Mayer-Wolf / Merzbach / Shwartz Stochastic Analysis

Liber Amicorum for Moshe Zakai
1. Auflage 2014
ISBN: 978-1-4832-1870-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 552 Seiten, Web PDF

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.

Autoren/Hrsg.

Mayer-Wolf, Eddy

Merzbach, Ely

Shwartz, Adam

Inhaltsverzeichnis

1;Front Cover;1
2;Stochastic Analysis: Liber Amicorum for Moshe Zakai;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Foreword;12
7;Publications by Moshe Zakai;16
8;Chapter 1. Conformally Invariant and Reflection Positive Random Fields in Two Dimensions;22
8.1;1. Introduction;22
8.2;2. Representation theory, invariant measures;24
8.3;3. Invariant test functions and regularity. Existence;26
8.4;4. Reflection positivity;30
9;Chapter 2. Real time architectures for the Zakai equation and applications;36
9.1;1. Introduction;36
9.2;2. Review of Basic Sequential Detection Problems;40
9.3;3. Numerical Solution for Scalar x;43
9.4;4. Real Time Architectures for Scalar and Two Dimensional x;46
9.5;5. Multigrid Algorithms for the Zakai Equation;47
9.6;6. Architectures for Implementing MG in Real-Time;53
9.7;References;58
10;Chapter 3. Quadratic Approximation by Linear Systems Controlled From Partial Observations;60
10.1;I. Introduction;60
10.2;2. The Problem;61
10.3;3. Predicted Miss;64
10.4;4. Solution;65
10.5;5. Example: A Guidance Problem;68
10.6;6. Example: A Disturbance Rejection Problem;69
10.7;7. Acknowledgements;70
10.8;References;71
11;Chapter 4. A model of stochastic differential equation in Hilbert spaces applicable to Navier Stokes equation in dimension 2;72
11.1;1 Introduction;72
11.2;2 Setting of the problem;72
11.3;3 Approximation scheme;78
11.4;4 Convergence;85
11.5;Bibliography;93
12;Chapter 5. Wavelets as Attractors of Random Dynamical Systems;96
12.1;Bibliography;108
13;Chapter 6. Markov Properties for Certain Random Fields;112
13.1;1 Introduction;112
13.2;2 Markov Properties and Conditional Independence;113
13.3;3 Markov Properties in the Plane;119
13.4;4 Markov Properties of the Poisson Sheet;123
13.5;Bibliography;130
14;Chapter 7. The Anatomy of a Low-Noise Jump Filter: Part I;132
14.1;1. Introduction;132
14.2;2. Preliminary definitions;133
14.3;3. The conditional laws of St;134
14.4;4. The conditional relative density of St;143
14.5;5. References;145
15;Chapter 8. On the Value of Information in Controlled Diffusion Processes;146
15.1;1 Introduction;146
15.2;2 Problem Formulation;149
15.3;3 Subspace Constraints and Lagrange Multipliers;153
15.4;4 Application to the Control Problem;154
15.5;5 Example: the LQG Problem;156
15.6;Bibliography;158
16;Chapter 9. Orthogonal Martingale Representation;160
16.1;1 Introduction;160
16.2;2 Orthogonal Projection;165
16.3;Bibliography;172
17;Chapter 10. Nonlinear Filtering with Small Observation Noise: Piecewise Monotone Observations;174
17.1;1 Introduction;174
17.2;2 Problem formulation;176
17.3;3 Sequential quadratic variation test;179
17.4;4 Mean decision time;187
17.5;5 Asymptotic optimal filters;188
17.6;Bibliography;189
18;Chapter 11. Closed Form Characteristic Functions for Certain Random Variables Related to Brownian Motion;190
18.1;1 Introduction;191
18.2;2 Derivation of Characteristic Functions
for the Examples;193
18.3;3 Proof of the Theorem;199
18.4;4 Closing Remarks;205
18.5;Bibliography;206
19;Chapter 12. Adaptedness and Existence of Occupation Densities for Stochastic Integral Processes in the Second Wiener Chaos;210
19.1;1 Introduction;210
19.2;2 Notations and Conventions;212
19.3;3 A Finite Dimensional Approximation;213
19.4;4 Stochastic interpretation of Berman's condition;221
19.5;5 The analytical integral criterion;225
19.6;Bibliography;233
20;Chapter 13. A Skeletal Theory of Filtering;234
20.1;1 Introduction;234
20.2;2 Basic ingredients of the theory;235
20.3;3 Liftings and skeletons;242
20.4;4 Bayes formula in the white noise theory;244
20.5;5 The Zakai equation in skeletal form;250
20.6;6. Measure–valued equations for the general optimal filter in the white noise theory;255
20.7;7 Consistency and robustness;260
20.8;8 Nonlinear prediction and smoothing in the white noise theory;261
20.9;References;264
21;Chapter 14. EQUILIBRIUM IN A SIMPLIFIED DYNAMIC, STOCHASTIC ECONOMY WITH HETEROGENEOUS AGENTS;266
21.1;1. Introduction;266
21.2;2. The Agents and their Endowments;270
21.3;4. The Financial Market;272
21.4;5. The Individual Agents' Optimization Problems;274
21.5;6. The Definition of Equilibrium;276
21.6;7. Solution of the nth Agent's Problem;276
21.7;8. Characterization of Equilibrium;280
21.8;9. The Representative Agent;281
21.9;10. The Equilibrium Financial Market;283
21.10;11. Examples;285
21.11;12. Existence and Uniqueness of Equilibrium;287
21.12;13. Variations of the Model;290
21.13;References;292
22;Chapter 15. Feynman-Kac Formula for a Degenerate Planar Diffusion and an Application in Stochastic Control;294
22.1;1 Introduction;294
22.2;2 A Degenerate, Planar Diffusion;295
22.3;3 The Feynman - Kac Formula;297
22.4;4 Analysis;299
22.5;5 Synthesis;306
22.6;6 A Control Problem With Partial Observations;311
22.7;7 Solution to the Control Problem;314
22.8;8 Acknowledgements;317
22.9;9. References;317
23;Chapter 16. On the Interior Smoothness of Harmonic Functions for Degenerate Diffusion Processes;318
23.1;1 Introduction;318
23.2;2 Notion of Quasi-derivative;320
23.3;3 Examples of Quasi-derivatives;323
23.4;4 Applications to the Study of Interior Smoothness of Harmonic Functions;327
23.5;Bibliography;331
24;Chapter 17. The Stability and Approximation Problems in Nonlinear Filtering Theory;332
24.1;1 Introduction;332
24.2;2 Approximation of the filter with discrete parameter;334
24.3;3 Asymptotic property of discrete filter;340
24.4;Bibliography;350
25;Chapter 18. Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE's;352
25.1;1 Introduction;352
25.2;2 Random evolutions;357
25.3;3 Numerical schemes;360
25.4;Bibliography;366
26;Chapter 19. Nonlinear Filtering for Singularly Perturbed Systems;368
26.1;1 Introduction;368
26.2;2 The Fixed-x Rescaled Fast Process and Assumptions;370
26.3;3 The Representation Theorem;373
26.4;4 The Filter For The Singularly Perturbed System;375
26.5;5 A Counterexample to the Averaged Filter;377
26.6;6 The Almost Optimality of the Averaged Filter;380
26.7;Bibliography;389
27;Chapter 20. Smooth s -Fields;392
27.1;1. Smooth functional;392
27.2;2. Smooth subalgebra;393
27.3;3. Functional calculus;395
27.4;4. Smooth s-field;396
27.5;5. Basic vector field;397
27.6;6. Basic differential form;401
27.7;BIBLIOGRAPHY;403
28;Chapter 21. Composition of Large Deviation Principles and Applications;404
28.1;1 Introduction;404
28.2;2 Composition of large deviation principles;405
28.3;3 Large deviations for anticipating stochastic differential equations;408
28.4;Bibliography;416
29;Chapter 22. Nonlinear Transformations of the Wiener Measure and Applications;418
29.1;Introduction;418
29.2;1 Nonlinear Transformations of the Wiener Measure;420
29.3;2 Stochastic Differential Equations with Boundary Conditions;433
29.4;References;450
30;Chapter 23. Finite Dimensional Approximate Filters in the case of High Signal–to–Noise Ratio;454
30.1;1 Introduction;454
30.2;2 Case A : Piecewise Monotone Observation Function;456
30.3;3 Case A : Piecewise Monotone Observation Function under the Detectability Assumption (DA2);460
30.4;4 Remarks on the problems with;467
30.5;Bibliography;468
31;Chapter 24. A Simple Proof of Uniqueness for Kushner and Zakai Equations;470
31.1;1 Introduction;470
31.2;2 Setting of the Problem. Notation;471
31.3;3 The Main Result;473
31.4;Bibliography;478
32;Chapter 25. Itô-Wiener expansions of holomorphic functions on the complex Wiener space;480
32.1;1 Introduction;480
32.2;2 Complex abstract Wiener space and holomorphic functions;481
32.3;3 Complex Hermite polynomials and complex Wiener chaos;484
32.4;4 Itô-Wiener expansions of holomorphic functions;488
32.5;Bibliography;493
33;Chapter 26. Limits of the Wong-Zakai Type with a Modified Drift Term;496
33.1;1 Introduction;496
33.2;2 Differential Equations with Inputs;498
33.3;3 Stochastic Ordinary Inputs;500
33.4;4 The Chen-Fliess Series;503
33.5;5 Construction of Approximating Processes;506
33.6;6 Proof of Convergence;511
33.7;Bibliography;514
34;Chapter 27. Donsker's d-functions in the Malliavin calculus;516
34.1;1 Introduction;516
34.2;2 A survey of Sobolev spaces in the Malliavin calculus;517
34.3;3 The exact Sobolev spaces to which Donsker's d-functions belong;520
34.4;Bibliography;522
35;Chapter 28. Implementing Boltzmann Machines;524
35.1;1 Introduction;524
35.2;2 Boltzmann Machine;526
35.3;3 Generalizing the Energy Function;529
35.4;4 Alternative Network Architectures;529
35.5;5 Acknowledgement;532
35.6;Bibliography;532
36;Chapter 29. Infinite Dimensionality Results for MAP Estimation;534
36.1;1 Introduction;534
36.2;2 Stochastic Calculus of Variations in Hilbert Space;539
36.3;3 Some properties of v(t, x), and a stochastic gradient representation;541
36.4;4 Conditions for finite dimensionality;547
36.5;5 Appendix;551
36.6;Bibliography;552