Chow / Yin / Mordukhovich Topics in Stochastic Analysis and Nonparametric Estimation

1. Auflage 2010
ISBN: 978-0-387-75111-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 214 Seiten

To honor Rafail Z. Khasminskii, on his seventy-fifth birthday, for his contributions to stochastic processes and nonparametric estimation theory an IMA participating institution conference entitled 'Conference on Asymptotic Analysis in Stochastic Processes, Nonparametric Estimation, and Related Problems' was held. This volume commemorates this special event. Dedicated to Professor Khasminskii, it consists of nine papers on various topics in probability and statistics.

Autoren/Hrsg.

Chow, Pao-Liu

Yin, George

Mordukhovich, Boris

Inhaltsverzeichnis

1;FOREWORD;6
2;DEDICATED TO PROFESSOR RAFAIL Z. KHASMINSKII ON THE OCCASION OF HIS SEVENTY-FIFTH BIRTHDAY
;7
3;PREFACE;8
4;Table of Contents
;10
5;PART I: ASYMPTOTIC ANALYSIS INVOLVING STOCHASTIC DIFFERENTIAL EQUATIONS
;11
5.1;SOME RECENT RESULTS ON AVERAGING PRINCIPLE;12
5.1.1;1. Non-perturbed system;12
5.1.2;2. Perturbations;17
5.1.3;3. Multiwell Hamiltonians;22
5.1.4;4. Systems with many degrees of freedom;25
5.1.5;REFERENCES;29
5.2;CRAMER'S THEOREM FOR NONNEGATIVE MULTIVARIATE POINT PROCESSES WITH INDEPENDENT INCREMENTS
;31
5.2.1;1. Introduction and main result;31
5.2.2;2. Counting random measure, its compensator. Laplace transform
;34
5.2.3;3. The proof of Theorem 1.2;35
5.2.3.1;3.1. The exponential tightness;35
5.2.3.2;3.2. The local LDP;36
5.2.4;REFERENCES;38
5.3;ON BOUNDED SOLUTIONS OF THE BALANCED GENERALIZED PANTOGRAPH EQUATION
;39
5.3.1;1. Introduction;39
5.3.2;2. Perturbative proof;44
5.3.3;3. Analytical proof;47
5.3.4;4. Probabilistic proof;48
5.3.5;5. Jump diffusions;51
5.3.6;6. L-harmonic functions
;52
5.3.7;7. Proof of the main results;54
5.3.8;REFERENCES;57
5.4;NUMERICAL METHODS FOR NON-ZERO-SUM STOCHASTIC DIFFERENTIAL GAMES: CONVERGENCE OF THE MARKOV CHAIN APPROXIMATION METHOD
;60
5.4.1;1. Introduction;60
5.4.2;2. The model;62
5.4.3;3. A discrete time approximation and randomized controls;64
5.4.4;4. Approximating the controls;66
5.4.5;5. Approximations to e-equilibrla
;69
5.4.6;6. The Markov chain approximation: Brief review and approximations;72
5.4.7;7. Approximating the chain: I;78
5.4.8;8. Convergence, Part I: An approximate equilibrium for the diffusion is also one for the chain
;80
5.4.9;9. Approximating the chain: II. Representations and approximations of the chain with control-independent driving noise
;84
5.4.10;10. Convergence: II. An approximate equilibrium for the chain is an approximate equilibrium for the diffusion
;91
5.4.11;REFERENCES;92
6;PART II: NONPARAMETRIC ESTIMATION;94
6.1;ON THE ESTIMATION OF AN ANALYTIC SPECTRAL DENSITY OUTSIDE OF THE OBSERVATION BAND
;95
6.1.1;1. Introduction;95
6.1.2;2. Construction of estimators. Proof of Theorems 1, 2.;98
6.1.3;3. Lower bounds. Proof of Theorem 1.3.;103
6.1.4;REFERENCES;113
6.2;ON ORACLE INEQUALITIES RELATED TO HIGH DIMENSIONAL LINEAR MODELS
;114
6.2.1;1. Introduction and main results;114
6.2.2;2. Proofs;120
6.2.2.1;2.1. Ordered processes and their properties;120
6.2.2.2;2.2. Some examples of ordered processes;123
6.2.2.3;2.3. Proof of Theorem 1.1;126
6.2.2.3.1;2.3.1. Proof of Theorem 1.2;129
6.2.3;REFERENCES;131
6.3;HYPOTHESIS TESTING UNDER COMPOSITE FUNCTIONS ALTERNATIVE
;132
6.3.1;1. Introduction;132
6.3.1.1;1.1. M inimax approach;133
6.3.1.2;1.2. Choice of parameter set;133
6.3.1.3;1.3. Composite functions;134
6.3.2;2. Test procedure and main results;136
6.3.2.1;2.1. Test procedure;136
6.3.2.2;2.2. Main results;137
6.3.2.3;2.3. Open problems;139
6.3.3;3. Proofs.;140
6.3.3.1;3.1. Proof of Theorem 1. I;140
6.3.3.2;3.2. Proof of Theorem 2;151
6.3.4;REFERENCES;158
7;PART III: STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
;160
7.1;ON PARABOLIC PDES AND SPDES IN SOBOLEV SPACES W2P WITHOUT AND WITH WEIGHTS
;161
7.1.1;1. Introduction;161
7.1.2;2. Interior estimates for SPDEs in the whole space;163
7.1.3;3. Interior estimates for solutions of parabolic PDEs in the whole space
;171
7.1.4;4. Local regularity near the boundary for SPDEs in half spaces;176
7.1.5;5. Local regularity near the boundary for parabolic PDEs in half spaces
;180
7.1.6;6. Auxiliary results;182
7.1.7;7. Ex istence and uniqueness for parabolic PDEs in half spaces
;193
7.1.8;8. Existence and uniqueness for SPDEs in half spaces;199
7.1.9;REFERENCES;206
7.2;STOCHASTIC PARABOLIC EQUATIONS OF FULL SECOND ORDER
;209
7.2.1;1. Introduction;209
7.2.2;2. Constructing a solution: an example;210
7.2.3;3. General constructions and the main result;214
7.2.4;REFERENCES;220
8;IMA SUMMER PROGRAMS;221
9;IMA “HOT TOPICS” WORKSHOPS;222
10;SPRINGER LECTURE NOTES FROM THE IMA;223