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 Diffusion ........................... 4
1.4 Examples of Switching Diffusions ........................ 5
1.5 Outline of the Book .................................... 21
Part I: Basic Properties, Recurrence, Ergodicity ............... 25
2 Switching Diffusion ......................................... 27
2.1 Introduction ........................................... 27
2.2 Switching Diffusions ................................... 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
3 Recurrence .................................................. 69
3.1 Introduction ........................................... 69
3.2 Formulation and Preliminaries .......................... 70
3.2.1 Switching Diffusion ............................. 70
3.2.2 Definitions 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 .......................................... 1ll
4.2 Ergodicity ............................................ 112
4.3 Feedback Controls for Weak Stabilization .............. 119
4.4 Ramifications ......................................... 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
5 Numerical Approximation to Invariant Measures .............. 159
6.1 Introduction .......................................... 159
6.2 Tightness of Approximation Sequences .................. 161
6.3 Convergence to Invariant Measures ..................... 165
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 Sufficient 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: Sufficient 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 ω-Limit Sets and Invariant Sets ................ 269
9.4.2 Switching Diffusions ........................... 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
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 Diffusions .... 301
11.1 Introduction .......................................... 301
11.2 Formulation ........................................... 303
11.3 Asymptotic Expansions ................................. 306
11.3.1 Construction of  0(S,t,i) and ψ0(S,τ,i) ........ 308
11.3.2 Construction of 1(S,t,i) and ψ1(5,τ,i) ........ 309
11.3.3 Construction of k(S,t) and ψ1(S,τ) ............ 313
11.4 Asymptotic Error Bounds ............................... 317
11.5 Notes ................................................. 321
12 Two-Time-Scale Switching Jump Diffusions ................... 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 Diffusion ................................ 339
12.4 Discussion and Remarks ................................ 348
12.5 Remarks on Numerical Solutions for Switching Jump
Diffusions ............................................ 349
12.6 Notes ................................................. 352
A Appendix ................................................... 355
A.l Discrete-Time Markov Chains ........................... 355
A.2 Continuous-Time Markov Chains ......................... 358
A.3 Fredholm Alternative and Ramification ................. 362
A.4 Martingales, Gaussian Processes, and Diffusions ....... 366
A.4.1 Martingales .................................... 366
A.4.2 Gaussian Processes and Diffusion Processes ..... 369
A.5 Weak Convergence ...................................... 371
A.6 Hybrid Jump Diffusion ................................. 376
A.7 Miscellany ............................................ 377
A.8 Notes ................................................. 378
References ................................................. 379
Index ......................................................... 392
|
