Stochastic Differential Equations: Theory and Applications, a Volume in Honor of Professor Boris L Rozovskii (Interdisciplinary Mathematical Sciences) (Interdisciplinary Mathematical Sciences)

This volume consists of 15 articles written by experts in stochastic analysis. The first paper in the volume, Stochastic Evolution Equations by N V Krylov and B L Rozovskii, was originally published in Russian in 1979. After more than a quarter-century, this paper remains a standard reference in the field of stochastic partial differential equations (SPDEs) and continues to attract the attention of mathematicians of all generations. Together with a short but thorough introduction to SPDEs, it presents a number of optimal, and essentially unimprovable, results about solvability for a large class of both linear and non-linear equations.The other papers in this volume were specially written for the occasion of Prof Rozovskii's 60th birthday. They tackle a wide range of topics in the theory and applications of stochastic differential equations, both ordinary and with partial derivatives. Contents 22 Preface 8 Boris Rozovskii 10 Publications of B. L. Rozovskii 12 1. Stochastic Evolution Equations N. V. Krylov and B. L. Rozovskii 24 1. Introduction 25 1.1. It^o Equations in Banach Spaces 25 1.2. Examples of Stochastic Evolution Equations 25 1.2.1. Linear Equation of Filtring of Di usion Processes 25 1.2.2. Equations of Population Genetics 26 1.2.3. System of Navier-Stokes Equations with Random External Forces 27 1.2.4. Equation of the Free Field 27 1.3. Stochastic Evolution Equations with Bounded Coe cients and Linear Stochastic Evolution Equations 28 1.4. Nonlinear Stochastic Evolution Equations 30 1.5. Content and Organization of the Work 31 2. Stochastic Integration in Hilbert Spaces 32 2.1. Introduction 32 2.2. Stochastic Integrals in Hilbert Spaces 33 2.3. It^o's Formula for the Square of the Norm 39 2.4. Proof of Theorem 2.16 43 3. It^o Stochastic Equations in Banach Spaces and the Meth Monotonicity 50 3.1. Introduction 50 3.2. Assumptions and the Main Results 52 3.3. It^o Equations in Rd 57 3.4. Uniqueness Theorem: A Priori Estimates and Finite-Dimensional Approximations 66 3.5. Existence of Solution and the Markov Property: Passing to the Limit by the Method of Monotonicity 72 4. It^o Stochastic Partial Di er tial Equations 77 4.1. Introduction 77 4.2. First Boundary-Value Problem for Nonlinear Stochastic Parabolic Equations 80 4.3. Cauchy Problem for Linear Second-Order Equations 84 References 88 2. Predictability of the Burgers Dynamics Under Model Uncertainty D. Bl omker and J. Duan 94 1 Introduction 94 2 Linear Theory 96 2.1 Mean Energy 97 2.2 Correlation Function 98 3 Nonlinear Theory 104 3.1 Body forcing - Mean energy bounds 104 3.2 Point forcing - Mean energy bounds 107 3.3 Body forcing - Transient behavior 109 3.4 Trace class noise: Additive vs. multiplicative body noises 110 Acknowledgments 111 References 111 3. Asymptotics for the Space-Time Wigner Transform with Applications to Imaging L. Borcea, G. Papanicolaou, and C. Tsogka 114 1. Introduction 114 2. The parabolic approximation 115 3. Scaling and the asymptotic regime 116 4. The It^o-Liouville equation for the Wigner transform 118 4.1. The white noise limit 118 4.2. The high frequency limit and the space-time Wigner transform 119 4.3. Statement of the strong lateral diversity limit 121 4.4. The mean space-time Wigner transform 122 5. Self-averaging of the smoothed space-time Wigner transform, in the strong lateral diversity limit 123 6. Application to imaging 124 6.1. Migration 125 6.2. Coherent interferometric imaging 127 Acknowledgments 132 References 132 4. The Korteweg-de Vries Equation with Multiplicative Homogeneous Noise A. de Bouard and A. Debussche 136 1. Introduction and statement of the results 136 2. Preliminaries and existence for a truncated equation 139 3. Global existence 151 References 155 5. On Stochastic Burgers Equation Driven by a Fractional Laplacian and Space-Time White Noise Z. Brze zniak and L. Debbi 158 1. Introduction 158 2. Existence of global solutions to approximating equations 161 3. Global solutions to Burgers equations 171 4. Proof of uniqueness 176 A.1. Proof of Lemma 2.3 179 B.1. Gronwall Lemma 180 C.1. Some estimates on stopped stochastic convolutions 180 D.1. Pointwise multiplication in Sobolev spaces 187 References 188 6. Stochastic Control Methods for the Problem of Optimal Compensation of Executives A. Cadenillas, J. Cvitani c, and F. Zapatero 192 1. Introduction 193 2. The Executive 194 2.1. Stock Dynamics 194 2.2. The Problem of the Executive 196 2.3. Optimal E ort and Choice of Projects 196 3. The Company 199 3.1. The Problem of the Company 199 3.2. Optimal Strike Price 199 4. Numerical Computations of the Strike Price 202 5. Price of the Options 205 6. The Case of Additional Cash Compensation 209 7. Conclusions 217 Acknowledgements 218 References 218 7. The Freidlin-Wentzell LDP with Rapidly Growing Coe cients P. Chigansky and R. Liptser 220 1. Introduction 220 2. Notations and the main result 221 3. Preliminaries 223 4. The proof of C-exponential tightness 224 4.1. Auxiliary lemma 224 4.2. The proof of (3.1) 225 4.3. The proof of (3.2) 226 5. Local LDP upper bound 227 6. Local LDP lower bound 231 6.1. Nonsingular a(x) 231 6.2. General a(x) 234 A.1. Exponential estimates for martingales 236 A.2. Pseudoinverse of nonnegative de nite matrices 237 A.3. Exponential negligibility of X"; t 237 Acknowledgement 240 References 240 8. On the Convergence Rates of a General Class of Weak Approximations of SDEs D. Crisan and S. Ghazali 244 1. Introduction 244 2. Preliminaries 247 3. The Main Theorem 250 4. An Application to Filtering 262 5. Some Auxiliary Results 266 Acknowledgments 269 References 269 9. Flow Properties of Di erential Equations Driven by Fractional Brownian Motion L. Decreusefond and D. Nualart 272 1. Introduction 272 2. Preliminaries 274 3. Stochastic di erential equations driven by an fBm 276 4. Flow of homeomorphisms 281 Acknowledgement 284 References 284 10. Regularity of Transition Semigroups Associated to a 3D Stochastic Navier-Stokes Equation F. Flandoli and M. Romito 286 1. Introduction 286 2. Preliminaries 288 2.1. Notations 288 2.2. De nitions, assumptions and known results 289 3. The Log-Lipschitz estimate 291 3.1. Probability of blow-up 294 3.2. Derivative of the regularised problem 294 4. Equivalence of all transition probabilities 295 5. Conclusion and remarks 297 A.1. An exponential tail estimate for the Stokes problem 298 A.2. The deterministic equation 300 References 302 11. Rate of Convergence of Implicit Approximations for Stochastic Evolution Equations I. Gy ongy and A. Millet 304 1. Introduction 304 2. Preliminaries and the approximation scheme 307 3. Convergence esults 314 4. Examples 324 4.1. Quasilinear stochastic PDEs 324 4.2. Linear stochastic PDEs 331 Acknowledgments 332 References 333 12. Maximum Principle for SPDEs and Its Applications N. V. Krylov 334 1. Introduction 334 2. The maximum principle 335 3. Auxiliary results 338 4. Proof of Theorems 2.5 and 2.6 344 5. Auxiliary functions 348 6. Continuity of solutions of SPDEs 352 Acknowledgements 360 References 360 13. On Delay Estimation and Testing for Di usion Type Processes Yu. A. Kutoyants 362 1. Introduction 362 2. Estimation (small noise asymptotics) _ 366 2.1. Asymptotic expansion 368 2.2. Generalizations 369 3. Hypotheses Testing (small noise asymptotics) 371 4. Estimation (large samples asymptotics) 372 5. Hypotheses Testing (large samples asymptotics) 375 6. Discussion 376 References 377 14. On Cauchy-Dirichlet Problem for Linear Integro-Di erential Equation in Weighted Sobolev Spaces R. Mikulevicius and H. Pragarauskas 380 1. Introduction 380 2. Notation and main result 382 3. Proof of the main results 384 3.1. Proof of Theorem 2.1 388 3.2. Proof of Theorem 2.2 391 Acknowledgement 397 References 397 15. Strict Solutions of Kolmogorov Equations in Hilbert Spaces and Applications G. Da Prato 398 1. Introduction 398 2. Existence of cores 402 3. Invariant measures 404 4. Application 406 4.1. Estimates for Xx(t; x) 407 4.2. Estimates for Xxx(t; x) 409 4.3. Estimates of TR[Xx;x(t; x)] 410 4.4. Estimates of P ' 411 References 413 Author Index 414 Subject Index 416 Stochastic evolution equations -- Predictability of the burgers dynamics under model uncertainty -- Asymptotics for the space-time Wigner transform with applications to imaging -- The Korteweg-de Vries equation with multiplicative homogeneous noise -- On stochastic burgers equation driven by a fractional Laplacian and space-time white noise -- Stochastic control methods for the problem of optimal compensation of executives -- The Freidlin-Wentzell LDP with rapidly growing coefficients -- On the convergence rates of a general class of weak approximations of SDEs -- Flow properties of differential equations driven by fractional Brownian motion -- Regularity of transition semigroups associated to a 3D stochastic Navier-Stokes equation -- Rate of convergence of implicit approximations for stochastic evolution equations -- Maximum principle for SPDEs and its applications -- On delay estimation and testing for diffusion type processes -- On Cauchy-Dirichlet problem for linear integro-differential equation in weighted Sobolev spaces -- Strict solutions of Kolmogorov equations in Hilbert spaces and applications