Stochastic differential equations: theory and applications

Peter H. Baxendale, Peter H. Baxendale; Sergey V. Lototsky9789812706621, 981-270-662-3

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.

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Table of contents :
Contents……Page 22
Preface……Page 8
Boris Rozovskii……Page 10
Publications of B. L. Rozovskii……Page 12
1. Stochastic Evolution Equations N. V. Krylov and B. L. Rozovskii……Page 24
1.2.1. Linear Equation of Filtring of Di usion Processes……Page 25
1.2.2. Equations of Population Genetics……Page 26
1.2.4. Equation of the Free Field……Page 27
1.3. Stochastic Evolution Equations with Bounded Coe cients and Linear Stochastic Evolution Equations……Page 28
1.4. Nonlinear Stochastic Evolution Equations……Page 30
1.5. Content and Organization of the Work……Page 31
2.1. Introduction……Page 32
2.2. Stochastic Integrals in Hilbert Spaces……Page 33
2.3. It^o’s Formula for the Square of the Norm……Page 39
2.4. Proof of Theorem 2.16……Page 43
3.1. Introduction……Page 50
3.2. Assumptions and the Main Results……Page 52
3.3. It^o Equations in Rd……Page 57
3.4. Uniqueness Theorem: A Priori Estimates and Finite-Dimensional Approximations……Page 66
3.5. Existence of Solution and the Markov Property: Passing to the Limit by the Method of Monotonicity……Page 72
4.1. Introduction……Page 77
4.2. First Boundary-Value Problem for Nonlinear Stochastic Parabolic Equations……Page 80
4.3. Cauchy Problem for Linear Second-Order Equations……Page 84
References……Page 88
1 Introduction……Page 94
2 Linear Theory……Page 96
2.1 Mean Energy……Page 97
2.2 Correlation Function……Page 98
3.1 Body forcing – Mean energy bounds……Page 104
3.2 Point forcing – Mean energy bounds……Page 107
3.3 Body forcing – Transient behavior……Page 109
3.4 Trace class noise: Additive vs. multiplicative body noises……Page 110
References……Page 111
1. Introduction……Page 114
2. The parabolic approximation……Page 115
3. Scaling and the asymptotic regime……Page 116
4.1. The white noise limit……Page 118
4.2. The high frequency limit and the space-time Wigner transform……Page 119
4.3. Statement of the strong lateral diversity limit……Page 121
4.4. The mean space-time Wigner transform……Page 122
5. Self-averaging of the smoothed space-time Wigner transform, in the strong lateral diversity limit……Page 123
6. Application to imaging……Page 124
6.1. Migration……Page 125
6.2. Coherent interferometric imaging……Page 127
References……Page 132
1. Introduction and statement of the results……Page 136
2. Preliminaries and existence for a truncated equation……Page 139
3. Global existence……Page 151
References……Page 155
1. Introduction……Page 158
2. Existence of global solutions to approximating equations……Page 161
3. Global solutions to Burgers equations……Page 171
4. Proof of uniqueness……Page 176
A.1. Proof of Lemma 2.3……Page 179
C.1. Some estimates on stopped stochastic convolutions……Page 180
D.1. Pointwise multiplication in Sobolev spaces……Page 187
References……Page 188
6. Stochastic Control Methods for the Problem of Optimal Compensation of Executives A. Cadenillas, J. Cvitani c, and F. Zapatero……Page 192
1. Introduction……Page 193
2.1. Stock Dynamics……Page 194
2.3. Optimal E ort and Choice of Projects……Page 196
3.2. Optimal Strike Price……Page 199
4. Numerical Computations of the Strike Price……Page 202
5. Price of the Options……Page 205
6. The Case of Additional Cash Compensation……Page 209
7. Conclusions……Page 217
References……Page 218
1. Introduction……Page 220
2. Notations and the main result……Page 221
3. Preliminaries……Page 223
4.1. Auxiliary lemma……Page 224
4.2. The proof of (3.1)……Page 225
4.3. The proof of (3.2)……Page 226
5. Local LDP upper bound……Page 227
6.1. Nonsingular a(x)……Page 231
6.2. General a(x)……Page 234
A.1. Exponential estimates for martingales……Page 236
A.3. Exponential negligibility of X”; t……Page 237
References……Page 240
1. Introduction……Page 244
2. Preliminaries……Page 247
3. The Main Theorem……Page 250
4. An Application to Filtering……Page 262
5. Some Auxiliary Results……Page 266
References……Page 269
1. Introduction……Page 272
2. Preliminaries……Page 274
3. Stochastic di erential equations driven by an fBm……Page 276
4. Flow of homeomorphisms……Page 281
References……Page 284
1. Introduction……Page 286
2.1. Notations……Page 288
2.2. De nitions, assumptions and known results……Page 289
3. The Log-Lipschitz estimate……Page 291
3.2. Derivative of the regularised problem……Page 294
4. Equivalence of all transition probabilities……Page 295
5. Conclusion and remarks……Page 297
A.1. An exponential tail estimate for the Stokes problem……Page 298
A.2. The deterministic equation……Page 300
References……Page 302
1. Introduction……Page 304
2. Preliminaries and the approximation scheme……Page 307
3. Convergence esults……Page 314
4.1. Quasilinear stochastic PDEs……Page 324
4.2. Linear stochastic PDEs……Page 331
Acknowledgments……Page 332
References……Page 333
1. Introduction……Page 334
2. The maximum principle……Page 335
3. Auxiliary results……Page 338
4. Proof of Theorems 2.5 and 2.6……Page 344
5. Auxiliary functions……Page 348
6. Continuity of solutions of SPDEs……Page 352
References……Page 360
1. Introduction……Page 362
2. Estimation (small noise asymptotics) _……Page 366
2.1. Asymptotic expansion……Page 368
2.2. Generalizations……Page 369
3. Hypotheses Testing (small noise asymptotics)……Page 371
4. Estimation (large samples asymptotics)……Page 372
5. Hypotheses Testing (large samples asymptotics)……Page 375
6. Discussion……Page 376
References……Page 377
1. Introduction……Page 380
2. Notation and main result……Page 382
3. Proof of the main results……Page 384
3.1. Proof of Theorem 2.1……Page 388
3.2. Proof of Theorem 2.2……Page 391
References……Page 397
1. Introduction……Page 398
2. Existence of cores……Page 402
3. Invariant measures……Page 404
4. Application……Page 406
4.1. Estimates for Xx(t; x)……Page 407
4.2. Estimates for Xxx(t; x)……Page 409
4.3. Estimates of TR[Xx;x(t; x)]……Page 410
4.4. Estimates of P ‘……Page 411
References……Page 413
Author Index……Page 414
Subject Index……Page 416