Stochastic Analysis (eBook)

Liber Amicorum for Moshe Zakai

Eddy Mayer-Wolf, Ely Merzbach, Adam Shwartz (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
552 Seiten
Elsevier Science (Verlag)
978-1-4832-1870-0 (ISBN)

Infos

Stochastic Analysis: Liber Amicorum for Moshe Zakai focuses on stochastic differential equations, nonlinear filtering, two-parameter martingales, Wiener space analysis, and related topics. The selection first ponders on conformally invariant and reflection positive random fields in two dimensions; real time architectures for the Zakai equation and applications; and quadratic approximation by linear systems controlled from partial observations. Discussions focus on predicted miss, review of basic sequential detection problems, multigrid algorithms for the Zakai equation, invariant test functions and regularity, and reflection positivity. The text then takes a look at a model of stochastic differential equation in Hubert spaces applicable to Navier Stokes equation in dimension 2; wavelets as attractors of random dynamical systems; and Markov properties for certain random fields. The publication examines the anatomy of a low-noise jump filter, nonlinear filtering with small observation noise, and closed form characteristic functions for certain random variables related to Brownian motion. Topics include derivation of characteristic functions for the examples, proof of the theorem, sequential quadratic variation test, asymptotic optimal filters, mean decision time, and asymptotic optimal filters. The selection is a valuable reference for researchers interested in stochastic analysis.

Inhalt

Front Cover 1
Stochastic Analysis: Liber Amicorum for Moshe Zakai 4
Copyright Page 5
Table of Contents 6
Preface 10
Foreword 12
Publications by Moshe Zakai 16
Chapter 1. Conformally Invariant and Reflection Positive Random Fields in Two Dimensions 22
1. Introduction 22
2. Representation theory, invariant measures 24
3. Invariant test functions and regularity. Existence 26
4. Reflection positivity 30
Chapter 2. Real time architectures for the Zakai equation and applications 36
1. Introduction 36
2. Review of Basic Sequential Detection Problems 40
3. Numerical Solution for Scalar x 43
4. Real Time Architectures for Scalar and Two Dimensional x 46
5. Multigrid Algorithms for the Zakai Equation 47
6. Architectures for Implementing MG in Real-Time 53
References 58
Chapter 3. Quadratic Approximation by Linear Systems Controlled From Partial Observations 60
I. Introduction 60
2. The Problem 61
3. Predicted Miss 64
4. Solution 65
5. Example: A Guidance Problem 68
6. Example: A Disturbance Rejection Problem 69
7. Acknowledgements 70
References 71
Chapter 4. A model of stochastic differential equation in Hilbert spaces applicable to Navier Stokes equation in dimension 2 72
1 Introduction 72
2 Setting of the problem 72
3 Approximation scheme 78
4 Convergence 85
Bibliography 93
Chapter 5. Wavelets as Attractors of Random Dynamical Systems 96
Bibliography 108
Chapter 6. Markov Properties for Certain Random Fields 112
1 Introduction 112
2 Markov Properties and Conditional Independence 113
3 Markov Properties in the Plane 119
4 Markov Properties of the Poisson Sheet 123
Bibliography 130
Chapter 7. The Anatomy of a Low-Noise Jump Filter: Part I 132
1. Introduction 132
2. Preliminary definitions 133
3. The conditional laws of St 134
4. The conditional relative density of St 143
5. References 145
Chapter 8. On the Value of Information in Controlled Diffusion Processes 146
1 Introduction 146
2 Problem Formulation 149
3 Subspace Constraints and Lagrange Multipliers 153
4 Application to the Control Problem 154
5 Example: the LQG Problem 156
Bibliography 158
Chapter 9. Orthogonal Martingale Representation 160
1 Introduction 160
2 Orthogonal Projection 165
Bibliography 172
Chapter 10. Nonlinear Filtering with Small Observation Noise: Piecewise Monotone Observations 174
1 Introduction 174
2 Problem formulation 176
3 Sequential quadratic variation test 179
4 Mean decision time 187
5 Asymptotic optimal filters 188
Bibliography 189
Chapter 11. Closed Form Characteristic Functions for Certain Random Variables Related to Brownian Motion 190
1 Introduction 191
2 Derivation of Characteristic Functions 
193 
3 Proof of the Theorem 199
4 Closing Remarks 205
Bibliography 206
Chapter 12. Adaptedness and Existence of Occupation Densities for Stochastic Integral Processes in the Second Wiener Chaos 210
1 Introduction 210
2 Notations and Conventions 212
3 A Finite Dimensional Approximation 213
4 Stochastic interpretation of Berman's condition 221
5 The analytical integral criterion 225
Bibliography 233
Chapter 13. A Skeletal Theory of Filtering 234
1 Introduction 234
2 Basic ingredients of the theory 235
3 Liftings and skeletons 242
4 Bayes formula in the white noise theory 244
5 The Zakai equation in skeletal form 250
6. Measure–valued equations for the general optimal filter in the white noise theory 255
7 Consistency and robustness 260
8 Nonlinear prediction and smoothing in the white noise theory 261
References 264
Chapter 14. EQUILIBRIUM IN A SIMPLIFIED DYNAMIC, STOCHASTIC ECONOMY WITH HETEROGENEOUS AGENTS 266
1. Introduction 266
2. The Agents and their Endowments 270
4. The Financial Market 272
5. The Individual Agents' Optimization Problems 274
6. The Definition of Equilibrium 276
7. Solution of the nth Agent's Problem 276
8. Characterization of Equilibrium 280
9. The Representative Agent 281
10. The Equilibrium Financial Market 283
11. Examples 285
12. Existence and Uniqueness of Equilibrium 287
13. Variations of the Model 290
References 292
Chapter 15. Feynman-Kac Formula for a Degenerate Planar Diffusion and an Application in Stochastic Control 294
1 Introduction 294
2 A Degenerate, Planar Diffusion 295
3 The Feynman - Kac Formula 297
4 Analysis 299
5 Synthesis 306
6 A Control Problem With Partial Observations 311
7 Solution to the Control Problem 314
8 Acknowledgements 317
9. References 317
Chapter 16. On the Interior Smoothness of Harmonic Functions for Degenerate Diffusion Processes 318
1 Introduction 318
2 Notion of Quasi-derivative 320
3 Examples of Quasi-derivatives 323
4 Applications to the Study of Interior Smoothness of Harmonic Functions 327
Bibliography 331
Chapter 17. The Stability and Approximation Problems in Nonlinear Filtering Theory 332
1 Introduction 332
2 Approximation of the filter with discrete parameter 334
3 Asymptotic property of discrete filter 340
Bibliography 350
Chapter 18. Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE's 352
1 Introduction 352
2 Random evolutions 357
3 Numerical schemes 360
Bibliography 366
Chapter 19. Nonlinear Filtering for Singularly Perturbed Systems 368
1 Introduction 368
2 The Fixed-x Rescaled Fast Process and Assumptions 370
3 The Representation Theorem 373
4 The Filter For The Singularly Perturbed System 375
5 A Counterexample to the Averaged Filter 377
6 The Almost Optimality of the Averaged Filter 380
Bibliography 389
Chapter 20. Smooth s -Fields 392
1. Smooth functional 392
2. Smooth subalgebra 393
3. Functional calculus 395
4. Smooth s-field 396
5. Basic vector field 397
6. Basic differential form 401
BIBLIOGRAPHY 403
Chapter 21. Composition of Large Deviation Principles and Applications 404
1 Introduction 404
2 Composition of large deviation principles 405
3 Large deviations for anticipating stochastic differential equations 408
Bibliography 416
Chapter 22. Nonlinear Transformations of the Wiener Measure and Applications 418
Introduction 418
1 Nonlinear Transformations of the Wiener Measure 420
2 Stochastic Differential Equations with Boundary Conditions 433
References 450
Chapter 23. Finite Dimensional Approximate Filters in the case of High Signal–to–Noise Ratio 454
1 Introduction 454
2 Case A : Piecewise Monotone Observation Function 456
3 Case A : Piecewise Monotone Observation Function under the Detectability Assumption (DA2) 460
4 Remarks on the problems with 467
Bibliography 468
Chapter 24. A Simple Proof of Uniqueness for Kushner and Zakai Equations 470
1 Introduction 470
2 Setting of the Problem. Notation 471
3 The Main Result 473
Bibliography 478
Chapter 25. Itô-Wiener expansions of holomorphic functions on the complex Wiener space 480
1 Introduction 480
2 Complex abstract Wiener space and holomorphic functions 481
3 Complex Hermite polynomials and complex Wiener chaos 484
4 Itô-Wiener expansions of holomorphic functions 488
Bibliography 493
Chapter 26. Limits of the Wong-Zakai Type with a Modified Drift Term 496
1 Introduction 496
2 Differential Equations with Inputs 498
3 Stochastic Ordinary Inputs 500
4 The Chen-Fliess Series 503
5 Construction of Approximating Processes 506
6 Proof of Convergence 511
Bibliography 514
Chapter 27. Donsker's d-functions in the Malliavin calculus 516
1 Introduction 516
2 A survey of Sobolev spaces in the Malliavin calculus 517
3 The exact Sobolev spaces to which Donsker's d-functions belong 520
Bibliography 522
Chapter 28. Implementing Boltzmann Machines 524
1 Introduction 524
2 Boltzmann Machine 526
3 Generalizing the Energy Function 529
4 Alternative Network Architectures 529
5 Acknowledgement 532
Bibliography 532
Chapter 29. Infinite Dimensionality Results for MAP Estimation 534
1 Introduction 534
2 Stochastic Calculus of Variations in Hilbert Space 539
3 Some properties of v(t, x), and a stochastic gradient representation 541
4 Conditions for finite dimensionality 547
5 Appendix 551
Bibliography 552

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• ISBN-10: 1-4832-1870-8 / 1483218708
• ISBN-13: 978-1-4832-1870-0 / 9781483218700