Modeling with Itô stochastic differential equations:
Saved in:
| Format: | Electronic eBook |
|---|---|
| Language: | English |
| Published: |
Dordrecht
Springer
2007
|
| Series: | Mathematical Modelling: Theory and Applications
22 |
| Subjects: | |
| Online Access: | BTU01 TUM01 UBA01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (XII, 228 S.) |
| ISBN: | 9781402059520 9781402059537 |
| DOI: | 10.1007/978-1-4020-5953-7 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface xi
1 Random Variables 1
1.1 Introduction 1
1.2 Probability Space 2
1.3 Random Variable, Probability Distribution 4
1.4 Expectation 7
1.5 Multiple Random Variables 9
1.6 A Hilbert Space of Random Variables 13
1.7 Convergence of Sequences of Random Variables 17
1.8 Computer Generation of Random Numbers 20
1.9 Monte Carlo 23
Exercises 27
Computer Programs 30
2 Stochastic Processes 33
2.1 Introduction 33
2.2 Discrete Stochastic Processes 34
2.3 Continuous Stochastic Processes 39
2.4 A Hilbert Space of Stochastic Processes 45
2.5 Computer Generation of Stochastic Processes 50
2.6 Examples of Stochastic Processes 52
Exercises 55
Computer Programs 59
3 Stochastic Integration 63
3.1 Introduction 63
3.2 Integrals of the Form fa f(s, u})ds 63
3.3 Ito Stochastic Integrals 67
3.4 Approximation of Stochastic Integrals 72
3.5 Stochastic Differentials and Ito s Formula 74
vii
viii Contents
3.6 Stratonovich Stochastic Integrals 80
3.7 Multidimensional Ito s Formula 82
Exercises 83
Computer Programs 87
4 Stochastic Differential Equations 89
4.1 Introduction 89
4.2 Existence of a Unique Solution 91
4.3 Properties of Solutions to Stochastic Differential Equations ... 93
4.4 Ito s Formula and Exact Solutions 95
4.5 Approximating Stochastic Differential Equations 99
4.6 Systems of Stochastic Differential Equations 107
4.7 Forward Kolmogorov (Fokker Planck) Equation 109
4.8 Stability Ill
4.9 Parameter Estimation for Stochastic Differential Equations ... 118
4.9.1 A maximum likelihood estimation method 118
4.9.2 A nonparametric estimation method 121
Exercises 123
Computer Programs 127
5 Modeling 135
5.1 Introduction 135
5.2 Population Biology Examples 145
5.2.1 General model of two interacting populations 145
5.2.2 Epidemic model and predator prey model 147
5.2.3 Persistence time estimation 150
5.2.4 A population model with a time delay 152
5.2.5 A model including environmental variability 153
5.3 Physical Systems 156
5.3.1 Mechanical vibration 156
5.3.2 Seed dispersal 158
5.3.3 Ion transport 160
5.3.4 Nuclear reactor kinetics 161
5.3.5 Precipitation 165
5.3.6 Chemical reactions 166
5.3.7 Cotton fiber breakage 169
5.4 Some Stochastic Finance Models 174
5.4.1 A stock price model 174
5.4.2 Option pricing 177
5.4.3 Interest rates 180
5.5 A Goodness of Fit Test for an SDE Model 183
5.6 Alternate Equivalent SDE Models 186
Exercises 193
Computer Programs 199
Contents ix
References 217
Basic Notation 223
Index 225
|
| adam_txt |
Contents
Preface xi
1 Random Variables 1
1.1 Introduction 1
1.2 Probability Space 2
1.3 Random Variable, Probability Distribution 4
1.4 Expectation 7
1.5 Multiple Random Variables 9
1.6 A Hilbert Space of Random Variables 13
1.7 Convergence of Sequences of Random Variables 17
1.8 Computer Generation of Random Numbers 20
1.9 Monte Carlo 23
Exercises 27
Computer Programs 30
2 Stochastic Processes 33
2.1 Introduction 33
2.2 Discrete Stochastic Processes 34
2.3 Continuous Stochastic Processes 39
2.4 A Hilbert Space of Stochastic Processes 45
2.5 Computer Generation of Stochastic Processes 50
2.6 Examples of Stochastic Processes 52
Exercises 55
Computer Programs 59
3 Stochastic Integration 63
3.1 Introduction 63
3.2 Integrals of the Form fa f(s, u})ds 63
3.3 Ito Stochastic Integrals 67
3.4 Approximation of Stochastic Integrals 72
3.5 Stochastic Differentials and Ito's Formula 74
vii
viii Contents
3.6 Stratonovich Stochastic Integrals 80
3.7 Multidimensional Ito's Formula 82
Exercises 83
Computer Programs 87
4 Stochastic Differential Equations 89
4.1 Introduction 89
4.2 Existence of a Unique Solution 91
4.3 Properties of Solutions to Stochastic Differential Equations . 93
4.4 Ito's Formula and Exact Solutions 95
4.5 Approximating Stochastic Differential Equations 99
4.6 Systems of Stochastic Differential Equations 107
4.7 Forward Kolmogorov (Fokker Planck) Equation 109
4.8 Stability Ill
4.9 Parameter Estimation for Stochastic Differential Equations . 118
4.9.1 A maximum likelihood estimation method 118
4.9.2 A nonparametric estimation method 121
Exercises 123
Computer Programs 127
5 Modeling 135
5.1 Introduction 135
5.2 Population Biology Examples 145
5.2.1 General model of two interacting populations 145
5.2.2 Epidemic model and predator prey model 147
5.2.3 Persistence time estimation 150
5.2.4 A population model with a time delay 152
5.2.5 A model including environmental variability 153
5.3 Physical Systems 156
5.3.1 Mechanical vibration 156
5.3.2 Seed dispersal 158
5.3.3 Ion transport 160
5.3.4 Nuclear reactor kinetics 161
5.3.5 Precipitation 165
5.3.6 Chemical reactions 166
5.3.7 Cotton fiber breakage 169
5.4 Some Stochastic Finance Models 174
5.4.1 A stock price model 174
5.4.2 Option pricing 177
5.4.3 Interest rates 180
5.5 A Goodness of Fit Test for an SDE Model 183
5.6 Alternate Equivalent SDE Models 186
Exercises 193
Computer Programs 199
Contents ix
References 217
Basic Notation 223
Index 225 |
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| spelling | Modeling with Itô stochastic differential equations by Edward Allen Dordrecht Springer 2007 1 Online-Ressource (XII, 228 S.) txt rdacontent c rdamedia cr rdacarrier Mathematical Modelling: Theory and Applications 22 Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s DE-604 Allen, Edward Sonstige oth Erscheint auch als Druck-Ausgabe, Hardcover 1-402-05952-3 Erscheint auch als Druck-Ausgabe, Hardcover 9781402059520 Mathematical Modelling: Theory and Applications 22 (DE-604)BV011613239 22 https://doi.org/10.1007/978-1-4020-5953-7 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015632619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Modeling with Itô stochastic differential equations Mathematical Modelling: Theory and Applications Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| subject_GND | (DE-588)4057621-8 |
| title | Modeling with Itô stochastic differential equations |
| title_auth | Modeling with Itô stochastic differential equations |
| title_exact_search | Modeling with Itô stochastic differential equations |
| title_exact_search_txtP | Modeling with Itô stochastic differential equations |
| title_full | Modeling with Itô stochastic differential equations by Edward Allen |
| title_fullStr | Modeling with Itô stochastic differential equations by Edward Allen |
| title_full_unstemmed | Modeling with Itô stochastic differential equations by Edward Allen |
| title_short | Modeling with Itô stochastic differential equations |
| title_sort | modeling with ito stochastic differential equations |
| topic | Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Stochastische Differentialgleichung |
| url | https://doi.org/10.1007/978-1-4020-5953-7 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015632619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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