Introduction to stochastic calculus with applications:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Imperial College Press
2012
|
| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Incl. bibliogr. references (S. 407-412) and index |
| Physical Description: | XIV, 438 S. graph. Darst. |
| ISBN: | 9781848168312 1848168314 9781848168329 1848168322 |
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| 100 | 1 | |a Klebaner, Fima C. |e Verfasser |0 (DE-588)173351743 |4 aut | |
| 245 | 1 | 0 | |a Introduction to stochastic calculus with applications |c Fima C. Klebaner |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a London |b Imperial College Press |c 2012 | |
| 300 | |a XIV, 438 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Titel: Introduction to stochastic calculus with applications
Autor: Klebaner, Fima C
Jahr: 2012
Contents
Preface xi
1. Preliminaries From Calculus 1
1.1 Functions in Calculus ................... 1
1.2 Variation of a Function .................. 4
1.3 Riemann Integrai and Stieltjes Integrai.......... 9
1.4 Lebesgue s Method of Integration............. 14
1.5 Differentials and Integrals................. 14
1.6 Taylor s Formula and Other Results........... 15
2. Concepts of Probability Theory 21
2.1 Discrete Probability Model................ 21
2.2 Continuous Probability Model............... 28
2.3 Expectation and Lebesgue Integrai............ 33
2.4 Transforms and Convergence............... 37
2.5 Independence and Covariance............... 39
2.6 Normal (Gaussian) Distributions............. 41
2.7 Conditional Expectation.................. 43
2.8 Stochastic Processes in Continuous Time ........ 47
3. Basic Stochastic Processes 55
3.1 Brownian Motion...................... 56
3.2 Properties of Brownian Motion Paths .......... 63
3.3 Three Martingales of Brownian Motion ......... 65
3.4 Markov Property of Brownian Motion.......... 67
3.5 Hitting Times and Exit Times .............. 69
3.6 Maximum and Minimum of Brownian Motion...... 71
3.7 Distribution of Hitting Times............... 73
3.8 Reflection Principle and Joint Distributions....... 74
3.9 Zeros of Brownian Motion - Arcsine Law........ 75
3.10 Size of Increments of Brownian Motion.......... 78
3.11 Brownian Motion in Higher Dimensions......... 81
3.12 RandomWalk........................ 81
3.13 Stochastic Integrai in Discrete Time........... 83
3.14 Poisson Process....................... 86
3.15 Exercises .......................... 88
4. Brownian Motion Calculus 91
4.1 Definition of Ito Integrai.................. 91
4.2 Ito Integrai Process..................... 100
4.3 Ito Integrai and Gaussian Processes ........... 103
4.4 Itò s Formula for Brownian Motion............ 106
4.5 Ito Processes and Stochastic Differentials ........ 108
4.6 Itó s Formula for Ito Processes.............. 112
4.7 Ito Processes in Higher Dimensions............ 118
4.8 Exercises .......................... 121
5. Stochastic Differential Equations 123
5.1 Definition of Stochastic Differential
Equations (SDEs) ..................... 123
5.2 Stochastic Exponential and Logarithm.......... 129
5.3 Solutions to Linear SDEs................. 131
5.4 Existence and Uniqueness of Strong Solutions...... 134
5.5 Markov Property of Solutions............... 136
5.6 Weak Solutions to SDEs.................. 137
5.7 Construction of Weak Solutions.............. 139
5.8 Backward and Forward Equations ............ 144
5.9 Stratonovich Stochastic Calculus............. 146
5.10 Exercises .......................... 148
6. Diffusion Processes 151
6.1 Martingales and Dynkin s Formula............ 151
6.2 Calculation of Expectations and PDEs.......... 155
6.3 Time-Homogeneous Diffusions............... 159
6.4 Exit Times from an Interval................ 163
6.5 Representation of Solutions of ODES........... 167
6.6 Explosion.......................... 168
6.7 Recurrence and Transience ................ 170
6.8 Diffusion on an Interval.................. 171
6.9 Stationary Distributions.................. 172
6.10 Multi-dimensional SDEs.................. 175
6.11 Exercises .......................... 183
7. Martingales 185
7.1 Definitions ......................... 185
7.2 Uniform Integrability.................... 187
7.3 Martingale Convergence.................. 189
7.4 Optional Stopping..................... 191
7.5 Localization and Locai Martingales............ 197
7.6 Quadratic Variation of Martingales............ 200
7.7 Martingale Inequalities................... 203
7.8 Continuous Martingales - Change of Time....... 205
7.9 Exercises .......................... 211
8. Calculus For Semimartingales 213
8.1 Semimartingales...................... 213
8.2 Predictable Processes ................... 214
8.3 Doob-Meyer Decomposition................ 215
8.4 Integrals with Respect to Semimartingales........ 217
8.5 Quadratic Variation and Covariation........... 220
8.6 Itó s Formula for Continuous Semimartingales...... 222
8.7 Locai Times......................... 224
8.8 Stochastic Exponential................... 226
8.9 Compensators and Sharp Bracket Process........ 230
8.10 Itó s Formula for Semimartingales ............ 236
8.11 Stochastic Exponential and Logarithm.......... 238
8.12 Martingale (Predictable) Representations........ 239
8.13 Elements of the General Theory ............. 242
8.14 Random Measures and Canonical Decomposition .... 246
8.15 Exercises .......................... 249
9. Pure Jump Processes 251
9.1 Definitions ......................... 251
9.2 Pure Jump Process Filtration............... 252
9.3 Itó s Formula for Processes of Finite Variation ..... 253
9.4 Counting Processes..................... 254
9.5 Markov Jump Processes.................. 261
9.6 Stochastic Equation for Jump Processes......... 264
9.7 Generators and Dynkin s Formula ............ 265
9.8 Explosions in Markov Jump Processes.......... 267
9.9 Exercises .......................... 268
10. Change of Probability Measure 269
10.1 Change of Measure for Random Variables........ 269
10.2 Change of Measure on a General Space ......... 273
10.3 Change of Measure for Processes............. 276
10.4 Change of Wiener Measure................ 281
10.5 Change of Measure for Point Processes.......... 283
10.6 Likelihood Functions.................... 284
10.7 Exercises .......................... 287
11. Applications in Finance: Stock and FX Options 289
11.1 Financial Derivatives and Arbitrage ........... 289
11.2 A Finite Market Model .................. 295
11.3 Semimartingale Market Model .............. 299
11.4 Diffusion and the Black-Scholes Model.......... 304
11.5 Change of Numeraire.................... 312
11.6 Currency (FX) Options.................. 315
11.7 Asian, Lookback, and Barrier Options.......... 318
11.8 Exercises .......................... 321
12. Applications in Finance: Bonds, Rates, and Options 325
12.1 Bonds and the Yield Curve................ 325
12.2 Models Adapted to Brownian Motion .......... 327
12.3 Models Based on the Spot Rate.............. 328
12.4 Merton s Model and Vasicek s Model........... 329
12.5 Heath-Jarrow-Morton (HJM) Model........... 333
12.6 Forward Measures - Bond as a Numeraire....... 338
12.7 Options, Caps, and Floors................. 341
12.8 Brace-Gatarek-Musiela (BGM) Model.......... 343
12.9 Swaps and Swaptions ................... 347
12.10 Exercises .......................... 349
13. Applications in Biology 353
13.1 Feller s Branching Diffusion................ 353
13.2 Wright-Fisher Diffusion.................. 357
13.3 Birth-Death Processes................... 359
13.4 Growth of Birth-Death Processes............. 363
13.5 Extinction, Probability, and Time to Exit........ 366
13.6 Processes in Genetics.................... 369
13.7 Birth-Death Processes in Many Dimensions....... 375
13.8 Cancer Models....................... 377
13.9 Branching Processes.................... 379
13.10 Stochastic Lotka-Volterra Model............. 386
13.11 Exercises .......................... 393
14. Applications in Engineering and Physics 395
14.1 Filtering........................... 395
14.2 Random Oscillators .................... 402
14.3 Exercises .......................... 408
Solutions to Selected Exercises 411
References 429
Index 435
|
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| edition | 3. ed. |
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| spelling | Klebaner, Fima C. Verfasser (DE-588)173351743 aut Introduction to stochastic calculus with applications Fima C. Klebaner 3. ed. London Imperial College Press 2012 XIV, 438 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Incl. bibliogr. references (S. 407-412) and index Stochastic analysis Calculus Stochastisches Integral (DE-588)4126478-2 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 s DE-604 Stochastisches Integral (DE-588)4126478-2 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024411712&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Klebaner, Fima C. Introduction to stochastic calculus with applications Stochastic analysis Calculus Stochastisches Integral (DE-588)4126478-2 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| subject_GND | (DE-588)4126478-2 (DE-588)4132272-1 |
| title | Introduction to stochastic calculus with applications |
| title_auth | Introduction to stochastic calculus with applications |
| title_exact_search | Introduction to stochastic calculus with applications |
| title_full | Introduction to stochastic calculus with applications Fima C. Klebaner |
| title_fullStr | Introduction to stochastic calculus with applications Fima C. Klebaner |
| title_full_unstemmed | Introduction to stochastic calculus with applications Fima C. Klebaner |
| title_short | Introduction to stochastic calculus with applications |
| title_sort | introduction to stochastic calculus with applications |
| topic | Stochastic analysis Calculus Stochastisches Integral (DE-588)4126478-2 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| topic_facet | Stochastic analysis Calculus Stochastisches Integral Stochastische Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024411712&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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