Yosida approximations of stochastic differential equations in infinite dimensions and applications

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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 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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,
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