Yin / Zhu Hybrid Switching Diffusions

Properties and Applications
2010
ISBN: 978-1-4419-1105-6
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 63, 398 Seiten

Reihe: Stochastic Modelling and Applied Probability

This book encompasses the study of hybrid switching di usion processes and their applications. The word \hybrid' 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.

Autoren/Hrsg.

Yin, G. George

Zhu, Chao

Inhaltsverzeichnis

1;Contents;7
2;Preface;11
3;Conventions;14
4;Glossary of Symbols;15
5;1 Introduction and Motivation;17
5.1;1.1 Introduction;17
5.2;1.2 Motivation;17
5.3;1.3 What Is a Switching Di usion;20
5.4;1.4 Examples of Switching Di usions;21
5.5;1.5 Outline of the Book;37
6;Part I Basic Properties, Recurrence, Ergodicity;41
7;2 Switching Diffusion;42
7.1;2.1 Introduction;42
7.2;2.2 Switching Di usions;42
7.3;2.3 Regularity;48
7.4;2.4 Weak Continuity;53
7.5;2.5 Feller Property;56
7.6;2.6 Strong Feller Property;67
7.7;2.7 Continuous and Smooth Dependence on theInitial Data x;71
7.8;2.8 A Remark Regarding NonhomogeneousMarkov Processes;80
7.9;2.9 Notes;82
8;3 Recurrence;83
8.1;3.1 Introduction;83
8.2;3.2 Formulation and Preliminaries;84
8.2.1;3.2.1 Switching Di usion;84
8.2.2;3.2.2 De nitions of Recurrence and Positive Recurrence;86
8.2.3;3.2.3 Preparatory Results;86
8.3;3.3 Recurrence and Transience;92
8.3.1;3.3.1 Recurrence;92
8.3.2;3.3.2 Transience;96
8.4;3.4 Positive and Null Recurrence;99
8.4.1;3.4.1 General Criteria for Positive Recurrence;99
8.4.2;3.4.2 Path Excursions;103
8.4.3;3.4.3 Positive Recurrence under Linearization;103
8.4.4;3.4.4 Null Recurrence;107
8.5;3.5 Examples;108
8.6;3.6 Proofs of Several Results;114
8.7;3.7 Notes;122
9;4 Ergodicity;124
9.1;4.1 Introduction;124
9.2;4.2 Ergodicity;125
9.3;4.3 Feedback Controls for Weak Stabilization;132
9.4;4.4 Rami cations;138
9.5;4.5 Asymptotic Distribution;142
9.6;4.6 Notes;146
10;Part II Numerical Solutions and Approximation;148
11;5 Numerical Approximation;149
11.1;5.1 Introduction;149
11.2;5.2 Formulation;150
11.3;5.3 Numerical Algorithms;151
11.4;5.4 Convergence of the Algorithm;152
11.4.1;5.4.1 Moment Estimates;152
11.4.2;5.4.2 Weak Convergence;156
11.5;5.5 Examples;163
11.6;5.6 Discussions and Remarks;164
11.6.1;5.6.1 Remarks on Rates of Convergence;165
11.6.2;5.6.2 Remarks on Decreasing Stepsize Algorithms;167
11.7;5.7 Notes;168
12;6 Numerical Approximation to Invariant Measures;170
12.1;6.1 Introduction;170
12.2;6.2 Tightness of Approximation Sequences;172
12.3;6.3 Convergence to Invariant Measures;176
12.4;6.4 Proof: Convergence of Algorithm;180
12.5;6.5 Notes;189
13;Part III Stability;191
14;7 Stability;192
14.1;7.1 Introduction;192
14.2;7.2 Formulation and Auxiliary Results;193
14.3;7.3 p-Stability;197
14.3.1;7.3.1 Stability;197
14.3.2;7.3.2 Auxiliary Results;202
14.3.3;7.3.3 Necessary and Su cient Conditions for p-Stability;210
14.4;7.4 Stability and Instability of Linearized Systems;212
14.5;7.5 Examples;217
14.6;7.6 Notes;224
15;8 Stability of Switching ODEs;225
15.1;8.1 Introduction;225
15.2;8.2 Formulation and Preliminary Results;227
15.2.1;8.2.1 Problem Setup;227
15.2.2;8.2.2 Preliminary Results;228
15.3;8.3 Stability and Instability: Su cient Conditions;235
15.4;8.4 A Sharper Result;239
15.5;8.5 Remarks on Liapunov Exponent;244
15.5.1;8.5.1 Stability under General Setup;244
15.5.2;8.5.2 Invariant Density;246
15.6;8.6 Examples;249
15.7;8.7 Notes;255
16;9 Invariance Principles;259
16.1;9.1 Introduction;259
16.2;9.2 Formulation;259
16.3;9.3 Invariance (I): A Sample Path Approach;261
16.3.1;9.3.1 Invariant Sets;262
16.3.2;9.3.2 Linear Systems;271
16.4;9.4 Invariance (II): A Measure-Theoretic Approach;273
16.4.1;9.4.1 !-Limit Sets and Invariant Sets;277
16.4.2;9.4.2 Switching Di usions;283
16.5;9.5 Notes;288
17;Part IV Two-Time-Scale Modeling and Applications;290
18;10 Positive Recurrence: Weakly Connected Ergodic Classes;291
18.1;10.1 Introduction;291
18.2;10.2 Problem Setup and Notation;291
18.3;10.3 Weakly Connected, Multiergodic-ClassSwitching Processes;292
18.3.1;10.3.1 Preliminary;293
18.3.2;10.3.2 Weakly Connected, Multiple Ergodic ClassesSuppose;294
18.3.3;10.3.3 Inclusion of Transient Discrete Events;303
19;11 Stochastic Volatility Using Regime-Switching Di usions;307
19.1;11.1 Introduction;307
19.2;11.2 Formulation;309
19.3;11.3 Asymptotic Expansions;312
19.3.1;11.3.1 Construction of '0(S; t; i) and 0(S; ; i);314
19.3.2;11.3.2 Construction of '1(S; t; i) and 1(S; ; i);315
19.3.3;11.3.3 Construction of 'k(S; t) and k(S; );319
19.4;11.4 Asymptotic Error Bounds;323
19.5;11.5 Notes;327
20;12 Two-Time-Scale Switching JumpDi usions;328
20.1;12.1 Introduction;328
20.2;12.2 Fast-Varying Switching;331
20.2.1;12.2.1 Fast-Varying Markov Chain Model;331
20.2.2;12.2.2 Limit System;334
20.3;12.3 Fast-Varying Di usion;344
20.4;12.4 Discussion and Remarks;353
20.5;12.5 Remarks on Numerical Solutions forSwitching Jump Di usions;354
20.6;12.6 Notes;357
21;Appendix A;359
21.1;A.1 Discrete-Time Markov Chains;359
21.2;A.2 Continuous-Time Markov Chains;362
21.3;A.3 Fredholm Alternative and Rami cation;366
21.4;A.4 Martingales, Gaussian Processes, andDi usions;370
21.4.1;A.4.1 Martingales;370
21.4.2;A.4.2 Gaussian Processes and Di usion Processes;373
21.5;A.5 Weak Convergence;375
21.6;A.6 Hybrid Jump Di usion;380
21.7;A.7 Miscellany;381
21.8;A.8 Notes;382
22;References;383
23;Index;396