Markov processes and applications: algorithms, networks, genome and finance
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester
Wiley/Dunod
2008
|
| Edition: | 1. publ. |
| Series: | Wiley series in probability and statistics
Wiley-Dunod series |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [295]-296) and index |
| Physical Description: | XII, 296, 12 S. graph. Darst. |
| ISBN: | 9780470772713 0470772719 |
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MARC
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| 100 | 1 | |a Pardoux, Etienne |d 1947- |e Verfasser |0 (DE-588)111361117 |4 aut | |
| 245 | 1 | 0 | |a Markov processes and applications |b algorithms, networks, genome and finance |c Etienne Pardoux |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Chichester |b Wiley/Dunod |c 2008 | |
| 300 | |a XII, 296, 12 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Wiley series in probability and statistics | |
| 490 | 0 | |a Wiley-Dunod series | |
| 500 | |a Includes bibliographical references (p. [295]-296) and index | ||
| 650 | 4 | |a Markov processes | |
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Record in the Search Index
| _version_ | 1804140697801981952 |
|---|---|
| adam_text | Contents
Preface
xi
1
Simulations and the Monte Carlo method
1
1.1
Description of the method
...................... 2
1.2
Convergence theorems
........................ 3
1.3
Simulation of random variables
................... 5
1.4
Variance reduction techniques
.................... 9
1.5
Exercises
............................... 13
2
Markov chains
17
2.1
Definitions and elementary properties
................ 17
2.2
Examples
............................... 21
2.2.1
Random walk in
E
=
Td
.................. 21
2.2.2
Bienaymé-Galton-Watson
process
............. 21
2.2.3
A discrete time queue
.................... 22
2.3
Strong Markov property
....................... 22
2.4
Recurrent and transient states
.................... 24
2.5
The irreducible and recurrent case
.................. 27
2.6
The aperiodic case
.......................... 32
2.7
Reversible Markov chain
....................... 38
2.8
Rate of convergence to equilibrium
................. 39
2.8.1
The reversible finite state case
................ 39
2.8.2
The general case
....................... 42
2.9
Statistics of Markov chains
...................... 42
2.10
Exercises
............................... 43
3
Stochastic algorithms
57
3.1
Markov chain Monte Carlo
...................... 57
3.1.1
An application
........................ 59
3.1.2
The Ising model
....................... 61
3.1.3
Bayesian analysis of images
................. 63
3.1.4
Heated chains
......................... 64
3.2
Simulation of the invariant probability
............... 64
3.2.1
Perfect simulation
...................... 65
CONTENTS
3.2.2
Coupling from the past
................... 68
3.3
Rate of convergence towards the invariant probability
....... 70
3.4
Simulated annealing
......................... 73
3.5
Exercises
............................... 75
Markov chains and the genome
77
4.1
Reading
DNA............................. 77
4.1.1
CpG islands
......................... 78
4.1.2
Detection of the genes in a prokaryotic genome
...... 79
4.2
The i.i.d. model
............................ 79
4.3
The Markov model
.......................... 80
4.3.1
Application to CpG islands
................. 80
4.3.2
Search for genes in a prokaryotic genome
......... 81
4.3.3
Statistics of Markov chains Mk
............... 82
4.3.4
Phased Markov chains
.................... 82
4.3.5
Locally homogeneous Markov chains
............ 82
4.4
Hidden Markov models
........................ 84
4.4.1
Computation of the likelihood
................ 85
4.4.2
The Viterbi algorithm
.................... 86
4.4.3
Parameter estimation
..................... 87
4.5
Hidden semi-Markov model
..................... 92
4.5.1
Limitations of the hidden Markov model
.......... 92
4.5.2
What is a semi-Markov chain?
............... 92
4.5.3
The hidden semi-Markov model
............... 93
4.5.4
The semi-Markov Viterbi algorithm
............. 94
4.5.5
Search for genes in a prokaryotic genome
......... 95
4.6
Alignment of two sequences
..................... 97
4.6.1
The Needleman
-Wunsch
algorithm
............. 98
4.6.2
Hidden Markov model alignment algorithm
........ 99
4.6.3
A posteriori probability distribution of the alignment
... 102
4.6.4
A posteriori probability of a given match
.......... 104
4.7
A multiple alignment algorithm
................... 105
4.8
Exercises
............................... 107
Control and filtering of Markov chains
109
5.1
Deterministic optimal control
.................... 109
5.2
Control of Markov chains
......................
Ill
5.3
Linear quadratic optimal control
...................
Ill
5.4
Filtering of Markov chains
...................... 113
5.5
The Kalman-Bucy filter
....................... 115
5.5.1
Motivation
.......................... 115
5.5.2
Solution of the filtering problem
.............. 116
5.6
Linear-quadratic control with partial observation
.......... 120
5.7
Exercises
............................... 121
CONTENTS
vii
The
Poisson
process
123
6.1
Point processes and counting processes
............... 123
6.2
The
Poisson
process
......................... 124
6.3
The Markov property
......................... 127
6.4
Large time behaviour
......................... 130
6.5
Exercises
............................... 132
Jump Markov processes
135
7.1
General facts
............................. 135
7.2
Infinitesimal generator
........................ 139
7.3
The strong Markov property
..................... 142
7.4
Embedded Markov chain
....................... 144
7.5
Recurrent and transient states
.................... 147
7.6
The irreducible recurrent case
.................... 148
7.7
Reversibility
.............................. 153
7.8
Markov models of evolution and phylogeny
............ 154
7.8.1
Models of evolution
..................... 156
7.8.2
Likelihood methods in phylogeny
.............. 160
7.8.3
The Bayesian approach to phylogeny
............ 163
7.9
Application to discretized partial differential equations
....... 166
7.10
Simulated annealing
......................... 167
7.11
Exercises
............................... 173
Queues and networks
179
8.1
M/M/l queue
............................ 179
8.2
M/M/l/K queue
........................... 182
8.3
M/M/s queue
............................ 182
8.4
M/M/s/s queue
........................... 184
8.5
Repair shop
.............................. 185
8.6
Queues in series
........................... 185
8.7
Λί
/G/oo
queue
............................ 186
8.8
M/G/l queue
............................ 187
8.8.1
An embedded chain
..................... 187
8.8.2
The positive recurrent case
................. 188
8.9
Open Jackson network
........................ 190
8.10
Closed Jackson network
....................... 194
8.11
Telephone network
.......................... 196
8.12
Kelly networks
............................ 199
8.12.1
Single queue
......................... 199
8.12.2
Multi-class network
..................... 202
8.13
Exercises
............................... 203
Introduction to mathematical finance
205
9.1
Fundamental concepts
........................ 205
9.1.1
Option
............................ 206
viii CONTENTS
9.1.2 Arbitrage........................... 206
9.1.3
Viable
and complete markets
................ 207
9.2
European options in the discrete model
............... 208
9.2.1
The model
.......................... 208
9.2.2
Admissible strategy
..................... 208
9.2.3
Martingales
.......................... 210
9.2.4
Viable and complete market
................. 211
9.2.5
Call and put pricing
..................... 213
9.2.6
The Black-Scholes formula
................. 214
9.3
The Black-Scholes model and formula
............... 216
9.3.1
Introduction to stochastic calculus
.............. 217
9.3.2
Stochastic differential equations
............... 223
9.3.3
The Feynman-Kac formula
................. 225
9.3.4
The Black-Scholes partial differential equation
....... 225
9.3.5
The Black-Scholes formula
(2)............... 228
9.3.6
Generalization of the Black-Scholes model
......... 228
9.3.7
The Black-Scholes formula
(3)............... 229
9.3.8
Girsanov s theorem
...................... 232
9.3.9
Markov property and partial differential equation
..... 233
9.3.10
Contingent claim on several underlying stocks
....... 235
9.3.11
Viability and completeness
.................. 237
9.3.12
Remarks on effective computation
............. 238
9.3.13
Historical and implicit volatility
............... 239
9.4
American options in the discrete model
............... 239
9.4.1
Snell envelope
........................ 240
9.4.2
Doob s decomposition
.................... 242
9.4.3
Snell envelope and Markov chain
.............. 244
9.4.4
Back to American options
.................. 244
9.4.5
American and European options
............... 245
9.4.6
American options and Markov model
............ 245
9.5
American options in the Black-Scholes model
........... 246
9.6
Interest rate and bonds
........................ 247
9.6.1
Future interest rate
...................... 247
9.6.2
Future interest rate and bonds
................ 248
9.6.3
Option based on a bond
................... 250
9.6.4
An interest rate model
.................... 251
9.7
Exercises
............................... 252
10
Solutions to selected exercises
257
10.1
Chapter
1............................... 257
10.2
Chapter
2............................... 262
10.3
Chapters
............................... 275
10.4
Chapter
4............................... 277
10.5
Chapters
............................... 278
10.6
Chapter
6............................... 279
CONTENTS ix
10.7
Chapter
7............................... 282
10.8
Chapter
8............................... 289
10.9
Chapter
9............................... 291
Reference
295
Index
297
Notations
The following notations will be used throughout this book.
IN
= {0, 1, 2,...}
stands for the set of positive integers, including
0.
IN*
— {1, 2,...}
stands for the set of positive integers,
0
excluded.
|
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| id | DE-604.BV035769162 |
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| indexdate | 2024-07-09T22:04:06Z |
| institution | BVB |
| isbn | 9780470772713 0470772719 |
| language | English |
| lccn | 2008043729 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018628893 |
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| owner | DE-703 DE-11 DE-578 DE-824 DE-29T DE-20 |
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| physical | XII, 296, 12 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | Wiley/Dunod |
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| series2 | Wiley series in probability and statistics Wiley-Dunod series |
| spelling | Pardoux, Etienne 1947- Verfasser (DE-588)111361117 aut Markov processes and applications algorithms, networks, genome and finance Etienne Pardoux 1. publ. Chichester Wiley/Dunod 2008 XII, 296, 12 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Wiley-Dunod series Includes bibliographical references (p. [295]-296) and index Markov processes Markov-Entscheidungsprozess (DE-588)4168927-6 gnd rswk-swf Markov-Entscheidungsprozess (DE-588)4168927-6 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628893&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pardoux, Etienne 1947- Markov processes and applications algorithms, networks, genome and finance Markov processes Markov-Entscheidungsprozess (DE-588)4168927-6 gnd |
| subject_GND | (DE-588)4168927-6 |
| title | Markov processes and applications algorithms, networks, genome and finance |
| title_auth | Markov processes and applications algorithms, networks, genome and finance |
| title_exact_search | Markov processes and applications algorithms, networks, genome and finance |
| title_full | Markov processes and applications algorithms, networks, genome and finance Etienne Pardoux |
| title_fullStr | Markov processes and applications algorithms, networks, genome and finance Etienne Pardoux |
| title_full_unstemmed | Markov processes and applications algorithms, networks, genome and finance Etienne Pardoux |
| title_short | Markov processes and applications |
| title_sort | markov processes and applications algorithms networks genome and finance |
| title_sub | algorithms, networks, genome and finance |
| topic | Markov processes Markov-Entscheidungsprozess (DE-588)4168927-6 gnd |
| topic_facet | Markov processes Markov-Entscheidungsprozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628893&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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