Stochastic differential equations with Markovian switching:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Imperial College Press [u.a.]
2006
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 409 S. |
| ISBN: | 1860947018 |
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Record in the Search Index
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| adam_text | WI TH MARKOVIA N S W I T C H I N G XUERONG MAO UNIVERSITY OF
STRATHCLYDE, UK CHENGGUIYUAN UNIVERSITY OF WALES SWANSEA, UK ICP
IMPERIAL COLLEGE PRESS CONTENTS PREFACE VII NOTATION XV 1. BROWNIAN
MOTIONS AND STOCHASTIC INTEGRALS 1 1.1 INTRODUCTION 1 1.2 PROBABILITY
THEORY 4 1.3 STOCHASTIC PROCESSES 12 1.4 BROWNIAN MOTIONS 18 1.5
STOCHASTIC INTEGRALS 23 1.6 ITO S FORMULA 38 1.7 MARKOV PROCESSES 43 1.8
GENERALISED ITO S FORMULA 47 1.9 EXERCISES 49 2. INEQUALITIES 51 2.1
INTRODUCTION 51 2.2 FREQUENTLY USED INEQUALITIES 51 2.3 GRONWALL-TYPE
INEQUALITIES 54 2.4 MATRICES AND INEQUALITIES 58 2.5 LINEAR MATRIX
INEQUALITIES 62 2.6 M-MATRIX INEQUALITIES 67 2.7 STOCHASTIC INEQUALITIES
69 2.8 EXERCISES 75 3. STOCHASTIC DIFFERENTIAL EQUATIONS WITH MARKOVIAN
SWITCHING 77 XI XII SDES WITH MARKOVIAN SWITCHING 3.1 INTRODUCTION 77
3.2 STOCHASTIC DIFFERENTIAL EQUATIONS 77 3.3 EXISTENCE AND UNIQUENESS OF
SOLUTIONS 81 3.4 SDES WITH MARKOVIAN SWITCHING 88 3.5 L P -ESTIMATES 96
3.6 ALMOST SURELY ASYMPTOTIC ESTIMATES 101 3.7 SOLUTIONS AS MARKOV
PROCESSES 104 3.8 EXERCISES 110 4. APPROXIMATE SOLUTIONS 111 4.1
INTRODUCTION 111 4.2 EULER-MARUYAMA S APPROXIMATIONS 111 4.2.1 GLOBAL
LIPSCHITZ CASE 114 4.2.2 LOCAL LIPSCHITZ CASE 118 4.2.3 MORE ON LOCAL
LIPSCHITZ CASE 121 4.3 CARATHEODORY S APPROXIMATIONS 126 4.4 SPLIT-STEP
BACKWARD EULER SCHEME 134 4.5 BACKWARD EULER SCHEME 146 4.6 STOCHASTIC
THETA METHOD 149 4.7 EXERCISES 154 5. BOUNDEDNESS AND STABILITY 155 5.1
INTRODUCTION 155 5.2 ASYMPTOTIC BOUNDEDNESS 157 5.3 EXPONENTIAL
STABILITY 164 5.3.1 NONLINEAR JUMP SYSTEMS 178 5.3.2 MULTI-DIMENSIONAL
LINEAR EQUATIONS 180 5.3.3 SCALAR LINEAR EQUATIONS 182 5.3.4 EXAMPLES
187 5.4 MOMENT AND ALMOST SURE ASYMPTOTIC STABILITY 191 5.5 STABILITY IN
PROBABILITY 204 5.6 ASYMPTOTIC STABILITY IN DISTRIBUTION 210 5.7
EXERCISES 226 6. NUMERICAL METHODS FOR ASYMPTOTIC PROPERTIES 229 6.1
INTRODUCTION 229 6.2 EULER-MARUYAMA S METHOD AND EXPONENTIAL STABILITY .
. . 230 6.3 EULER-MARUYAMA S METHOD AND LYAPUNOV EXPONENTS . . . 239
CONTENTS XIII 6.4 GENERALISED RESULTS AND STOCHASTIC THETA METHOD 241
6.5 ASYMPTOTIC STABILITY IN DISTRIBUTION OF THE EM METHOD: CONSTANT STEP
SIZE 249 6.5.1 STABILITY IN DISTRIBUTION OF THE EM METHOD 249 6.5.2
SUFRCIENT CRITERIA FOR ASSUMPTIONS 6.16-6.18 . . . . 256 6.5.3
CONVERGENCE OF STATIONARY DISTRIBUTIONS 265 6.6 ASYMPTOTIC STABILITY IN
DISTRIBUTION OF THE EM METHOD: VARIABLE STEP SIZES 267 6.7 EXERCISES 270
7. STOCHASTIC DIFFERENTIAL DELAY EQUATIONS WITH MARKOVIAN SWITCHING 271
7.1 INTRODUCTION 271 7.2 STOCHASTIC DIFFERENTIAL DELAY EQUATIONS 273 7.3
SDDES WITH MARKOVIAN SWITCHING 277 7.4 MOMENT PROPERTIES 282 7.5
ASYMPTOTIC BOUNDEDNESS 285 7.6 EXPONENTIAL STABILITY 289 7.7 APPROXIMATE
SOLUTIONS 294 7.8 EXERCISES 300 8. STOCHASTIC FUNCTIONAL DIFFERENTIAL
EQUATIONS WITH MARKO- VIAN SWITCHING 301 8.1 INTRODUCTION 301 8.2
STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 301 8.3 SFDES WITH
MARKOVIAN SWITCHING 303 8.4 BOUNDEDNESS 305 8.5 ASYMPTOTIC STABILITY 308
8.6 RAZUMIKHIN-TYPE THEOREMS ON STABILITY 311 8.7 EXAMPLES 314 8.8
EXERCISES 317 9. STOCHASTIC INTERVAL SYSTEMS WITH MARKOVIAN SWITCHING
319 9.1 INTRODUCTION 319 9.2 INTERVAL MATRICES 320 9.3 SDISS WITH
MARKOVIAN SWITCHING 322 9.4 RAZUMIKHIN TECHNOLOGY ON SDISS 328 9.4.1
DELAY INDEPENDENT CRITERIA 328 9.4.2 DELAY DEPENDENT CRITERIA 334 XIV
SDES WITH MARKOVIAN SWITCHING 9.4.3 EXAMPLES 341 9.5 SISS WITH MARKOVIAN
SWITCHING 346 9.6 EXERCISES 349 10. APPLICATIONS 351 10.1 INTRODUCTION
351 10.2 STOCHASTIC POPULATION DYNAMICS 351 10.2.1 GLOBAL POSITIVE
SOLUTIONS 353 10.2.2 ULTIMATE BOUNDEDNESS 356 10.2.3 MOMENT AVERAGE IN
TIME 358 10.3 STOCHASTIC FINANCIAL MODELLING 360 10.3.1 NON-NEGATIVE
SOLUTIONS 361 10.3.2 THE EM APPROXIMATIONS 363 10.3.3 STOCHASTIC
VOLATILITY MODEL 370 10.3.4 OPTIONS UNDER STOCHASTIC VOLATILITY 375 10.4
STOCHASTIC STABILISATION AND DESTABILISATION 379 10.5 STOCHASTIC NEURAL
NETWORKS 387 10.6 EXERCISES 394 BIBLIOGRAPHICAL NOTES 395 BIBLIOGRAPHY
397 INDEX 407
|
| adam_txt |
WI TH MARKOVIA N S W I T C H I N G XUERONG MAO UNIVERSITY OF
STRATHCLYDE, UK CHENGGUIYUAN UNIVERSITY OF WALES SWANSEA, UK ICP
IMPERIAL COLLEGE PRESS CONTENTS PREFACE VII NOTATION XV 1. BROWNIAN
MOTIONS AND STOCHASTIC INTEGRALS 1 1.1 INTRODUCTION 1 1.2 PROBABILITY
THEORY 4 1.3 STOCHASTIC PROCESSES 12 1.4 BROWNIAN MOTIONS 18 1.5
STOCHASTIC INTEGRALS 23 1.6 ITO'S FORMULA 38 1.7 MARKOV PROCESSES 43 1.8
GENERALISED ITO'S FORMULA 47 1.9 EXERCISES 49 2. INEQUALITIES 51 2.1
INTRODUCTION 51 2.2 FREQUENTLY USED INEQUALITIES 51 2.3 GRONWALL-TYPE
INEQUALITIES 54 2.4 MATRICES AND INEQUALITIES 58 2.5 LINEAR MATRIX
INEQUALITIES 62 2.6 M-MATRIX INEQUALITIES 67 2.7 STOCHASTIC INEQUALITIES
69 2.8 EXERCISES 75 3. STOCHASTIC DIFFERENTIAL EQUATIONS WITH MARKOVIAN
SWITCHING 77 XI XII SDES WITH MARKOVIAN SWITCHING 3.1 INTRODUCTION 77
3.2 STOCHASTIC DIFFERENTIAL EQUATIONS 77 3.3 EXISTENCE AND UNIQUENESS OF
SOLUTIONS 81 3.4 SDES WITH MARKOVIAN SWITCHING 88 3.5 L P -ESTIMATES 96
3.6 ALMOST SURELY ASYMPTOTIC ESTIMATES 101 3.7 SOLUTIONS AS MARKOV
PROCESSES 104 3.8 EXERCISES 110 4. APPROXIMATE SOLUTIONS 111 4.1
INTRODUCTION 111 4.2 EULER-MARUYAMA'S APPROXIMATIONS 111 4.2.1 GLOBAL
LIPSCHITZ CASE 114 4.2.2 LOCAL LIPSCHITZ CASE 118 4.2.3 MORE ON LOCAL
LIPSCHITZ CASE 121 4.3 CARATHEODORY'S APPROXIMATIONS 126 4.4 SPLIT-STEP
BACKWARD EULER SCHEME 134 4.5 BACKWARD EULER SCHEME 146 4.6 STOCHASTIC
THETA METHOD 149 4.7 EXERCISES 154 5. BOUNDEDNESS AND STABILITY 155 5.1
INTRODUCTION 155 5.2 ASYMPTOTIC BOUNDEDNESS 157 5.3 EXPONENTIAL
STABILITY 164 5.3.1 NONLINEAR JUMP SYSTEMS 178 5.3.2 MULTI-DIMENSIONAL
LINEAR EQUATIONS 180 5.3.3 SCALAR LINEAR EQUATIONS 182 5.3.4 EXAMPLES
187 5.4 MOMENT AND ALMOST SURE ASYMPTOTIC STABILITY 191 5.5 STABILITY IN
PROBABILITY 204 5.6 ASYMPTOTIC STABILITY IN DISTRIBUTION 210 5.7
EXERCISES 226 6. NUMERICAL METHODS FOR ASYMPTOTIC PROPERTIES 229 6.1
INTRODUCTION 229 6.2 EULER-MARUYAMA'S METHOD AND EXPONENTIAL STABILITY .
. . 230 6.3 EULER-MARUYAMA'S METHOD AND LYAPUNOV EXPONENTS . . . 239
CONTENTS XIII 6.4 GENERALISED RESULTS AND STOCHASTIC THETA METHOD 241
6.5 ASYMPTOTIC STABILITY IN DISTRIBUTION OF THE EM METHOD: CONSTANT STEP
SIZE 249 6.5.1 STABILITY IN DISTRIBUTION OF THE EM METHOD 249 6.5.2
SUFRCIENT CRITERIA FOR ASSUMPTIONS 6.16-6.18 . . . . 256 6.5.3
CONVERGENCE OF STATIONARY DISTRIBUTIONS 265 6.6 ASYMPTOTIC STABILITY IN
DISTRIBUTION OF THE EM METHOD: VARIABLE STEP SIZES 267 6.7 EXERCISES 270
7. STOCHASTIC DIFFERENTIAL DELAY EQUATIONS WITH MARKOVIAN SWITCHING 271
7.1 INTRODUCTION 271 7.2 STOCHASTIC DIFFERENTIAL DELAY EQUATIONS 273 7.3
SDDES WITH MARKOVIAN SWITCHING 277 7.4 MOMENT PROPERTIES 282 7.5
ASYMPTOTIC BOUNDEDNESS 285 7.6 EXPONENTIAL STABILITY 289 7.7 APPROXIMATE
SOLUTIONS 294 7.8 EXERCISES 300 8. STOCHASTIC FUNCTIONAL DIFFERENTIAL
EQUATIONS WITH MARKO- VIAN SWITCHING 301 8.1 INTRODUCTION 301 8.2
STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 301 8.3 SFDES WITH
MARKOVIAN SWITCHING 303 8.4 BOUNDEDNESS 305 8.5 ASYMPTOTIC STABILITY 308
8.6 RAZUMIKHIN-TYPE THEOREMS ON STABILITY 311 8.7 EXAMPLES 314 8.8
EXERCISES 317 9. STOCHASTIC INTERVAL SYSTEMS WITH MARKOVIAN SWITCHING
319 9.1 INTRODUCTION 319 9.2 INTERVAL MATRICES 320 9.3 SDISS WITH
MARKOVIAN SWITCHING 322 9.4 RAZUMIKHIN TECHNOLOGY ON SDISS 328 9.4.1
DELAY INDEPENDENT CRITERIA 328 9.4.2 DELAY DEPENDENT CRITERIA 334 XIV
SDES WITH MARKOVIAN SWITCHING 9.4.3 EXAMPLES 341 9.5 SISS WITH MARKOVIAN
SWITCHING 346 9.6 EXERCISES 349 10. APPLICATIONS 351 10.1 INTRODUCTION
351 10.2 STOCHASTIC POPULATION DYNAMICS 351 10.2.1 GLOBAL POSITIVE
SOLUTIONS 353 10.2.2 ULTIMATE BOUNDEDNESS 356 10.2.3 MOMENT AVERAGE IN
TIME 358 10.3 STOCHASTIC FINANCIAL MODELLING 360 10.3.1 NON-NEGATIVE
SOLUTIONS 361 10.3.2 THE EM APPROXIMATIONS 363 10.3.3 STOCHASTIC
VOLATILITY MODEL 370 10.3.4 OPTIONS UNDER STOCHASTIC VOLATILITY 375 10.4
STOCHASTIC STABILISATION AND DESTABILISATION 379 10.5 STOCHASTIC NEURAL
NETWORKS 387 10.6 EXERCISES 394 BIBLIOGRAPHICAL NOTES 395 BIBLIOGRAPHY
397 INDEX 407 |
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| id | DE-604.BV023045750 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:22:29Z |
| indexdate | 2024-07-09T21:09:43Z |
| institution | BVB |
| isbn | 1860947018 |
| language | English |
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| owner_facet | DE-29T DE-20 DE-188 |
| physical | XVIII, 409 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Imperial College Press [u.a.] |
| record_format | marc |
| spelling | Mao, Xuerong 1957- Verfasser (DE-588)1145722040 aut Stochastic differential equations with Markovian switching Xuerong Mao ; Chenggui Yuan London Imperial College Press [u.a.] 2006 XVIII, 409 S. txt rdacontent n rdamedia nc rdacarrier Markov, Processus de Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s Markov-Prozess (DE-588)4134948-9 s DE-604 Yuan, Chenggui Verfasser (DE-588)1041769113 aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016249206&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mao, Xuerong 1957- Yuan, Chenggui Stochastic differential equations with Markovian switching Markov, Processus de Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd Markov-Prozess (DE-588)4134948-9 gnd |
| subject_GND | (DE-588)4057621-8 (DE-588)4134948-9 |
| title | Stochastic differential equations with Markovian switching |
| title_auth | Stochastic differential equations with Markovian switching |
| title_exact_search | Stochastic differential equations with Markovian switching |
| title_exact_search_txtP | Stochastic differential equations with Markovian switching |
| title_full | Stochastic differential equations with Markovian switching Xuerong Mao ; Chenggui Yuan |
| title_fullStr | Stochastic differential equations with Markovian switching Xuerong Mao ; Chenggui Yuan |
| title_full_unstemmed | Stochastic differential equations with Markovian switching Xuerong Mao ; Chenggui Yuan |
| title_short | Stochastic differential equations with Markovian switching |
| title_sort | stochastic differential equations with markovian switching |
| topic | Markov, Processus de Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd Markov-Prozess (DE-588)4134948-9 gnd |
| topic_facet | Markov, Processus de Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung Markov-Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016249206&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT maoxuerong stochasticdifferentialequationswithmarkovianswitching AT yuanchenggui stochasticdifferentialequationswithmarkovianswitching |