Stochastic differential equations: an introduction with applications
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2007
|
| Edition: | 6. ed., corr. 4. print. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [351] - 359 |
| Physical Description: | XXIX, 369 S. graph. Darst. |
| ISBN: | 9783540047582 3540047581 |
Staff View
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| 245 | 1 | 0 | |a Stochastic differential equations |b an introduction with applications |c Bernt Øksendal |
| 250 | |a 6. ed., corr. 4. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XXIX, 369 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
1
Introduction
............................................... 1
1.1
Stochastic Analogs of Classical Differential Equations
........ 1
1.2
Filtering Problems
....................................... 2
1.3
Stochastic Approach to Deterministic Boundary Value
Problems
............................................... 3
1.4
Optimal Stopping
....................................... 3
1.5
Stochastic Control
....................................... 4
1.6
Mathematical Finance
................................... 4
2
Some Mathematical Preliminaries
......................... 7
2.1
Probability Spaces, Random Variables and Stochastic Processes
7
2.2
An Important Example: Brownian Motion
.................. 12
Exercises
................................................... 15
3
Ito
Integrals
............................................... 21
3.1
Construction of the
Ito
Integral
........................... 21
3.2
Some properties of the
Ito
integral
......................... 30
3.3
Extensions of the
Ito
integral
............................. 34
Exercises
................................................... 37
4
The
Ito
Formula and the Martingale Representation
Theorem
.................................................. 43
4.1
The 1-dimensional
Ito
formula
............................ 43
4.2
The Multi-dimensional
Ito
Formula
........................ 48
4.3
The Martingale Representation Theorem
................... 49
Exercises
................................................... 54
5
Stochastic Differential Equations
.......................... 63
5.1
Examples and Some Solution Methods
..................... 63
5.2
An Existence and Uniqueness Result
....................... 68
5.3
Weak and Strong Solutions
............................... 72
Exercises
................................................... 74
6
The Filtering Problem
..................................... 83
6.1
Introduction
............................................ 83
6.2
The 1-Dimensional Linear Filtering Problem
................ 85
6.3
The Multidimensional Linear Filtering Problem
.............104
Exercises
...................................................105
7
Diffusions: Basic Properties
................................113
7.1
The Markov Property
....................................113
7.2
The Strong Markov Property
.............................116
7.3
The Generator of an
Ito
Diffusion
.........................121
7.4
The Dynkin Formula
.....................................124
7.5
The Characteristic Operator
..............................126
Exercises
...................................................128
8
Other Topics in Diffusion Theory
..........................139
8.1
Kolmogorov s Backward Equation. The Resolvent
...........139
8.2
The Feynman-Kac Formula. Killing
........................143
8.3
The Martingale Problem
.................................146
8.4
When is an
Ito
Process a Diffusion?
........................148
8.5
Random Time Change
...................................153
8.6
The Girsanov Theorem
...................................159
Exercises
...................................................168
9
Applications to Boundary Value Problems
.................177
9.1
The Combined Dirichlet-Poisson Problem. Uniqueness
........177
9.2
The Dirichlet Problem. Regular Points
.....................181
9.3
The
Poisson
Problem
....................................192
Exercises
...................................................199
10
Application to Optimal Stopping
..........................207
10.1
The Time-Homogeneous Case
.............................207
10.2
The Time-Inhomogeneous Case
...........................220
10.3
Optimal Stopping Problems Involving an Integral
............224
10.4
Connection with Variational Inequalities
....................226
Exercises
...................................................230
11
Application to Stochastic Control
..........................237
11.1
Statement of the Problem
................................237
11.2
The Hamilton-Jacobi-Bellman Equation
....................239
11.3
Stochastic control problems with terminal conditions
.........253
Exercises
...................................................254
12
Application to Mathematical Finance
......................263
12.1
Market, portfolio and arbitrage
............................263
12.2
Attainability and Completeness
...........................273
12.3
Option Pricing
..........................................280
Exercises
...................................................300
Appendix A: Normal Random Variables
.......................307
Appendix B: Conditional Expectation
.........................311
Appendix C: Uniform Integrability and Martingale
Convergence
...............................................313
Appendix D: An Approximation Result
........................317
Solutions and Additional Hints to Some of the Exercises
.......321
References
.....................................................351
List of Frequently Used Notation and Symbols
................361
Index
..........................................................365
|
| adam_txt |
Contents
1
Introduction
. 1
1.1
Stochastic Analogs of Classical Differential Equations
. 1
1.2
Filtering Problems
. 2
1.3
Stochastic Approach to Deterministic Boundary Value
Problems
. 3
1.4
Optimal Stopping
. 3
1.5
Stochastic Control
. 4
1.6
Mathematical Finance
. 4
2
Some Mathematical Preliminaries
. 7
2.1
Probability Spaces, Random Variables and Stochastic Processes
7
2.2
An Important Example: Brownian Motion
. 12
Exercises
. 15
3
Ito
Integrals
. 21
3.1
Construction of the
Ito
Integral
. 21
3.2
Some properties of the
Ito
integral
. 30
3.3
Extensions of the
Ito
integral
. 34
Exercises
. 37
4
The
Ito
Formula and the Martingale Representation
Theorem
. 43
4.1
The 1-dimensional
Ito
formula
. 43
4.2
The Multi-dimensional
Ito
Formula
. 48
4.3
The Martingale Representation Theorem
. 49
Exercises
. 54
5
Stochastic Differential Equations
. 63
5.1
Examples and Some Solution Methods
. 63
5.2
An Existence and Uniqueness Result
. 68
5.3
Weak and Strong Solutions
. 72
Exercises
. 74
6
The Filtering Problem
. 83
6.1
Introduction
. 83
6.2
The 1-Dimensional Linear Filtering Problem
. 85
6.3
The Multidimensional Linear Filtering Problem
.104
Exercises
.105
7
Diffusions: Basic Properties
.113
7.1
The Markov Property
.113
7.2
The Strong Markov Property
.116
7.3
The Generator of an
Ito
Diffusion
.121
7.4
The Dynkin Formula
.124
7.5
The Characteristic Operator
.126
Exercises
.128
8
Other Topics in Diffusion Theory
.139
8.1
Kolmogorov's Backward Equation. The Resolvent
.139
8.2
The Feynman-Kac Formula. Killing
.143
8.3
The Martingale Problem
.146
8.4
When is an
Ito
Process a Diffusion?
.148
8.5
Random Time Change
.153
8.6
The Girsanov Theorem
.159
Exercises
.168
9
Applications to Boundary Value Problems
.177
9.1
The Combined Dirichlet-Poisson Problem. Uniqueness
.177
9.2
The Dirichlet Problem. Regular Points
.181
9.3
The
Poisson
Problem
.192
Exercises
.199
10
Application to Optimal Stopping
.207
10.1
The Time-Homogeneous Case
.207
10.2
The Time-Inhomogeneous Case
.220
10.3
Optimal Stopping Problems Involving an Integral
.224
10.4
Connection with Variational Inequalities
.226
Exercises
.230
11
Application to Stochastic Control
.237
11.1
Statement of the Problem
.237
11.2
The Hamilton-Jacobi-Bellman Equation
.239
11.3
Stochastic control problems with terminal conditions
.253
Exercises
.254
12
Application to Mathematical Finance
.263
12.1
Market, portfolio and arbitrage
.263
12.2
Attainability and Completeness
.273
12.3
Option Pricing
.280
Exercises
.300
Appendix A: Normal Random Variables
.307
Appendix B: Conditional Expectation
.311
Appendix C: Uniform Integrability and Martingale
Convergence
.313
Appendix D: An Approximation Result
.317
Solutions and Additional Hints to Some of the Exercises
.321
References
.351
List of Frequently Used Notation and Symbols
.361
Index
.365 |
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| discipline | Mathematik Wirtschaftswissenschaften |
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| id | DE-604.BV023045079 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:22:12Z |
| indexdate | 2024-07-09T21:09:42Z |
| institution | BVB |
| isbn | 9783540047582 3540047581 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016248543 |
| oclc_num | 890557615 |
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| physical | XXIX, 369 S. graph. Darst. |
| publishDate | 2007 |
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| publisher | Springer |
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| series2 | Universitext |
| spelling | Øksendal, Bernt K. 1945- Verfasser (DE-588)128742054 aut Stochastic differential equations an introduction with applications Bernt Øksendal 6. ed., corr. 4. print. Berlin [u.a.] Springer 2007 XXIX, 369 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Literaturverz. S. [351] - 359 Stochastische Differentialgleichung Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016248543&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 Stochastische Differentialgleichung 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 Bernt Øksendal |
| title_fullStr | Stochastic differential equations an introduction with applications Bernt Øksendal |
| title_full_unstemmed | Stochastic differential equations an introduction with applications Bernt Øksendal |
| title_short | Stochastic differential equations |
| title_sort | stochastic differential equations an introduction with applications |
| title_sub | an introduction with applications |
| topic | Stochastische Differentialgleichung Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Stochastische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016248543&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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