Stochastic calculus and financial applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
[2009]
|
| Edition: | 4. print |
| Series: | Applications of mathematics
45 |
| Subjects: | |
| Links: | http://www.gbv.de/dms/hebis-darmstadt/toc/12285151X.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018619049&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Literaturverz. S. [293] - 295 |
| Physical Description: | IX, 300 S. 25 cm |
| ISBN: | 0387950168 9780387950167 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1819263934816845824 |
|---|---|
| adam_text | IMAGE 1
J. MICHAEL STEELE
STOCHASTIC CALCULUS AND
FINANCIAL APPLICATIONS
SPRINGER
IMAGE 2
CONTENTS
PREFACE V
1. RANDOM WALK AND FIRST STEP ANALYSIS 1
1.1. FIRST STEP ANALYSIS 1
1.2. TIME AND INFINITY 2
1.3. TOSSING AN UNFAIR COIN 5
1.4. NUMERICAL CALCULATION AND INTUITION 7
1.5. FIRST STEPS WITH GENERATING FUNCTIONS 7
1.6. EXERCISES 9
FIRST MARTINGALE STEPS 11
2.1. CLASSIC EXAMPLES 11
2.2. NEW MARTINGALES FROM OLD 13
2.3. REVISITING THE OLD RUINS 15
2.4. SUBMARTINGALES 17
2.5. DOOB S INEQUALITIES 19
2.6. MARTINGALE CONVERGENCE 22
2.7. EXERCISES 26
3. BROWNIAN MOTION 29
3.1. COVARIANCES AND CHARACTERISTIC FUNCTIONS 30
3.2. VISIONS OF A SERIES APPROXIMATION 33
3.3. TWO WAVELETS 35
3.4. WAVELET REPRESENTATION OF BROWNIAN MOTION 36
3.5. SCALING AND INVERTING BROWNIAN MOTION 40
3.6. EXERCISES 41
4. MARTINGALES: THE NEXT STEPS 43
4.1. FOUNDATION STONES 43
4.2. CONDITIONAL EXPECTATIONS 44
4.3. UNIFORM INTEGRABILITY 47
4.4. MARTINGALES IN CONTINUOUS TIME 50
4.5. CLASSIC BROWNIAN MOTION MARTINGALES 55
4.6. EXERCISES 58
IMAGE 3
VIII CONTENTS
5. RICHNESS OF PATHS 61
5.1. QUANTITATIVE SMOOTHNESS 61
5.2. NOT TOO SMOOTH 63
5.3. TWO REFLECTION PRINCIPLES 66
5.4. THE INVARIANCE PRINCIPLE AND DONSKER S THEOREM 70
5.5. RANDOM WALKS INSIDE BROWNIAN MOTION 72
5.6. EXERCISES 77
6. ITO INTEGRATION 79
6.1. DEFINITION OF THE ITO INTEGRAL: FIRST TWO STEPS 79
6.2. THIRD STEP: J TO S INTEGRAL AS A PROCESS 82
6.3. THE INTEGRAL SIGN: BENEFITS AND COSTS 85
6.4. AN EXPLICIT CALCULATION 85
6.5. PATHWISE INTERPRETATION OF ITO INTEGRALS 87
6.6. APPROXIMATION IN 7I 2 90
6.7. EXERCISES 93
7. LOCALIZATION AND ITO S INTEGRAL 95
7.1. ITO S INTEGRAL ON OC 95
7.2. AN INTUITIVE REPRESENTATION 99
7.3. WHY JUST OC ? 102
7.4. LOCAL MARTINGALES AND HONEST ONES 103
7.5. ALTERNATIVE FIELDS AND CHANGES OF TIME 106
7.6. EXERCISES 109
8. ITO S FORMULA I LL
8.1. ANALYSIS AND SYNTHESIS I LL
8.2. FIRST CONSEQUENCES AND ENHANCEMENTS 115
8.3. VECTOR EXTENSION AND HARMONIC FUNCTIONS 120
8.4. FUNCTIONS OF PROCESSES 123
8.5. THE GENERAL ITO FORMULA 126
8.6. QUADRATIC VARIATION 128
8.7. EXERCISES 134
9. STOCHASTIC DIFFERENTIAL EQUATIONS 137
9.1. MATCHING ITO S COEFFICIENTS 137
9.2. ORNSTEIN-UHLENBECK PROCESSES 138
9.3. MATCHING PRODUCT PROCESS COEFFICIENTS 139
9.4. EXISTENCE AND UNIQUENESS THEOREMS 142
9.5. SYSTEMS OF SDES 148
9.6. EXERCISES 149
10. ARBITRAGE AND SDES 153
10.1. REPLICATION AND THREE EXAMPLES OF ARBITRAGE 153
10.2. THE BLACK-SCHOLES MODEL 156
10.3. THE BLACK-SCHOLES FORMULA 158
10.4. TWO ORIGINAL DERIVATIONS 160
10.5. THE PERPLEXING POWER OF A FORMULA 165
10.6. EXERCISES 167
IMAGE 4
CONTENTS IX
11. THE DIFFUSION EQUATION 169
11.1. THE DIFFUSION OF MICE . . 169
11.2. SOLUTIONS OF THE DIFFUSION EQUATION 172
11.3. UNIQUENESS OF SOLUTIONS 178
11.4. HOW TO SOLVE THE BLACK-SCHOLES PDE 182
11.5. UNIQUENESS AND THE BLACK-SCHOLES PDE 187
11.6. EXERCISES 189
12. REPRESENTATION THEOREMS 191
12.1. STOCHASTIC INTEGRAL REPRESENTATION THEOREM 191
12.2. THE MARTINGALE REPRESENTATION THEOREM 196
12.3. CONTINUITY OF CONDITIONAL EXPECTATIONS 201
12.4. REPRESENTATION VIA TIME CHANGE 203
12.5. LEVY S CHARACTERIZATION OF BROWNIAN MOTION 204
12.6. BEDROCK APPROXIMATION TECHNIQUES 206
12.7. EXERCISES 211
13. GIRSANOV THEORY 213
13.1. IMPORTANCE SAMPLING 213
13.2. TILTING A PROCESS 215
13.3. SIMPLEST GIRSANOV THEOREM 218
13.4. CREATION OF MARTINGALES 221
13.5. SHIFTING THE GENERAL DRIFT 222
13.6. EXPONENTIAL MARTINGALES AND NOVIKOV S CONDITION 225
13.7. EXERCISES 229
14. ARBITRAGE AND MARTINGALES 233
14.1. REEXAMINATION OF THE BINOMIAL ARBITRAGE 233
14.2. THE VALUATION FORMULA IN CONTINUOUS TIME 235
14.3. THE BLACK-SCHOLES FORMULA VIA MARTINGALES 241
14.4. AMERICAN OPTIONS 244
14.5. SELF-FINANCING AND SELF-DOUBT 246
14.6. ADMISSIBLE STRATEGIES AND COMPLETENESS 252
14.7. PERSPECTIVE ON THEORY AND PRACTICE 257
14.8. EXERCISES 259
15. THE FEYNMAN-KAC CONNECTION 263
15.1. FIRST LINKS 263
15.2. THE FEYNMAN-KAC CONNECTION FOR BROWNIAN MOTION 265
15.3. LEVY S ARCSIN LAW 267
15.4. THE FEYNMAN-KAC CONNECTION FOR DIFFUSIONS 270
15.5. FEYNMAN-KAC AND THE BLACK-SCHOLES PDES 271
15.6. EXERCISES 274
APPENDIX I. MATHEMATICAL TOOLS 277
APPENDIX II. COMMENTS AND CREDITS 285
BIBLIOGRAPHY 293
INDEX 297
|
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| discipline | Mathematik |
| edition | 4. print |
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| id | DE-604.BV035759137 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T13:59:32Z |
| institution | BVB |
| isbn | 0387950168 9780387950167 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018619049 |
| oclc_num | 903010920 |
| open_access_boolean | 1 |
| owner | DE-83 DE-19 DE-BY-UBM |
| owner_facet | DE-83 DE-19 DE-BY-UBM |
| physical | IX, 300 S. 25 cm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Applications of mathematics |
| series2 | Applications of mathematics |
| spellingShingle | Steele, John Michael 1949- Stochastic calculus and financial applications Applications of mathematics Stochastic analysis Business mathematics Finanzmathematik Stochastische Analysis Stochastische Analysis (DE-588)4132272-1 gnd Stochastische Optimierung (DE-588)4057625-5 gnd Finanzmathematik (DE-588)4017195-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd |
| subject_GND | (DE-588)4132272-1 (DE-588)4057625-5 (DE-588)4017195-4 (DE-588)4057630-9 (DE-588)4066472-7 |
| title | Stochastic calculus and financial applications |
| title_auth | Stochastic calculus and financial applications |
| title_exact_search | Stochastic calculus and financial applications |
| title_full | Stochastic calculus and financial applications J. Michael Steele |
| title_fullStr | Stochastic calculus and financial applications J. Michael Steele |
| title_full_unstemmed | Stochastic calculus and financial applications J. Michael Steele |
| title_short | Stochastic calculus and financial applications |
| title_sort | stochastic calculus and financial applications |
| topic | Stochastic analysis Business mathematics Finanzmathematik Stochastische Analysis Stochastische Analysis (DE-588)4132272-1 gnd Stochastische Optimierung (DE-588)4057625-5 gnd Finanzmathematik (DE-588)4017195-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd |
| topic_facet | Stochastic analysis Business mathematics Finanzmathematik Stochastische Analysis Stochastische Optimierung Stochastischer Prozess Wirtschaftsmathematik |
| url | http://www.gbv.de/dms/hebis-darmstadt/toc/12285151X.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018619049&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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