Applications Stochastic Modelling and Applied Probability

Hybrid Switching Diffusions:Properties and Applications

G. George Yin • Chao Zhu

2010

394pages


"This book is written for applied mathematicians, probabilists, systems engineers, control scientists, operations researchers, and financial analysts among others. The results presented in the book are useful to researchers working in stochastic modeling, systems theory, and applications in which continuous dynamics and discrete events are intertwined. The book can be served as a reference for researchers and practitioners in the aforementioned areas. Selected materials from the book may also be used in a graduate-level course on stochastic processes and applications."






Contents
Preface xi
Conventions xv
Glossary of Symbols xvii
1 Introduction and Motivation 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 What Is a Switching Di usion . . . . . . . . . . . . . . . . . 4
1.4 Examples of Switching Di usions . . . . . . . . . . . . . . . 5
1.5 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . 21
Part I: Basic Properties, Recurrence, Ergodicity 25
2 Switching Di usion 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Switching Di usions . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Weak Continuity . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Feller Property . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Strong Feller Property . . . . . . . . . . . . . . . . . . . . . 52
2.7 Continuous and Smooth Dependence on the Initial Data x . 56
2.8 A Remark Regarding Nonhomogeneous Markov Processes . 65
2.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
vii
viii
3 Recurrence 69
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Formulation and Preliminaries . . . . . . . . . . . . . . . . 70
3.2.1 Switching Di usion . . . . . . . . . . . . . . . . . . . 70
3.2.2 De nitions of Recurrence and Positive Recurrence . 72
3.2.3 Preparatory Results . . . . . . . . . . . . . . . . . . 72
3.3 Recurrence and Transience . . . . . . . . . . . . . . . . . . 78
3.3.1 Recurrence . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.2 Transience . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Positive and Null Recurrence . . . . . . . . . . . . . . . . . 85
3.4.1 General Criteria for Positive Recurrence . . . . . . . 85
3.4.2 Path Excursions . . . . . . . . . . . . . . . . . . . . 89
3.4.3 Positive Recurrence under Linearization . . . . . . . 89
3.4.4 Null Recurrence . . . . . . . . . . . . . . . . . . . . 93
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6 Proofs of Several Results . . . . . . . . . . . . . . . . . . . . 100
3.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4 Ergodicity 111
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3 Feedback Controls for Weak Stabilization . . . . . . . . . . 119
4.4 Rami cations . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5 Asymptotic Distribution . . . . . . . . . . . . . . . . . . . . 129
4.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Part II: Numerical Solutions and Approximation 135
5 Numerical Approximation 137
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.3 Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . 139
5.4 Convergence of the Algorithm . . . . . . . . . . . . . . . . . 140
5.4.1 Moment Estimates . . . . . . . . . . . . . . . . . . . 140
5.4.2 Weak Convergence . . . . . . . . . . . . . . . . . . . 144
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6 Discussions and Remarks . . . . . . . . . . . . . . . . . . . 152
5.6.1 Remarks on Rates of Convergence . . . . . . . . . . 153
5.6.2 Remarks on Decreasing Stepsize Algorithms . . . . . 155
5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6 Numerical Approximation to Invariant Measures 159
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2 Tightness of Approximation Sequences . . . . . . . . . . . . 161
6.3 Convergence to Invariant Measures . . . . . . . . . . . . . . 165
Contents
ix
6.4 Proof: Convergence of Algorithm . . . . . . . . . . . . . . . 169
6.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Part III: Stability 181
7 Stability 183
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2 Formulation and Auxiliary Results . . . . . . . . . . . . . . 184
7.3 p-Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.3.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.3.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . 193
7.3.3 Necessary and Sucient Conditions for p-Stability . 201
7.4 Stability and Instability of Linearized Systems . . . . . . . 203
7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8 Stability of Switching ODEs 217
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.2 Formulation and Preliminary Results . . . . . . . . . . . . . 219
8.2.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . 219
8.2.2 Preliminary Results . . . . . . . . . . . . . . . . . . 220
8.3 Stability and Instability: Sucient Conditions . . . . . . . . 227
8.4 A Sharper Result . . . . . . . . . . . . . . . . . . . . . . . . 231
8.5 Remarks on Liapunov Exponent . . . . . . . . . . . . . . . 236
8.5.1 Stability under General Setup . . . . . . . . . . . . . 236
8.5.2 Invariant Density . . . . . . . . . . . . . . . . . . . . 238
8.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9 Invariance Principles 251
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.3 Invariance (I): A Sample Path Approach . . . . . . . . . . . 253
9.3.1 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . 254
9.3.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . 263
9.4 Invariance (II): A Measure-Theoretic Approach . . . . . . . 265
9.4.1 !-Limit Sets and Invariant Sets . . . . . . . . . . . . 269
9.4.2 Switching Di usions . . . . . . . . . . . . . . . . . . 275
9.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Part IV: Two-time-scale Modeling and Applications 283
10 Positive Recurrence: Weakly Connected Ergodic Classes 285
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
10.2 Problem Setup and Notation . . . . . . . . . . . . . . . . . 285
10.3 Weakly Connected, Multiergodic-Class Switching Processes 286
Contents
x
10.3.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . 287
10.3.2 Weakly Connected, Multiple Ergodic Classes . . . . 288
10.3.3 Inclusion of Transient Discrete Events . . . . . . . . 297
10.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
11 Stochastic Volatility Using Regime-Switching Di usions 301
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
11.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
11.3 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 306
11.3.1 Construction of '0(S; t; i) and  0(S; ; i) . . . . . . . 308
11.3.2 Construction of '1(S; t; i) and  1(S; ; i) . . . . . . . 309
11.3.3 Construction of 'k(S; t) and  k(S;  ) . . . . . . . . . 313
11.4 Asymptotic Error Bounds . . . . . . . . . . . . . . . . . . . 317
11.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
12 Two-Time-Scale Switching Jump Di usions 323
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
12.2 Fast-Varying Switching . . . . . . . . . . . . . . . . . . . . . 326
12.2.1 Fast-Varying Markov Chain Model . . . . . . . . . . 326
12.2.2 Limit System . . . . . . . . . . . . . . . . . . . . . . 329
12.3 Fast-Varying Di usion . . . . . . . . . . . . . . . . . . . . . 339
12.4 Discussion and Remarks . . . . . . . . . . . . . . . . . . . . 348
12.5 Remarks on Numerical Solutions for Switching Jump Di usions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
12.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
A Appendix 355
A.1 Discrete-Time Markov Chains . . . . . . . . . . . . . . . . . 355
A.2 Continuous-Time Markov Chains . . . . . . . . . . . . . . . 358
A.3 Fredholm Alternative and Rami cation . . . . . . . . . . . 362
A.4 Martingales, Gaussian Processes, and Di usions . . . . . . . 366
A.4.1 Martingales . . . . . . . . . . . . . . . . . . . . . . . 366
A.4.2 Gaussian Processes and Di usion Processes . . . . . 369
A.5 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . 371
A.6 Hybrid Jump Di usion . . . . . . . . . . . . . . . . . . . . . 376
A.7 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
A.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
References 379
Index 392