Numerical methods for stochastic processes:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1994
|
| Schriftenreihe: | Wiley series in probability and mathematical statistics / Applied probability and statistics
A Wiley interscience publication |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 359 S. graph. Darst. |
| ISBN: | 0471546410 |
Internformat
MARC
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| 100 | 1 | |a Bouleau, Nicolas |d 1945- |e Verfasser |0 (DE-588)11295006X |4 aut | |
| 245 | 1 | 0 | |a Numerical methods for stochastic processes |c Nicolas Bouleau ; Dominique Lépingle |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1994 | |
| 300 | |a XVII, 359 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 mathematical statistics / Applied probability and statistics | |
| 490 | 0 | |a A Wiley interscience publication | |
| 650 | 0 | 7 | |a Monte-Carlo-Simulation |0 (DE-588)4240945-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Stochastischer Prozess |0 (DE-588)4057630-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Stochastischer Prozess |0 (DE-588)4057630-9 |D s |
| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Monte-Carlo-Simulation |0 (DE-588)4240945-7 |D s |
| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Lépingle, Dominique |e Verfasser |4 aut | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-006164434 | ||
Datensatz im Suchindex
| _version_ | 1804123711547113472 |
|---|---|
| adam_text | Contents
Preface xiii
1. Preliminaries 1
A Set Theory and General Topology 1
A.I Some Notations 1
A.2 A Summary of General Topology 3
A.3 Functional Analysis 4
B Probability Theory 5
B.I Measure Spaces 5
B.2 Schwartz Distributions 10
B.3 Random Variables 11
C Random Processes 14
C.I General Random Processes 14
C.2 Markov Chains 16
C.3 Ergodic Theory 18
C.4 Discrete Parameter Martingales 18
C.5 Continuous Martingales 20
C.6 Stochastic Integral 24
D Wiener Levy Calculus 26
D.I Brownian Motion 26
D.2 Wiener Space 32
2. Computation of Expectations in Finite Dimension 34
A Mathematical Framework of Simulation 34
A.I Narrow Convergence 34
A.2 Narrow Convergence and |a Riemann Integrable
Functions 35
A.3 Application to Simulation 38
A.4 Distribution Functions 40
vii
ViH CONTENTS
A.5 Uniformly Distributed Sequences and Discrepancy 44
B The Monte Carlo Method 46
B.I Algorithms for Simulation 46
B.2 Glivenko Cantelli Theorem 48
B.3 Central Limit Theorem for Empirical
Distributions 50
B.4 Stopping Criteria and Asymptotic Rate 54
B.5 Rate of Convergence of Stochastic Approximation
Algorithms 58
C Low Discrepancy Sequences 67
C.I Koksma Hlawka Inequality 67
C.2 Lower Bounds of Discrepancy 69
C.3 Irrational Translations on the Torus 71
C.4 Other Usual Sequences with Low Discrepancy 75
C.5 Low Discrepancy Sequences and Simulation 82
C.6 Complement: Boreloid Sequences 86
D Numerical Computation of Conditional Expectation 95
D.I Complements on Conditional Expectation 95
D.2 Polynomial Approximation 98
D.3 Point by Point Calculation 106
D.4 Comparison of Both Methods on an Example 110
3. Simulation of Random Processes 113
A Integration in Large or Infinite Dimensions 113
A.I Changing Dimension 113
A.2 Simulation of a Non Riemann Integrable Random
Variable 114
A.3 Effective Computation of the Expectation of
Random Variables in L1 116
A.4 Integration of Functionals of Stochastic Processes 118
A.5 The Shift Method 121
A.6 Rate of Convergence for the Shift Method 125
A.7 On the Shift Method in Finite Dimension 130
A.8 Complement: Quasi every Sequence Is Shift
Uniformly Distributed 133
B Representations of Stationary Fields 142
B.I Wiener Integral 143
B.2 Spectral Representation of Stationary (Wide Sense)
Fields 147
B.3 Linear Transformations of Stationary Fields 150
B.4 Second Order Linear Differential Equations 153
CONTENTS IX
B.5 Multidimensional Processes and Fields 156
B.6 Real Valued Fields 158
B.7 Strictly Stationary Fields 160
B.8 Space and Time Discretizations 167
B.9 Spectrum Discretization 172
B.10 Simulating Stationary Processes with Stationary
Germs 173
C Markov Processes 181
C. 1 Jump Processes 182
C.2 Iterative or Recursive Simulation 183
C.3 Simulation of a Brownian Motion Until Hitting a
Hyperplane 189
D Processes with Stationary Independent Increments 198
D.I Definition and First Properties 198
D.2 Convolution Semi groups on Rd 200
D.3 Convolution Semi Groups on 1R+ 202
D.4 Examples 203
D.5 Subordinators 205
D.6 Stable PSIIs 209
D.7 Application of the Subordination in Bochner
Sense to Simulation 213
D.8 The Bernstein Monoid 219
E Point Processes 221
E.I Setting Out the Problem 221
E.2 Setting Up the Models 221
E.3 Examples of Probability Distributions 224
E.4 Computation of Physical Quantities 227
4. Deterministic Resolution of Some Markovian Problems 230
A Elements in Markovian Potential Theory 230
A. 1 Notation and Setting: Feller Processes 230
A.2 Fundamental Representations 235
A.3 Feynman Kac Formula 237
B Balayage Algorithms 241
B.I Poincare s Balayage 241
B.2 The Harmonic Kernel of a Markov Chain 244
B.3 Numerical Balayage 246
C Reduced Function Algorithm 250
C. 1 Reduced Function Algorithms with Respect to a
Sub Markov Kernel 250
C.2 The Filling Scheme and the Reduite Property 252
X CONTENTS
C.3 Optimal Stopping 253
C.4 Computation of Reduced Functions with Respect
to Semi groups 255
D The Carre du Champ Operator 262
D. 1 The Extended Generator and Its Domain 263
D.2 Application to Hedging Strategies in Incomplete
Markovian Markets 265
5. Stochastic Differential Equations and Brownian Functionals 269
A Lipschitzian Stochastic Differential Equations: Ito s
Theorem 269
A. 1 Notation and Lemmas 269
A.2 Lipschitz hypotheses 270
A.3 Ito s Theorem 270
A.4 Two Properties of the Solution 273
B Discretization of SDEs 274
B.I Euler Scheme: Convergence and Rate Estimates 274
B.2 Example: Wiener Iterated Integrals 278
B.3 Example of a Multistep Scheme 279
B.4 Do Not Use the Shift Method Along the Sample
Paths 283
C Irregularity of Brownian Functionals 284
C. 1 Lack of Continuity of Stochastic Integrals 284
C.2 Non Riemann Integrable Stochastic Integrals 287
C.3 Other Topologies on Wiener Space 293
C.4 Importance of LP Estimates 293
C.5 Sequential Integration by Picking Smooth Functions
in the Wiener Space 294
D Simulatable Functionals 296
D.I Simulatable Functionals on {0,1}N 296
D.2 Simulatable Functionals on [0,1]N 297
D.3 Regularity Properties of Simulatable Functionals 299
D.4 Simulatable Functionals on a General Probability
Space 300
D.5 The Case of Wiener Space 302
E Symbolic Expansions of Solutions to SDEs 306
E.I Ito Stratonovitch transformation 306
E.2 Hu Meyer Formulas 306
E.3 Isobe Sato Formula 310
E.4 Expansion of the Solution of an SDE as a Series
of Iterated Stochastic Integrals 314
CONTENTS XI
F Application of the Shift Method to Multiple Wiener
Integrals and to Solutions of SDEs 319
F.I First Factorization of the Wiener Space 320
F.2 Functions of Solutions of Lipschitz SDEs 321
F.3 Functions of Multiple Wiener Integrals 323
F.4 Other Factorizations of Wiener Space 326
Notes 330
References 337
Index 353
|
| any_adam_object | 1 |
| author | Bouleau, Nicolas 1945- Lépingle, Dominique |
| author_GND | (DE-588)11295006X |
| author_facet | Bouleau, Nicolas 1945- Lépingle, Dominique |
| author_role | aut aut |
| author_sort | Bouleau, Nicolas 1945- |
| author_variant | n b nb d l dl |
| building | Verbundindex |
| bvnumber | BV009263008 |
| classification_rvk | SK 820 |
| classification_tum | MAT 679f MAT 605f |
| ctrlnum | (OCoLC)844189374 (DE-599)BVBBV009263008 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV009263008 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:34:07Z |
| institution | BVB |
| isbn | 0471546410 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006164434 |
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| physical | XVII, 359 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Wiley |
| record_format | marc |
| series2 | Wiley series in probability and mathematical statistics / Applied probability and statistics A Wiley interscience publication |
| spelling | Bouleau, Nicolas 1945- Verfasser (DE-588)11295006X aut Numerical methods for stochastic processes Nicolas Bouleau ; Dominique Lépingle New York [u.a.] Wiley 1994 XVII, 359 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and mathematical statistics / Applied probability and statistics A Wiley interscience publication Monte-Carlo-Simulation (DE-588)4240945-7 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Monte-Carlo-Simulation (DE-588)4240945-7 s Lépingle, Dominique Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006164434&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bouleau, Nicolas 1945- Lépingle, Dominique Numerical methods for stochastic processes Monte-Carlo-Simulation (DE-588)4240945-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4240945-7 (DE-588)4128130-5 (DE-588)4057630-9 |
| title | Numerical methods for stochastic processes |
| title_auth | Numerical methods for stochastic processes |
| title_exact_search | Numerical methods for stochastic processes |
| title_full | Numerical methods for stochastic processes Nicolas Bouleau ; Dominique Lépingle |
| title_fullStr | Numerical methods for stochastic processes Nicolas Bouleau ; Dominique Lépingle |
| title_full_unstemmed | Numerical methods for stochastic processes Nicolas Bouleau ; Dominique Lépingle |
| title_short | Numerical methods for stochastic processes |
| title_sort | numerical methods for stochastic processes |
| topic | Monte-Carlo-Simulation (DE-588)4240945-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Monte-Carlo-Simulation Numerisches Verfahren Stochastischer Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006164434&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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