Stochastic Differential Equations: Theory and Applications: A Volume in Honor of Professor Boris L Rozovskii

Description

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 Preface Boris Rozovskii Publications of B. L. Rozovskii 1. Stochastic Evolution Equations N. V. Krylov and B. L. Rozovskii 1. Introduction 1.1. It^o Equations in Banach Spaces 1.2. Examples of Stochastic Evolution Equations 1.2.1. Linear Equation of Filtring of Di usion Processes 1.2.2. Equations of Population Genetics 1.2.3. System of Navier-Stokes Equations with Random External Forces 1.2.4. Equation of the Free Field 1.3. Stochastic Evolution Equations with Bounded Coe cients and Linear Stochastic Evolution Equations 1.4. Nonlinear Stochastic Evolution Equations 1.5. Content and Organization of the Work 2. Stochastic Integration in Hilbert Spaces 2.1. Introduction 2.2. Stochastic Integrals in Hilbert Spaces 2.3. It^o's Formula for the Square of the Norm 2.4. Proof of Theorem 2.16 3. It^o Stochastic Equations in Banach Spaces and the Meth Monotonicity 3.1. Introduction 3.2. Assumptions and the Main Results 3.3. It^o Equations in Rd 3.4. Uniqueness Theorem: A Priori Estimates and Finite-Dimensional Approximations 3.5. Existence of Solution and the Markov Property: Passing to the Limit by the Method of Monotonicity 4. It^o Stochastic Partial Di er tial Equations 4.1. Introduction 4.2. First Boundary-Value Problem for Nonlinear Stochastic Parabolic Equations 4.3. Cauchy Problem for Linear Second-Order Equations References 2. Predictability of the Burgers Dynamics Under Model Uncertainty D. Bl omker and J. Duan 1 Introduction 2 Linear Theory 2.1 Mean Energy 2.2 Correlation Function 3 Nonlinear Theory 3.1 Body forcing - Mean energy bounds 3.2 Point forcing - Mean energy bounds 3.3 Body forcing - Transient behavior 3.4 Trace class noise: Additive vs. multiplicative body noises Acknowledgments References 3. Asymptotics for the Space-Time Wigner Transform with Applications to Imaging L. Borcea, G. Papanicolaou, and C. Tsogka 1. Introduction 2. The parabolic approximation 3. Scaling and the asymptotic regime 4. The It^o-Liouville equation for the Wigner transform 4.1. The white noise limit 4.2. The high frequency limit and the space-time Wigner transform 4.3. Statement of the strong lateral diversity limit 4.4. The mean space-time Wigner transform 5. Self-averaging of the smoothed space-time Wigner transform, in the strong lateral diversity limit 6. Application to imaging 6.1. Migration 6.2. Coherent interferometric imaging Acknowledgments References 4. The Korteweg-de Vries Equation with Multiplicative Homogeneous Noise A. de Bouard and A. Debussche 1. Introduction and statement of the results 2. Preliminaries and existence for a truncated equation 3. Global existence References 5. On Stochastic Burgers Equation Driven by a Fractional Laplacian and Space-Time White Noise Z. Brze zniak and L. Debbi 1. Introduction 2. Existence of global solutions to approximating equations 3. Global solutions to Burgers equations 4. Proof of uniqueness A.1. Proof of Lemma 2.3 B.1. Gronwall Lemma C.1. Some estimates on stopped stochastic convolutions D.1. Pointwise multiplication in Sobolev spaces References 6. Stochastic Control Methods for the Problem of Optimal Compensation of Executives A. Cadenillas, J. Cvitani c, and F. Zapatero 1. Introduction 2. The Executive 2.1. Stock Dynamics 2.2. The Problem of the Executive 2.3. Optimal E ort and Choice of Projects 3. The Company 3.1. The Problem of the Company 3.2. Optimal Strike Price 4. Numerical Computations of the Strike Price 5. Price of the Options 6. The Case of Additional Cash Compensation 7. Conclusions Acknowledgements References 7. The Freidlin-Wentzell LDP with Rapidly Growing Coe cients P. Chigansky and R. Liptser 1. Introduction 2. Notations and the main result 3. Preliminaries 4. The proof of C-exponential tightness 4.1. Auxiliary lemma 4.2. The proof of (3.1) 4.3. The proof of (3.2) 5. Local LDP upper bound 6. Local LDP lower bound 6.1. Nonsingular a(x) 6.2. General a(x) A.1. Exponential estimates for martingales A.2. Pseudoinverse of nonnegative de nite matrices A.3. Exponential negligibility of X"; t Acknowledgement References 8. On the Convergence Rates of a General Class of Weak Approximations of SDEs D. Crisan and S. Ghazali 1. Introduction 2. Preliminaries 3. The Main Theorem 4. An Application to Filtering 5. Some Auxiliary Results Acknowledgments References 9. Flow Properties of Di erential Equations Driven by Fractional Brownian Motion L. Decreusefond and D. Nualart 1. Introduction 2. Preliminaries 3. Stochastic di erential equations driven by an fBm 4. Flow of homeomorphisms Acknowledgement References 10. Regularity of Transition Semigroups Associated to a 3D Stochastic Navier-Stokes Equation F. Flandoli and M. Romito 1. Introduction 2. Preliminaries 2.1. Notations 2.2. De nitions, assumptions and known results 3. The Log-Lipschitz estimate 3.1. Probability of blow-up 3.2. Derivative of the regularised problem 4. Equivalence of all transition probabilities 5. Conclusion and remarks A.1. An exponential tail estimate for the Stokes problem A.2. The deterministic equation References 11. Rate of Convergence of Implicit Approximations for Stochastic Evolution Equations I. Gy ongy and A. Millet 1. Introduction 2. Preliminaries and the approximation scheme 3. Convergence esults 4. Examples 4.1. Quasilinear stochastic PDEs 4.2. Linear stochastic PDEs Acknowledgments References 12. Maximum Principle for SPDEs and Its Applications N. V. Krylov 1. Introduction 2. The maximum principle 3. Auxiliary results 4. Proof of Theorems 2.5 and 2.6 5. Auxiliary functions 6. Continuity of solutions of SPDEs Acknowledgements References 13. On Delay Estimation and Testing for Di usion Type Processes Yu. A. Kutoyants 1. Introduction 2. Estimation (small noise asymptotics) _ 2.1. Asymptotic expansion 2.2. Generalizations 3. Hypotheses Testing (small noise asymptotics) 4. Estimation (large samples asymptotics) 5. Hypotheses Testing (large samples asymptotics) 6. Discussion References 14. On Cauchy-Dirichlet Problem for Linear Integro-Di erential Equation in Weighted Sobolev Spaces R. Mikulevicius and H. Pragarauskas 1. Introduction 2. Notation and main result 3. Proof of the main results 3.1. Proof of Theorem 2.1 3.2. Proof of Theorem 2.2 Acknowledgement References 15. Strict Solutions of Kolmogorov Equations in Hilbert Spaces and Applications G. Da Prato 1. Introduction 2. Existence of cores 3. Invariant measures 4. Application 4.1. Estimates for Xx(t; x) 4.2. Estimates for Xxx(t; x) 4.3. Estimates of TR[Xx;x(t; x)] 4.4. Estimates of P ' References Author Index Subject Index