Random ordinary differential equations and their numerical solution:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore
Springer
[2017]
|
| Series: | Probability theory and stochastic modelling
Volume 85 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 250 Seiten Diagramme |
| ISBN: | 9789811062643 |
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MARC
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Record in the Search Index
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| adam_text | Contents
Part I Random and Stochastic Ordinary Differential Equations
1 Introduction........................................................... 3
1.1 Simple Numerical Schemes for RODEs.............................. 4
1.2 Taylor Expansions for RODEs..................................... 6
1.2.1 ODE Case................................................ 6
1.2.2 SODECase................................................ 7
1.2.3 RODE Case............................................... 8
1.3 RODEs with Bounded Noise....................................... 11
1.4 RODEs and Caratheodory ODEs.................................... 12
1.5 Endnotes....................................................... 13
2 Random Ordinary Differential Equations................................ 15
2.1 Existence and Uniqueness Theorems.............................. 16
2.1.1 Classical Assumptions.................................. 16
2.1.2 Measurability of Solutions............................. 20
2.1.3 Caratheodory Assumptions............................... 21
2.1.4 Positivity of Solutions................................ 24
2.2 RODEs with Canonical Noise.................................... 25
2.3 Endnotes....................................................... 26
3 Stochastic Differential Equations..................................... 29
3.1 Wiener Processes and Ito Integrals............................. 30
3.2 Ito Stochastic Differential Equations.......................... 31
3.3 The ltd Formula: The Stochastic Chain Rule..................... 33
3.4 Stratonovich SODEs............................................. 34
3.5 Relationship Between RODEs and SODEs........................... 35
3.5.1 Doss-Sussmann Transformation........................... 35
3.6 Endnotes..................................................... 36
xiii
xiv
Contents
4 Random Dynamical Systems.......................................... 37
4.1 Nontrivial Equilibrium Solutions............................... 37
4.2 Random Dynamical Systems....................................... 42
4.2.1 Random Attractors...................................... 43
4.2.2 Contractive Cocycles................................... 46
4.3 Endnotes....................................................... 47
5 Numerical Dynamics................................................ 49
5.1 Discretisation of Random Attractors............................ 49
5.2 Discretisation of a Random Hyperbolic Point.................... 53
5.3 Endnotes....................................................... 58
Part II Taylor Expansions
6 Taylor Expansions for Ordinary and Stochastic Differential
Equations........................................................... 61
6.1 Taylor Approximations for ODEs................................ 61
6.2 Taylor Approximations of Ito SODEs............................ 62
6.2.1 Multi-indices......................................... 64
6.2.2 Multiple Integrals of Stochastic Processes............ 64
6.2.3 Coefficient Functions................................. 65
6.2.4 Hierarchical and Remainder Sets....................... 66
6.3 General Itô-Taylor Expansions................................. 66
6.4 Strong Itô—Taylor Approximations.............................. 69
6.4.1 Examples............................................. 70
6.4.2 Pathwise Convergence.................................. 70
6.5 Endnotes...................................................... 72
7 Taylor Expansions for RODEs with Affine Noise.................... 73
7.1 An Illustrative Example....................................... 74
7.2 Affine-RODE Taylor Expansions................................. 76
7.3 General Affine-RODE-Taylor Approximations..................... 77
7.4 Endnotes.................................................... 79
8 Taylor-Like Expansions for General Random Ordinary
Differential Equations.............................................. 81
8.1 Preliminaries and Notation.................................... 81
8.1.1 Regularity of the Driving Stochastic Process.......... 82
8.1.2 Multi-index Notation.................................. 83
8.1.3 Iterated Integrals.................................... 84
8.1.4 Function Spaces....................................... 84
8.1.5 Iterated Differential Operators....................... 85
8.2 Integral Equation Expansions.................................. 86
8.3 RODE-Taylor Approximations.................................. 88
8.4 Essential RODE-Taylor Approximations.......................... 89
Contents xv
8.5 Examples ................................................... 91
8.6 Proof of Theorem 8.1........................................ 94
8.7 Endnotes.................................................. 98
Part III Numerical Schemes for Random Ordinary Differential
Equations
9 Numerical Methods for Ordinary and Stochastic Differential
Equations....................................................... 101
9.1 One-Step Numerical Schemes for ODEs........................ 101
9.2 One-Step Numerical Schemes for Ito SODEs................... 104
9.3 Strong Taylor Schemes for Ito SODEs........................ 105
9.4 Endnotes................................................... 108
10 It6-Taylor Schemes for RODEs with ltd Noise.................... 109
10.1 One-Step Schemes........................................... 110
10.1.1 Scalar Case........................................ 110
10.1.2 Vector Case........................................ Ill
10.1.3 Examples........................................... 113
10.1.4 Derivative-Free Explicit Strong Schemes............ 114
10.2 Implicit Strong Schemes.................................... 116
10.3 Multi-step Schemes......................................... 117
10.4 RODEs with Affine Noise.................................... 124
10.5 Proof of Theorem 10.1...................................... 125
10.6 Endnotes................................................... 127
11 Numerical Schemes for RODEs with Affine Noise.................. 129
11.1 Affine-RODE Taylor Schemes for Bounded Noise. ............. 130
11.2 Affine-RODEs with Special Structure........................ 133
11.2.1 Additive Noise..................................... 133
11.2.2 Commutative Noise.................................. 133
11.3 Affine-RODE Derivative-Free Schemes ....................... 134
11.3.1 Finite Difference Approximation of Derivatives... 134
11.3.2 Runge-Kutta Schemes for Affine-RODE................ 136
11.4 Linear Multi-step Methods for Affine RODEs................. 139
11.5 Endnotes................................................... 142
12 RODE-Taylor Schemes: General Case............................. 143
12.1 RODE-Taylor Schemes........................................ 143
12.1.1 The Essential RODE-Taylor Schemes.................. 146
12.2 Examples of the RODE-Taylor Schemes........................ 147
12.3 RODEs with Affine Noise.................................. 149
12.4 Other Numerical Schemes for RODEs.......................... 150
12.4.1 The Local Linearisation Scheme for RODEs........ 150
12.4.2 The Averaged Euler Scheme.......................... 152
XVI
Contents
12.4.3 Heuristic RODE-Taylor Schemes....................... 152
12.5 Endnotes................................................ 153
13 Numerical Stability............................................... 155
13.1 B-Stability of the Implicit Averaged Schemes .............. 156
13.2 B-Stability of the Implicit Multi-step Schemes............. 158
13.3 Endnotes................................................... 161
14 Stochastic Integrals: Simulation and Approximation................ 163
14.1 Calculating a Finer Approximation of the Same
Sample Path................................................ 164
14.2 Integral of a Wiener Process............................... 165
14.3 Integral of an Omstein-Uhlenbeck Process................... 165
14.4 Fractional Brownian Motion................................. 167
14.4.1 Riemann Integral of an fBm......................... 172
14.4.2 Riemann Sums Approximation.......................... 173
14.4.3 Comparison of Computational Costs................... 174
14.5 Integrals of Compound Poisson Processes.................... 177
14.6 Endnotes................................................. 179
Part IV Random Ordinary Differential Equations in the Life
Sciences
15 Comparative Simulations of Biological Systems..................... 183
15.1 Tumor Inhibition Model..................................... 183
15.2 Population Dynamics........................................ 185
15.3 Toggle Switch Model........................................ 187
15.4 Sea Shell Pattern Model.................................... 189
15.5 Endnotes.................................................. 192
16 Chemostat......................................................... 193
16.1 Random Chemostat Models.................................... 194
16.2 RDS Generated by Random Chemostat.......................... 196
16.3 Existence of a Random Attractor............................ 198
16.4 Endnotes................................................... 203
17 Immune System Virus Model......................................... 205
17.1 Properties of Solutions ................................... 206
17.2 Existence of Global Random Attractors...................... 208
17.3 Numerical Simulations...................................... 212
17.4 Endnotes................................................... 214
18 Random Markov Chains.............................................. 215
18.1 Random Environment......................................... 216
18.2 Positivity of Solutions of Linear RODEs.................... 217
18.3 Linear Random Dynamical Systems............................ 220
Contents
xvii
18.4 Random Attractor Under Discretisation
18.5 Endnotes.........................
Appendix A: Probability Spaces..............
Appendix B: Chain Rule for Affine RODEs
Appendix C: Covariance Matrix of a Fractional Brownian
Motion and Its Integral.......................
References...........................
Index...................
. . . 221
. . . 222
. . . 223
... 227
. . 231
. . 241
. . 247
|
| any_adam_object | 1 |
| author | Han, Xiaoying Kloeden, Peter E. 1949- |
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| id | DE-604.BV044537074 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:55:18Z |
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| isbn | 9789811062643 |
| language | English |
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| physical | xvii, 250 Seiten Diagramme |
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| series | Probability theory and stochastic modelling |
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| spelling | Han, Xiaoying (DE-588)1147610703 aut Random ordinary differential equations and their numerical solution Xiaoying Han, Peter E. Kloeden Singapore Springer [2017] © 2017 xvii, 250 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Probability theory and stochastic modelling Volume 85 Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Kloeden, Peter E. 1949- (DE-588)115479155 aut Erscheint auch als Online-Ausgabe 978-981-10-6265-0 Probability theory and stochastic modelling Volume 85 (DE-604)BV042008213 85 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029936219&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Han, Xiaoying Kloeden, Peter E. 1949- Random ordinary differential equations and their numerical solution Probability theory and stochastic modelling Differentialgleichung (DE-588)4012249-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4128130-5 |
| title | Random ordinary differential equations and their numerical solution |
| title_auth | Random ordinary differential equations and their numerical solution |
| title_exact_search | Random ordinary differential equations and their numerical solution |
| title_full | Random ordinary differential equations and their numerical solution Xiaoying Han, Peter E. Kloeden |
| title_fullStr | Random ordinary differential equations and their numerical solution Xiaoying Han, Peter E. Kloeden |
| title_full_unstemmed | Random ordinary differential equations and their numerical solution Xiaoying Han, Peter E. Kloeden |
| title_short | Random ordinary differential equations and their numerical solution |
| title_sort | random ordinary differential equations and their numerical solution |
| topic | Differentialgleichung (DE-588)4012249-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029936219&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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