Hybrid Switching Diffusions (eBook)
XVIII, 398 Seiten
Springer New York (Verlag)
978-1-4419-1105-6 (ISBN)
This book encompasses the study of hybrid switching di usion processes and their applications. The word /hybrid"e; signi es the coexistence of c- tinuous dynamics and discrete events, which is one of the distinct features of the processes under consideration. Much of the book is concerned with the interactions of the continuous dynamics and the discrete events. Our motivations for studying such processes originate from emerging and - isting applications in wireless communications, signal processing, queueing networks, production planning, biological systems, ecosystems, nancial engineering, and modeling, analysis, and control and optimization of lar- scale systems, under the in uence of random environments. Displaying mixture distributions, switching di usions may be described by the associated operators or by systems of stochastic di erential eq- tions together with the probability transition laws of the switching actions. We either have Markov-modulated switching di usions or processes with continuous state-dependent switching. The latter turns out to be much more challenging to deal with. Viewing the hybrid di usions as a number of di usions joined together by the switching process, they may be se- ingly not much di erent from their di usion counterpart. Nevertheless, the underlying problems become more di cult to handle, especially when the switching processes depend on continuous states. The di culty is due to the interaction of the discrete and continuous processes and the tangled and hybrid information pattern.
Contents 7
Preface 11
Conventions 14
Glossary of Symbols 15
1 Introduction and Motivation 17
1.1 Introduction 17
1.2 Motivation 17
1.3 What Is a Switching Di usion 20
1.4 Examples of Switching Di usions 21
1.5 Outline of the Book 37
Part I Basic Properties, Recurrence, Ergodicity 41
2 Switching Diffusion 42
2.1 Introduction 42
2.2 Switching Di usions 42
2.3 Regularity 48
2.4 Weak Continuity 53
2.5 Feller Property 56
2.6 Strong Feller Property 67
2.7 Continuous and Smooth Dependence on theInitial Data x 71
2.8 A Remark Regarding NonhomogeneousMarkov Processes 80
2.9 Notes 82
3 Recurrence 83
3.1 Introduction 83
3.2 Formulation and Preliminaries 84
3.2.1 Switching Di usion 84
3.2.2 De nitions of Recurrence and Positive Recurrence 86
3.2.3 Preparatory Results 86
3.3 Recurrence and Transience 92
3.3.1 Recurrence 92
3.3.2 Transience 96
3.4 Positive and Null Recurrence 99
3.4.1 General Criteria for Positive Recurrence 99
3.4.2 Path Excursions 103
3.4.3 Positive Recurrence under Linearization 103
3.4.4 Null Recurrence 107
3.5 Examples 108
3.6 Proofs of Several Results 114
3.7 Notes 122
4 Ergodicity 124
4.1 Introduction 124
4.2 Ergodicity 125
4.3 Feedback Controls for Weak Stabilization 132
4.4 Rami cations 138
4.5 Asymptotic Distribution 142
4.6 Notes 146
Part II Numerical Solutions and Approximation 148
5 Numerical Approximation 149
5.1 Introduction 149
5.2 Formulation 150
5.3 Numerical Algorithms 151
5.4 Convergence of the Algorithm 152
5.4.1 Moment Estimates 152
5.4.2 Weak Convergence 156
5.5 Examples 163
5.6 Discussions and Remarks 164
5.6.1 Remarks on Rates of Convergence 165
5.6.2 Remarks on Decreasing Stepsize Algorithms 167
5.7 Notes 168
6 Numerical Approximation to Invariant Measures 170
6.1 Introduction 170
6.2 Tightness of Approximation Sequences 172
6.3 Convergence to Invariant Measures 176
6.4 Proof: Convergence of Algorithm 180
6.5 Notes 189
Part III Stability 191
7 Stability 192
7.1 Introduction 192
7.2 Formulation and Auxiliary Results 193
7.3 p-Stability 197
7.3.1 Stability 197
7.3.2 Auxiliary Results 202
7.3.3 Necessary and Su cient Conditions for p-Stability 210
7.4 Stability and Instability of Linearized Systems 212
7.5 Examples 217
7.6 Notes 224
8 Stability of Switching ODEs 225
8.1 Introduction 225
8.2 Formulation and Preliminary Results 227
8.2.1 Problem Setup 227
8.2.2 Preliminary Results 228
8.3 Stability and Instability: Su cient Conditions 235
8.4 A Sharper Result 239
8.5 Remarks on Liapunov Exponent 244
8.5.1 Stability under General Setup 244
8.5.2 Invariant Density 246
8.6 Examples 249
8.7 Notes 255
9 Invariance Principles 259
9.1 Introduction 259
9.2 Formulation 259
9.3 Invariance (I): A Sample Path Approach 261
9.3.1 Invariant Sets 262
9.3.2 Linear Systems 271
9.4 Invariance (II): A Measure-Theoretic Approach 273
9.4.1 !-Limit Sets and Invariant Sets 277
9.4.2 Switching Di usions 283
9.5 Notes 288
Part IV Two-Time-Scale Modeling and Applications 290
10 Positive Recurrence: Weakly Connected Ergodic Classes 291
10.1 Introduction 291
10.2 Problem Setup and Notation 291
10.3 Weakly Connected, Multiergodic-ClassSwitching Processes 292
10.3.1 Preliminary 293
10.3.2 Weakly Connected, Multiple Ergodic ClassesSuppose 294
10.3.3 Inclusion of Transient Discrete Events 303
11 Stochastic Volatility Using Regime-Switching Di usions 307
11.1 Introduction 307
11.2 Formulation 309
11.3 Asymptotic Expansions 312
11.3.1 Construction of '0(S t
11.3.2 Construction of '1(S t
11.3.3 Construction of 'k(S t) and k(S
11.4 Asymptotic Error Bounds 323
11.5 Notes 327
12 Two-Time-Scale Switching JumpDi usions 328
12.1 Introduction 328
12.2 Fast-Varying Switching 331
12.2.1 Fast-Varying Markov Chain Model 331
12.2.2 Limit System 334
12.3 Fast-Varying Di usion 344
12.4 Discussion and Remarks 353
12.5 Remarks on Numerical Solutions forSwitching Jump Di usions 354
12.6 Notes 357
Appendix A 359
A.1 Discrete-Time Markov Chains 359
A.2 Continuous-Time Markov Chains 362
A.3 Fredholm Alternative and Rami cation 366
A.4 Martingales, Gaussian Processes, andDi usions 370
A.4.1 Martingales 370
A.4.2 Gaussian Processes and Di usion Processes 373
A.5 Weak Convergence 375
A.6 Hybrid Jump Di usion 380
A.7 Miscellany 381
A.8 Notes 382
References 383
Index 396
| Erscheint lt. Verlag | 3.10.2009 |
|---|---|
| Reihe/Serie | Stochastic Modelling and Applied Probability |
| Zusatzinfo | XVIII, 398 p. |
| Verlagsort | New York |
| Sprache | englisch |
| Themenwelt | Informatik ► Theorie / Studium ► Künstliche Intelligenz / Robotik |
| Mathematik / Informatik ► Mathematik ► Statistik | |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Technik ► Elektrotechnik / Energietechnik | |
| Technik ► Maschinenbau | |
| Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation | |
| Schlagworte | continuous dynamics • continuous-state-dependent switching process • discrete event • hybrid system • Markov Chain • measure • recurrence • stability • switching diffusion |
| ISBN-10 | 1-4419-1105-7 / 1441911057 |
| ISBN-13 | 978-1-4419-1105-6 / 9781441911056 |
| Haben Sie eine Frage zum Produkt? |
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