Acad Nauk SSSR, 58, 961–964, 1946 I I Gikhman and A V Skorokhod [1] Stochastic Differential Equations, Springer-Verlag, 1972 [2] The Theory of Stochastic Processes I, Springer-Verlag, NY, 1974 (Translated from the Russian by S Kotz) T E Govindan [1] Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77, 139–154, 2005 [2] Stability of stochastic differential equations in a Banach space, in Mathematical Theory of Control (Editors Mohan C Joshi and A.V Balakrishnan), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Vol 142, 161–171, 1992 396 Bibliography [3] Autonomous semilinear stochastic Volterra integrodifferential equations in Hilbert spaces, Dynamic Systems Appl 3, 51–74, 1994 [4] Semigroup theoretic approach to quasilinear stochastic hyperbolic Ito integrodifferential equations, J Ramanujan Math Soc 10, 31–49, 1995 [5] Weak convergence of probability measures of Yosida approximate mild solutions of neutral SPDEs, Statist Probab Letters 95, 26–32, 2014 [6] Existence and uniqueness of solutions of SPDEs in infinite dimensions, in Optimization, control, and applications of stochastic systems: in honor of Onésimo Hernández-Lerma, Editors Hernández-Hernández, D and Minjárez-Sosa, A., Birkhauser, USA, Chapter 7, 111–124, 2012 [7] Exponential stability in mean-square of parabolic quasilinear stochastic delay evolution equations, Stochastic Anal Appl 17, 443–461, 1999 [8] Existence and stability of solutions of stochastic semilinear functional differential equations, Stochastic Anal Appl 20, No 6, 1257–1280, 2002 [9] Stability and Optimal Control of Stochastic Dynamical Systems, Ph.D Thesis, Indian Institute of Technology, Bombay, India, February 1991 T E Govindan and N U Ahmed [1] A note on exponential state feedback stabilizability by a Razumikhin type theorem of mild solutions of SDEs with delay, Statist Probab Letters, 82, 1303–1309, 2012 [2] Robust stabilization with a general decay of mild solutions of stochastic evolution equations, Statist Probab Letters 83, 115–122, 2013 [3] On Yosida approximations of abstract McKean-Vlasov stochastic evolution equations, Stochastic Anal Appl., 33, 383–398, 2015 I Gyöngy and N V Krylov [1] On stochastic equations with respect to semimartingales II, Itô formula in Banach spaces, Stochastics 6, Nos 3–4, 153–173, 1981/1982 I Gyöngy [1] On stochastic equations with respect to semimartingales III, Stochastics 7, No 4, 231–254, 1982 O Hans [1] Inverse and adjoint transforms of linear bounded transforms, Trans 1st Prague Conf on Information Theory, Statist Decision Functions and Random Processes (1956), 127–133, 1957 [2] Reduzierende zufällige Transformationen, Czechoslovak Math J (82), 154–158, 1957 U G Haussmann [1] Asymptotic stability of the linear Ito’s equation in infinite dimensions, J Math Anal Appl 65, 219–235, 1978 E Hernández and H R Henriquez [1] Existence results for partial neutral functional differential equations with unbounded delay, J Math Anal Appl 221, 452–475, 1998 E Hille [1] Functional Analysis and Semi-groups, 3rd Print, Amer Math Soc Colloq Publ Vol 31, NY, 1948 E Hille and R S Phillips [1] Functional Analysis and Semi-groups, Amer Math Soc Colloq Publ Vol 31, NY, 1974 Bibliography 397 C J Himmelberg [1] Measurable relations, Fund Math 87, 53–72, 1975 S Hu and N S Papageorgiou [1] Handbook of Multivalued Analysis, Vol 1, Theory, Kluwer Academic Publ., Dordrecht/ Boston/London, 1997 Y Hu [1] On the solution of forward-backward SDEs with monotone and continuous coefficients, Nonlinear Anal 42, 1–12, 2000 A Ichikawa [1] Linear stochastic evolution equations in Hilbert space, J Differential Eqns 28, 266–277, 1978 [2] Dynamic programming approach to stochastic evolution equations, SIAM Journal on Control and Optimization, 17, 152–174, 1979 [3] Stability of semilinear stochastic evolution equations, J Math Anal Appl 90, 12–44, 1982 [4] Semilinear stochastic evolution equations: boundedness, stability and invariant measures, Stochastics 12, 1–39, 1984 [5] Optimal control of a linear stochastic evolution equation with state and control dependent noise, Proc IMA Conference, Recent Theoretical Developments in Control, Leicester, England, Academic Press, 1976 N Ikeda and S Watanabe [1] Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981 K Itô [1] Differential equations determining Markov processes, Zenkoku Shijo Sugaku Danwakai No 1077, 1352–1400, 1942 [2] Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, Vol 47 of CBMS Notes, SIAM, Baton Rouge, 1984 S Itoh [1] Nonlinear random equations with monotone operators in Banach spaces, Math Ann 236, No 2, 133–146, 1978 M C Joshi and R K Bose [1] Some Topics in Nonlinear Functional Analysis, Halsted Press, 1985 M C Joshi and T E Govindan [1] Local stability properties of a feed-back system- Deterministic and random, Stochastic Anal Appl 8, 397–430, 1990 G Kallianpur and J Xiong [1] Stochastic Differential Equations in Infinite Dimensional Spaces, Vol 26, Institute of Math Statist., USA, 1995 O Kallenberg [1] Foundations of Modern Probability, 2nd Edn., Probability and Its Applications, SpringerVerlag, NY, 2002 D Kannan and A T Bharucha-Reid [1] On a stochastic integrodifferential evolution equation of Volterra type, J Integral eqns 10, 351–379, 1985 398 Bibliography A Karamolegos and D Kravvaritis [1] Nonlinear random operator equations and inequalities in Banach spaces, Internat J Math Math Sci 15, 111–118, 1992 R Khasminskii [1] Stochastic Stability of Differential Equations, 2nd Edition, Springer, NY, 2012 C Knoche [1] Mild solutions of SPDEs driven by Poisson noise in infinite dimensions and their dependence on initial conditions, Ph.D thesis, Facultat fur Mathematik, Universitat Bielefeld, 2005 P Kotelenez [1] Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations, Springer, 2008 S G Kou [1] A jump-diffusion model for option pricing, Management Science 48, No 8, 1086–1101, 2002 D Kravvaritis [1] Nonlinear random equations involving operators of monotone type, J Math Anal Appl 114, No 2, 295–304, 1986 P Kree [1] Diffusion equation for multivalued stochastic differential equations, J Funct Anal 49, 73–90, 1982 P Kree and C Soize [1] Mathematics of Random Phenomena, Reidel Publishing Company, 1986 H H Kuo [1] On integral contractors, J Integral Eqns 1, 35–46, 1979 T G Kurtz and J Xiong [1] A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems, Communications Math Sci., 2, 325–358, 2004 P Li and N U Ahmed [1] Feedback stabilization of some nonlinear stochastic systems on Hilbert space, Nonlinear Anal.: Theory, Methods Appl 17, 31–43, 1991 J L Lions [1] Equations Differentielles Operationelles et Problemes aux Limites, Springer, Berlin, 1961 K Liu [1] Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics and Stochastics Reports 63, 1–26, 1998 [2] Stability of Infinite Dimensional Stochastic Differential Equations with Applications, CRC Press, USA, 2006 [3] Carathéodory approximate solutions for a class of semilinear stochastic evolution equations with time delays, J Math Anal Appl 220, 349–364, 1998 R Liu and V Mandrekar [1] Stochastic semilinear evolution equations: Lyapunov function, stability and ultimate boundedness, J Math Anal Appl 212, 537–553, 1997 Bibliography 399 W Liu and M Röckner [1] SPDE in Hilbert space with locally monotone coefficients, J Functional Anal 259, No 11, 2902–2922, 2010 W Liu and M Stephan [1] Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple, J Math Anal.Appl 410, 158–178, 2014 J Luo [1] Exponential stability for stochastic neutral partial functional differential equations, J Math Anal Appl., 355, 414–425, 2009 J Luo and K Liu [1] Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stochastic Proc Appl 118, 864–895, 2008 V Mandrekar and B Rüdiger [1] Stochastic Integration in Banach Spaces: Theory and Applications, Springer, 2015 X Mao and C Yuan [1] Stochastic Differential Equations with Markovian Switching, Imperial College, London, 2006 C Marinelli, C Prévơt and M Rưckner [1] Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, J Functional Anal 258, No 2, 616–649, 2010 H P McKean [1] A class of Markov processes associated with nonlinear parabolic equations, Proc N.A.S 56, 1907–1911, 1966 M A McKibben [1] Discovering Evolution Equations with Applications, Volume - Deterministic Equations, CRC Press, USA, 2011 [2] Discovering Evolution Equations with Applications, Volume 2- Stochastic Equations, CRC Press, USA, 2011 R C Merton [1] Option pricing when underlying stock returns are discontinuous, J Financial Economics 3, 125–144, 1976 [2] Optimum consumption and portfolio rules in a continuous-time model, J Economic Theory 3, 373–413, 1971 [3] Theory of rational option pricing, Bell J Economics and Management Science 4, 141–183, 1973 M Metivier and G Pistone [1] Une formula d’isometrie pour I’integrale stochatique et equations d’évolution lineares stochastiques, Z Warsch Verw Gebiete 33, 1–18, 1975 R K Miller and R L Wheeler [1] Asymptotic behavior for a linear Volterra integral equation in Hilbert space, J Differential Eqns 23, 270–284, 1977 400 Bibliography I S Molcanov [1] Theory of Random Sets, Springer, London, 2005 J R Norris [1] Markov Chains, Cambridge Univ Press, 1998 M M Olama, S M Djouadi and C D Charalambos [1] Stochastic power control for time-varying long-term fading wireless networks, Univ of Cyprus, Nicosia, Cyprus, preprint 2006/2007 W J Padgett [1] The method of random contractors and its applications to random nonlinear equations, in Probabilistic Analysis and Related Topics (Edn A.T Bharucha-Reid) Vol 3, Academic Press, NY, 1983 W J Padgett and A N V Rao [1] Solution of a stochastic integral equation using integral contractors, Information and Control 41, 56–66, 1979 E Pardoux [1] Equations aux Dérivées Partielles Stochastiques Non linéaires Monotones, Thése, Université Paris XI, 1975 [2] Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3, 127–167, 1979 V Paulauskas [1] On operator-norm approximation of some semigroups by quasi-sectorial operators, J Funct Anal 207, 58–67, 2004 A Pazy [1] Semigroups of Linear Operators and Applications to Partial Differential Equations, SpringerVerlag, NY, USA, 1983 S Peszat and J Zabczyk [1] Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equations Approach, Cambridge Univ Press, 2007 R Pettersson [1] Yosida approximations for multivalued stochastic differential equations, Stochastics Stochastics Reports, 52, 107–120, 1995 E Platen and N Bruti-Liberati [1] Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, 2010 C Prévơt and M Rưckner [1] A Concise Course on Stochastic Partial Differential Equations, Springer, 2007 P E Protter [1] Stochastic Integration and Differential Equations, 2nd Edn., Springer, 2004 B S Razumikhin [1] On stability of systems with a delay, Prikl Mat Meh 20, 500–512, 1956 Bibliography 401 [2] Application of Liapunov’s methods to problems in stability of systems with a delay, Automat i Telemeh 21, 740–749, 1960 M Röckner and T Zhang [1] Stochastic evolution equation of jump type: existence, uniqueness and large deviation principles, Potential Anal 26, 255–279, 2007 B L Rozovskii [1] Stochastic Evolution Equations, Linear Theory and Applications to Nonlinear Filtering, Kluwer, 1990 P A Samuelson [1] Proof that properly anticipated prices fluctuate randomly, Industrial Management Review 6, 41–49, 1965 M Stephan [1] Yosida Approximations for Multivalued Stochastic Differential Equations on Banach Spaces via a Gelfand Triple, Doctoral Dissertation, Universit at Bielefeld, 2012 A V Svishchuk and Yu I Kazmerchuk [1] Stability of stochastic delay equations of Itô form with jumps and Markovian switchings, and their applications in finance, Theor probab Math Statist 64, 167–178, 2002 H Tanabe [1] Equations of Evolution, Pitman Research Notes in Mathematics, 1979 T Taniguchi [1] Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics Stochastics Reports 53, 41–52, 1995 [2] Almost sure exponential stability for stochastic partial functional differential equations, Stochastic Anal Appl 16, 965–975, 1998 L Tubaro [1] An estimate of Burkholder type for stochastic processes defined by the stochastic integral, Stochastic Anal Appl 2, 187–192, 1984 C Tudor [1] Optimal control for an infinite-dimensional periodic problem under white noise perturbations, SIAM J Control Optimiz 28, 253–264, 1990 M Vilkiene [1] Another approach to asymptotic expansions for Euler’s approximations of semigroups, Lithuan Math J 46, No 2, 217–232, 2006 [2] Asymptotic expansions for Yosida approximations of semigroups, Liet Matem Rink., 48/49, 78–83, 2008 [3] Optimal convergence rate for Yosida approximations of bounded holomorphic semigroups Lithuan Math J 49, No 2, 234–239, 2009 [4] Investigations of the accuracy of approximations of semigroups, Doctoral Dissertation, Vilnius University, 2011 J Wu [1] Theory and Applications of Partial Functional Differential Equations, Springer, NY, 1996 402 Bibliography K Yosida [1] On the differentiability and representation of one parameter semi-groups of linear operators, J Math Soc Japan, 1, 15–21, 1948 [2] Functional Analysis, 6th Edn., Springer-Verlag, 1980 C Yuan and J Bao [1] On the exponential stability of switching-diffusion processes with jumps Quart Appl Math 71, No 2, 311–329, 2013 J Zabczyk [1] On stability of infinite dimensional stochastic systems, Probability Theory, (Z Ciesielski, Ed.), Vol 5, pp 273–281, Banach Center Publications, Warsaw, 1979 E Zeidler [1] Nonlinear Functional Analysis and its Applications, IIA, Linear Monotone Operators, Springer-Verlag, NY, 1990 [2] Nonlinear Functional Analysis and its Applications, IIB, Nonlinear Monotone Operators, Springer-Verlag, NY, 1990 Index Symbols C0 -semigroup, 12 differentiable semigroup, 27 exponentially stable, 13, 98, 185 of contractions, 13 pseudo-contraction, 13, 204, 214, 219, 317 uniformly bounded, 13 A a version, 35 adapted process, 35 admissible control, 334, 350 analytic semigroup, Ascoli-Arzella theorem, 367 asymptotic expansion, 21, 29 asymptotic expansions, 27 attainable set, 350 B Banach fixed point theorem, 100, 108 Banach-Alaoglu theorem, 141, 164 Bellman-Gronwall lemma, 32, 95, 119, 128, 134, 149, 159, 188, 227, 233 Bochner integrable, 34 Bolza problem, 356 Borel-Cantelli Lemma, 119, 248, 275, 341 bounded, 137, 222 bounded holomorphic semigroup, 22 bounded linear operators, 11, 15, 44 bounded stochastic integral contractor, 67, 154 is regular, 156 Burgers type equation, 282 Burkholder type inequality, 62, 90, 95, 152, 216 for a Poisson integral, 63 C c`adl`ag, 36 Cauchy, 91, 117, 255, 294 Cauchy’s formula, 33, 118 central limit theorem, 21 Chapman-Kolmogorov equation, 249 Chebyshev’s inequality, 251, 289, 341 closed, 17 weakly closed, 350 closed convex hull, 19 coercive, 20, 87, 137, 222, 238 compact set, 170 compact support, 167, 179 control law, 337 converges strongly, 33 converges weakly, 33 counting Poisson random measure, 10 covariance, 34 joint covariance, 35 D Davis’s inequality, 226 demicontinuous, 17, 19, 381 dense, 184 detectability, 338 Dhumels formula, 195 differentiable, 22 Dirichlet boundary condition, 263, 269, 354 © Springer International Publishing Switzerland 2016 T E Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling 79, DOI 10.1007/978-3-319-45684-3 403 404 dissipative, 187 duality mapping, 17, 377 Dunford-Pettis theorem, 160 dynamic programming method, 334 E electric circuit, empirical measure, empirical measure-valued process, ergodic property of Markov chains, 327 Euler beam equation, 303 Euler’s approximations, 21 evolution operator almost strong, 31, 152 mild, 30 quasi, 31 strong, 31, 338 expectation, 34 conditional, 35, 177 exponential martingale inequality with jumps, 324 exponential stability of moments of SEEs, 242 of moments of stochastic integrodifferentiable equations, 309 of moments of time-varying SEEs, 331 of sample paths of McKean-Vlasov equations, 302 of sample paths of SEEs, 249 of sample paths of SEEs with Markovian switching driven by Lévy martingales, 316, 322 exponential stabilizability of moments of SEEs, 259 of moments of SEEs with delay, 298 F factorization method, 60 Fatou’s lemma, 266 feedback control, 6, 182, 259, 334 PID feedback control, filtration, 39, 42 right-continuous filtration, 42 forward Kolmogorov equation, 167, 179 Fréchet derivative, 32, 65, 106, 162, 242, 309, 334 Fubini’s theorem, 142, 209, 364 G Gateaux derivative, 357 Gaussian law, 37 Index Gelfand triple, 45, 135 graph of the operator A, 17 H Hölder’s inequality, 61, 62, 115, 148, 317 Hahn-Banach theorem, 348 Hausdorff space, 161 heat equation, controlled stochastic heat equation, 260, 298 stochastic heat equation, 2, 70, 248, 310 hemicontinuous, 32 hereditary control, Hilbert-Schmidt operators, 46, 135, 170, 370 Hille-Yosida Theorem, 13, 16, 25, 82 hyperbolic equation, I improper Riemann integral, 14 increasing process, 42 independence, 35 infinitesimal generator of the semigroup, 12 integral operator, 153 integro-differential identity, 23, 25 interacting particle system , isometric property, 229 isometrically isomorphic, 185 Itô stochastic integral, 45 w.r.t a Q-Wiener process , 46 w.r.t a cylindrical Wiener process, 50 Itô’s formula, 54, 71, 106, 140, 224, 242, 273, 309, 315, 335 for a Q-Wiener process, 54 for a compensated Poisson process, 56 for a cylindrical Wiener process, 55 iterations, 109, 114 J Jordan decomposition, 98 K Kirchhoff’s law, L Lévy martingales, 211 Lévy process, 43 Lévy-Itô decomposition, 44 Lévy-Khinchine formula, 43 Lagrange problem, 361 law of large numbers, 345 Index Lebesgue dominated convergence theorem, 81, 101, 128, 134, 145, 159, 217, 243, 310 Levy’s Theorem, 37 linear growth condition, 93, 100, 108, 113, 204, 222, 241 Lipschitz condition, 67, 92, 93, 101, 108, 113, 136, 204, 222, 238, 241 one-sided, 137 local martingale property, 49 lower semicontinuous, 200, 353 lumped control systems, Lyapunov exponent, 263, 271, 313 Lyapunov function, 266 Lyapunov functional, 281 M Markov chain, 56 irreducible, 218 Markov inequality, 341 Markov property, 293, 339 Markovian switching, 211 martingale, 36 semimartingale, 357 submartingale, 36 Mazur theorem, 142 McKean-Vlasov theory, measurable, 35 Borel, 83, 113, 193 Effros, 379 progressively, 36, 136, 142, 146, 223 progressively Effros, 136, 237 strongly, 114 measurable selection, 379 measurable space, 33 measure characteristic, 10, 42 Dirac delta, 124, 164, 167 Dirac delta measure, 175 intensity, 43 invariant, 183, 337 Lévy, 317 Lebesgue, 42, 58, 197 stochastic vector, 171 metric, 98, 110 Fréchet metric, 123 Hausdorff metric, 366 Prohorov metric, 366 minimal selection, 145 mobile communication, 354 monotone, 17 maximal monotone, 17, 19, 136, 221, 238, 375 strictly, 378 405 multivalued map, 373 multivalued operator, 17 N necessary conditions of optimality, 356 Newton-Leibnitz formula, 24 non-decreasing process, 36, 237 nuclear operator, 34, 369 O obstacle avoidance problem, 365 optimal control, 345, 350, 353, 354, 362, 364, 365 optimal convergence rate, 26 optimal error bound, 21 optimality lemma, 334 option price dynamics, Ornstein-Uhlenbeck process, 185 Ornstein-Uhlenbeck semigroup, 183, 184 orthonormal basis, 40, 105, 369 P Poincaré’s inequality, 328 point function, 41 Poisson integral, 53 Poisson jumps, 203, 295 Poisson noise, 221 Poisson point process, 41 Poisson process, Poisson random measure, 40 compensated Poisson random measure, 10, 42 Polish space, 379 polynomial decay, 271 polynomial nonlinearities, 162 predictable process, 47, 60 probability distribution, 5, 97, 108 probability measure, 33 complete, 33 probability measure space, 33 Q quasi-generator, 31 quasi-leftcontinuous, 42 R Radon-Nikodym derivatives, 238 Radon-Nikodym property, 160 406 Radon-Nikodym theorem, 238 random contractor, 64 random evolution equation, 124 random inclusion, 381 random multivalued operator, 379 random operator, 44 random variable, 33 Bochner, 33 Pettis, 34 Razumikhin type theorem, 296 Razumikhin-Lyapunov function, 321 regularity, 194 regulator problem, 335 relaxed control, 197 resolvent of A, 13 of a maximal monotone operator A, 18 resolvent set of A, 13 Riccati equation, 335, 341 Riesz isomorphism, 234 Riesz theorem, 362 robustness in stability with a constant decay, 258 with a general decay, 263 S sample path continuity, 243 a modification, 245 sample path stability with a general decay, 272 Skorokold extension, 180 Sobolev embedding, 234 Sobolev space, 234 solution classical, 170 generalized, 162, 173 mild, 70, 75, 83, 93, 99, 106, 108, 113, 129, 153, 184, 205, 213, 219, 242, 261, 333 of a multivalued equation with Poisson noise, 222 of a multivalued equation with white noise, 136 of multivalued equation driven by Poisson noise with a general drift term, 237 periodic solution, 341 strong, 69, 74, 86, 93, 94, 99, 104, 105, 129, 135, 153, 157, 204, 213 weak, 170, 186 square bracket, 53 square root, 370 stabilizability, 338 stable in distribution, 249 of SEEs, 256 Index of SEEs with delay driven by Poisson jumps, 294 state dependent noise, 69 stochastic convolution integrals, 59 stochastic differential equations multivalued SDEs, 20 stochastic differential equations with Poisson jumps, 10 stochastic evolution equation, stochastic evolution equations with delay, stochastic evolution equations with variable delay, 92 stochastic Fubini theorem, 58, 60, 209 for Poisson integral, 58 stochastic integrodifferential equation, stochastic partial differential equations, neutral stochastic partial differential equation, stochastic porous media equations, 233 stochastic process, 35 stock price dynamics, stopping times, 44, 225, 228, 254, 291 strongly continuous semigroup, 12 subnet, 164 switching diffusion processes, 218 T target set, 365 Taylor series, 23, 326 terminal control problem, 352 theory of lifting, 160 tight, 251, 308 topological compactification, 161 of Stone-Cech, 161 trace, 35 transition semigroup, 183 Tychonoff space, 161 U uniformly continuous semigroup, 12, 15 uniformly continuous semigroup of contractions, 15, 22 uniformly convex, 17 upper semicontinuous, 353, 364 V variance, 34 variational method, 86 Volterra integrodifferential equation, Volterra series, 68 Index W wave equation, 269 weak convergence of induced probability measures of McKean-Vlasov equations, 301 of neutral SPDEs, 305 weak limit, 145 weakly compact subset, 350 Wiener process, 2, Q-Wiener process, 38 cylindrical Wiener process, 5, 40, 97, 182 Y Yosida approximation, 15, 60, 163, 200, 235, 251, 290, 338 of measure-valued McKean-Vlasov evolution equation, 186 of a periodic control problem, 346 of Itô stochastic integrodifferential equation, 134 of linear stochastic integrodifferential equation, 126 of McKean-Vlasov equation, 104 of McKean-Vlasov equation with multiplicative diffusion, 111 of multivalued SDEs with white noise, 138 407 of multivalued SPDEs driven by Poisson noise, 223 of multivalued SPDEs driven by Poisson noise with a general drift term, 239 of neutral SPDEs, 119 of SDEs with delay with Markovian switching driven by Lévy processes, 215 of SEEs driven by stochastic vector measures, 175 of SEEs with constant delay, 85 of SEEs with delay with Poisson jumps, 208 of SEEs with time-varying coefficients, 262 of SEEs with variable delay, 94 of semilinear stochastic evolution equations, 79 of semilinear stochastic integrodifferential equations, 132 of stochastic evolution equations, 72 of switching diffusion processes with Poisson Jumps, 220 of time-varying stochastic evolution equations, 157 of a multivalued operator, 17 Young’s inequality, 61, 152, 225, 226 ... study on Yosida approximations of stochastic differential equations in infinite dimensions and applications On the other hand, a study on stochastic differential equations (SDEs) in infinite dimensions. .. Springer International Publishing Switzerland 2016 T E Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic. .. International Publishing Switzerland 2016 T E Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling 79,