Topics in Stochastic Analysis and Nonparametric Estimation (eBook)

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