Stochastic differential equations: an introduction with applications
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1992
|
| Edition: | 3. ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 224 S. |
| ISBN: | 3540533354 0387533354 |
Staff View
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| 245 | 1 | 0 | |a Stochastic differential equations |b an introduction with applications |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1992 | |
| 300 | |a XIII, 224 S. | ||
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Record in the Search Index
| _version_ | 1804135852788416512 |
|---|---|
| adam_text | XI
Contents
I. INTRODUCTION 1
Some problems (1 6) where stochastic differential equations play an essential
role in the solution
II. SOME MATHEMATICAL PRELIMINARIES 5
Random variables, independence, stochastic processes 5
Kolmogorov s extension theorem 7
Brownian motion 7
Basic properties of Brownian motion 8
Versions of processes and Kolmogorov s continuity theorem 10
Exercises 11
III. ITO INTEGRALS 14
Mathematical interpretation of equations involving noise 14
The Ito integral 18
Some properties of the Ito integral 22
Martingales 22
Extensions of the Ito integral 25
Comparison between Ito and Stratonovich integrals 27
Exercises 29
IV. STOCHASTIC INTEGRALS AND THE ITO FORMULA 32
Stochastic integrals 32
The 1 dimensional Ito formula 32
The multi dimensional Ito formula 37
Exercises 38
V. STOCHASTIC DIFFERENTIAL EQUATIONS 44
The population growth model and other examples 44
Brownian motion on the unit circle 48
Existence and uniqueness theorem for stochastic differential equations 48
Weak and strong solutions 53
Exercises 54
VI. THE FILTERING PROBLEM 58
Statement of the general problem 59
The linear filtering problem 60
Step 1: Z linear and Z measurable estimates 63
Step 2: The innovation process 65
Step 3: The innovation process and Brownian motion 68
Step 4: An explicit formula for Xt 70
XII
Step 5: The stochastic differential equation for Xt 71
The 1 dimensional Kalman Bucy filter 73
Examples 74
The multi dimensional Kalman Bucy filter 79
Exercises 80
VII. DIFFUSIONS: BASIC PROPERTIES 86
Definition of an Ito diffusion 86
(A) The Markov property 86
(B) The strong Markov property 89
Hitting distribution, harmonic measure and the mean value property 93
(C) The generator of a diffusion 93
(D) The Dynkin formula 96
(E) The characteristic operator 97
Examples 99
Exercises 100
VIII. OTHER TOPICS IN DIFFUSION THEORY 105
(A) Kolmogorov s backward equation
The resolvent 105
(B) The Feynman Kac formula. Killing 108
(C) The martingale problem 110
(D) When is a stochastic integral a diffusion? 112
How to recognize a Brownian motion 117
(E) Random time change 117
Time change formula for Ito integrals 119
Examples: Brownian motion on the unit sphere 120
Harmonic and analytic functions 121
(F) The Cameron Martin Girsanov formula I 123
The Cameron Martin Girsanov transformation 126
The Cameron Martin Girsanov formula II 126
Exercises 127
IX. APPLICATIONS TO BOUNDARY VALUE PROBLEMS 133
(A) The Dirichlet problem 133
Regular points 135
Examples 135
The stochastic Dirichlet problem 138
Existence and uniqueness of solution 139
When is the solution of the stochastic Dirichlet problem also a solution of
the original Dirichlet problem? 141
Examples 143
(B) The Poisson problem 144
A stochastic version 145
XIII
Existence of solution 145
Uniqueness of solution 147
The combined Dirichlet Poisson problem 147
The Green measure 148
Exercises 150
X. APPLICATION TO OPTIMAL STOPPING 155
Statement of the problem 155
Least superharmonic majorants 159
Existence theorem for optimal stopping 161
Uniqueness theorem for optimal stopping 165
Examples: 1) Some stopping problems for Brownian motion 166
2) When is the right time to sell the stocks? 168
3) When to quit a contest 171
4) The marriage problem 173
Exercises 175
XI. APPLICATION TO STOCHASTIC CONTROL 180
Statement of the problem 180
The Hamilton Jacobi Bellman (HJB) equation 182
A converse of the HJB equation 185
Markov controls versus general adaptive controls 186
The linear regulator problem 187
An optimal portfolio selection problem 190
A simple problem where the optimal process is discontinuous 192
Exercises 194
APPENDIX A: NORMAL RANDOM VARIABLES 200
APPENDIX B: CONDITIONAL EXPECTATIONS 203
APPENDLX C: UNIFORM INTEGRABILITY AND MARTINGALE
CONVERGENCE 205
BIBLIOGRAPHY 208
LIST OF FREQUENTLY USED NOTATION AND SYMBOLS 215
INDEX 218
|
| adam_txt |
XI
Contents
I. INTRODUCTION 1
Some problems (1 6) where stochastic differential equations play an essential
role in the solution
II. SOME MATHEMATICAL PRELIMINARIES 5
Random variables, independence, stochastic processes 5
Kolmogorov's extension theorem 7
Brownian motion 7
Basic properties of Brownian motion 8
Versions of processes and Kolmogorov's continuity theorem 10
Exercises 11
III. ITO INTEGRALS 14
Mathematical interpretation of equations involving "noise" 14
The Ito integral 18
Some properties of the Ito integral 22
Martingales 22
Extensions of the Ito integral 25
Comparison between Ito and Stratonovich integrals 27
Exercises 29
IV. STOCHASTIC INTEGRALS AND THE ITO FORMULA 32
Stochastic integrals 32
The 1 dimensional Ito formula 32
The multi dimensional Ito formula 37
Exercises 38
V. STOCHASTIC DIFFERENTIAL EQUATIONS 44
The population growth model and other examples 44
Brownian motion on the unit circle 48
Existence and uniqueness theorem for stochastic differential equations 48
Weak and strong solutions 53
Exercises 54
VI. THE FILTERING PROBLEM 58
Statement of the general problem 59
The linear filtering problem 60
Step 1: Z linear and Z measurable estimates 63
Step 2: The innovation process 65
Step 3: The innovation process and Brownian motion 68
Step 4: An explicit formula for Xt 70
XII
Step 5: The stochastic differential equation for Xt 71
The 1 dimensional Kalman Bucy filter 73
Examples 74
The multi dimensional Kalman Bucy filter 79
Exercises 80
VII. DIFFUSIONS: BASIC PROPERTIES 86
Definition of an Ito diffusion 86
(A) The Markov property 86
(B) The strong Markov property 89
Hitting distribution, harmonic measure and the mean value property 93
(C) The generator of a diffusion 93
(D) The Dynkin formula 96
(E) The characteristic operator 97
Examples 99
Exercises 100
VIII. OTHER TOPICS IN DIFFUSION THEORY 105
(A) Kolmogorov's backward equation
The resolvent 105
(B) The Feynman Kac formula. Killing 108
(C) The martingale problem 110
(D) When is a stochastic integral a diffusion? 112
How to recognize a Brownian motion 117
(E) Random time change 117
Time change formula for Ito integrals 119
Examples: Brownian motion on the unit sphere 120
Harmonic and analytic functions 121
(F) The Cameron Martin Girsanov formula I 123
The Cameron Martin Girsanov transformation 126
The Cameron Martin Girsanov formula II 126
Exercises 127
IX. APPLICATIONS TO BOUNDARY VALUE PROBLEMS 133
(A) The Dirichlet problem 133
Regular points 135
Examples 135
The stochastic Dirichlet problem 138
Existence and uniqueness of solution 139
When is the solution of the stochastic Dirichlet problem also a solution of
the original Dirichlet problem? 141
Examples 143
(B) The Poisson problem 144
A stochastic version 145
XIII
Existence of solution 145
Uniqueness of solution 147
The combined Dirichlet Poisson problem 147
The Green measure 148
Exercises 150
X. APPLICATION TO OPTIMAL STOPPING 155
Statement of the problem 155
Least superharmonic majorants 159
Existence theorem for optimal stopping 161
Uniqueness theorem for optimal stopping 165
Examples: 1) Some stopping problems for Brownian motion 166
2) When is the right time to sell the stocks? 168
3) When to quit a contest 171
4) The marriage problem 173
Exercises 175
XI. APPLICATION TO STOCHASTIC CONTROL 180
Statement of the problem 180
The Hamilton Jacobi Bellman (HJB) equation 182
A converse of the HJB equation 185
Markov controls versus general adaptive controls 186
The linear regulator problem 187
An optimal portfolio selection problem 190
A simple problem where the optimal process is discontinuous 192
Exercises 194
APPENDIX A: NORMAL RANDOM VARIABLES 200
APPENDIX B: CONDITIONAL EXPECTATIONS 203
APPENDLX C: UNIFORM INTEGRABILITY AND MARTINGALE
CONVERGENCE 205
BIBLIOGRAPHY 208
LIST OF FREQUENTLY USED NOTATION AND SYMBOLS 215
INDEX 218 |
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| author | Øksendal, Bernt K. 1945- |
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| edition | 3. ed. |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:04:55Z |
| indexdate | 2024-07-09T20:47:06Z |
| institution | BVB |
| isbn | 3540533354 0387533354 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015120249 |
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| physical | XIII, 224 S. |
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| spelling | Øksendal, Bernt K. 1945- Verfasser (DE-588)128742054 aut Stochastic differential equations an introduction with applications 3. ed. Berlin [u.a.] Springer 1992 XIII, 224 S. txt rdacontent n rdamedia nc rdacarrier Universitext Equations différentielles stochastiques ram Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015120249&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Øksendal, Bernt K. 1945- Stochastic differential equations an introduction with applications Equations différentielles stochastiques ram Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| subject_GND | (DE-588)4057621-8 |
| title | Stochastic differential equations an introduction with applications |
| title_auth | Stochastic differential equations an introduction with applications |
| title_exact_search | Stochastic differential equations an introduction with applications |
| title_exact_search_txtP | Stochastic differential equations an introduction with applications |
| title_full | Stochastic differential equations an introduction with applications |
| title_fullStr | Stochastic differential equations an introduction with applications |
| title_full_unstemmed | Stochastic differential equations an introduction with applications |
| title_short | Stochastic differential equations |
| title_sort | stochastic differential equations an introduction with applications |
| title_sub | an introduction with applications |
| topic | Equations différentielles stochastiques ram Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Equations différentielles stochastiques Équations différentielles stochastiques Stochastic differential equations Stochastische Differentialgleichung |
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