Brownian motion and stochastic calculus:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Springer
[2007]
|
| Ausgabe: | 2. ed., 9. print. [Nachdr.] |
| Schriftenreihe: | Graduate texts in mathematics
113 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXIII, 470 S. graph. Darst. |
| ISBN: | 9780387976556 0387976558 3540976558 |
Internformat
MARC
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| 100 | 1 | |a Karatzas, Ioannis |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Brownian motion and stochastic calculus |c Ioannis Karatzas ; Steven E. Shreve |
| 250 | |a 2. ed., 9. print. [Nachdr.] | ||
| 264 | 1 | |a New York |b Springer |c [2007] | |
| 300 | |a XXIII, 470 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 700 | 1 | |a Shreve, Steven E. |e Verfasser |4 aut | |
| 830 | 0 | |a Graduate texts in mathematics |v 113 |w (DE-604)BV000000067 |9 113 | |
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Datensatz im Suchindex
| _version_ | 1804137136673259520 |
|---|---|
| adam_text | Contents
Preface
vii
Suggestions
for the Reader
xvii
Interdependence of the Chapters
xix
Frequently Used Notation
xxi
Chapter
1
Martingales, Stopping Times, and Filtrations
1
1.1.
Stochastic Processes and
σ
-Fields
I
1.2.
Stopping Times
6
1.3.
Continuous-Time Martingales
11
A. Fundamental inequalities
12
B. Convergence results
17
С
The optional sampling theorem
19
1.4.
The Doob-Meyer Decomposition
21
1.5.
Continuous, Square-Integrable Martingales
30
1.6.
Solutions to Selected Problems
38
1.7.
Notes
45
Chapter
2
Brownian Motion
47
2.1.
Introduction
47
2.2.
First Construction of Brownian Motion
49
A. The consistency theorem
49
B. The Kolmogorov-Centsov theorem
53
2.3.
Second Construction of Brownian Motion
56
2.4.
The Space C[0, oo), Weak Convergence, and Wiener Measure
59
A. Weak convergence
60
B.
Tightness
61
C.
Convergence of finite-dimensional distributions
64
D. The
invariance
principle and the Wiener measure
66
2.5.
The Markov Property
71
A. Brownian motion in several dimensions
72
B. Markov processes and Markov families
74
C. Equivalent formulations of the Markov property
75
2.6.
The Strong Markov Property and the Reflection Principle
79
A. The reflection principle
79
B. Strong Markov processes and families
81
C. The strong Markov property for Brownian motion
84
2.7.
Brownian Filtrations
89
A. Right-continuity of the augmented filtration for a
strong Markov process
90
B. A
universal filtration
93
C. The
Blumenthal
zero-one law
94
2.8.
Computations Based on Passage Times
94
A. Brownian motion and its running maximum
95
B. Brownian motion on a half-line
97
C. Brownian motion on a finite interval
97
D. Distributions involving last exit times
100
2.9.
The Brownian Sample Paths
103
A. Elementary properties
103
B. The zero set and the quadratic variation
104
C. Local maxima and points of increase
106
D. Nowhere differentiability
109
E. Law of the iterated logarithm 111
F. Modulus of continuity
114
2.10.
Solutions to Selected Problems
116
2.11.
Notes
126
Chapter
3
Stochastic Integration
128
3.1.
Introduction
128
3.2.
Construction of the Stochastic Integral
129
A. Simple processes and approximations
132
B. Construction and elementary properties of the integral
137
C. A
characterization of the integral
141
D. Integration with respect to continuous, local martingales
145
3.3.
The Change-of-Variable Formula
148
A. The
Ito
rule
149
B. Martingale characterization of Brownian motion
156
C. Bessel processes, questions of recurrence
158
D. Martingale moment inequalities
163
E. Supplementary exercises
167
3.4.
Representations of Continuous Martingales in Terms of
Brownian Motion
169
A. Continuous local martingales as stochastic integrals with
respect to Brownian motion
170
В.
Continuous local martingales as time-changed Brownian motions
С
A theorem of
F.
В.
Knight
D.
Brownian martingales as stochastic integrals
E. Brownian functional as stochastic integrals
3.5.
The Girsanov Theorem
A. The basic result
B. Proof and ramifications
C. Brownian motion with drift
D. The Novikov condition
3.6.
Local Time and a Generalized
Ito
Rule for Brownian Motion
A. Definition of local time and the Tanaka formula
B. The Trotter existence theorem
C. Reflected Brownian motion and the Skorohod equation
D. A generalized
Ito
rule for convex functions
E. The
Engelbert-Schmidt
zero-one law
3.7.
Local Time for Continuous Semimartingales
3.8.
Solutions to Selected Problems
3.9.
Notes
Chapter
4
Brownian Motion and Partial Differential Equations
4.1.
Introduction
4.2.
Harmonic Functions and the Dirichlet Problem
A. The mean-value property
B. The Dirichlet problem
С
Conditions for regularity
D. Integral formulas of
Poisson
E.
Supplementary exercises
4.3.
The One-Dimensional Heat Equation
A. The Tychonoff uniqueness theorem
B.
Nonnegative
solutions of the heat equation
С
Boundary crossing probabilities for Brownian motion
D. Mixed initial/boundary value problems
4.4.
The Formulas of Feynman and
Кас
A. The multidimensional formula
B. The one-dimensional formula
4.5.
Solutions to selected problems
4.6.
Notes
Chapter
5
Stochastic Differential Equations
5.1.
Introduction
5.2.
Strong Solutions
A. Definitions
B. The
Ito
theory
C. Comparison results and other refinements
D. Approximations of stochastic differential equations
E. Supplementary exercises
5.3.
Weak
Solutions
300
A. Two notions of uniqueness
301
B. Weak solutions by means of the Girsanov theorem
302
С
A digression on regular conditional probabilities
306
D. Results of Yamada and Watanabe on weak and strong solutions
308
5.4.
The Martingale Problem of Stroock and
Varadhan
311
A. Some fundamental martingales
312
B. Weak solutions and martingale problems
314
C. Well-posedness
and the strong Markov property
319
D. Questions of existence
323
E. Questions of uniqueness
325
F. Supplementary exercises
328
5.5.
A Study of the One-Dimensional Case
329
A. The method of time change
330
B. The method of removal of drift
339
С
Feller s test for explosions
342
D. Supplementary exercises
351
5.6.
Linear Equations
354
A. Gauss-Markov processes
355
B. Brownian bridge
358
C. The general, one-dimensional, linear equation
360
D. Supplementary exercises
361
5.7.
Connections with Partial Differential Equations
363
A. The Dirichlet problem
364
B. The Cauchy problem and a Feynman-Kac representation
366
C. Supplementary exercises
369
5.8.
Applications to Economics
371
A. Portfolio and consumption processes
371
B. Option pricing
376
C. Optimal consumption and investment (general theory)
379
D. Optimal consumption and investment (constant
coefficients)
381
5.9.
Solutions to Selected Problems
387
5.10.
Notes
394
Chapter
6
P. Levy s Theory of Brownian Local Time
399
6.1.
Introduction
399
6.2.
Alternate Representations of Brownian Local Time
400
A. The process of passage times
400
B. Poisson
random measures
403
C. Subordinators
405
D. The process of passage times revisited
411
E. The excursion and downcrossing representations of local time
414
6.3.
Two Independent Reflected Brownian Motions
418
A. The positive and negative parts of a Brownian motion
418
B. The first formula of D. Williams
421
С
The joint density of
(W(t), L(t),
Г+
(t))
423
6.4.
Elastic Brownian
Motion 425
A. The Feynman-Kac formulas for elastic Brownian motion
426
B. The Ray-Knight description of local time
430
С
The second formula of
D.
Williams
434
6.5.
An Application: Transition Probabilities of Brownian Motion with
Two-Valued Drift
437
6.6.
Solutions to Selected Problems
442
6.7.
Notes
445
Bibliography
447
Index
459
|
| adam_txt |
Contents
Preface
vii
Suggestions
for the Reader
xvii
Interdependence of the Chapters
xix
Frequently Used Notation
xxi
Chapter
1
Martingales, Stopping Times, and Filtrations
1
1.1.
Stochastic Processes and
σ
-Fields
I
1.2.
Stopping Times
6
1.3.
Continuous-Time Martingales
11
A. Fundamental inequalities
12
B. Convergence results
17
С
The optional sampling theorem
19
1.4.
The Doob-Meyer Decomposition
21
1.5.
Continuous, Square-Integrable Martingales
30
1.6.
Solutions to Selected Problems
38
1.7.
Notes
45
Chapter
2
Brownian Motion
47
2.1.
Introduction
47
2.2.
First Construction of Brownian Motion
49
A. The consistency theorem
49
B. The Kolmogorov-Centsov theorem
53
2.3.
Second Construction of Brownian Motion
56
2.4.
The Space C[0, oo), Weak Convergence, and Wiener Measure
59
A. Weak convergence
60
B.
Tightness
61
C.
Convergence of finite-dimensional distributions
64
D. The
invariance
principle and the Wiener measure
66
2.5.
The Markov Property
71
A. Brownian motion in several dimensions
72
B. Markov processes and Markov families
74
C. Equivalent formulations of the Markov property
75
2.6.
The Strong Markov Property and the Reflection Principle
79
A. The reflection principle
79
B. Strong Markov processes and families
81
C. The strong Markov property for Brownian motion
84
2.7.
Brownian Filtrations
89
A. Right-continuity of the augmented filtration for a
strong Markov process
90
B. A
"universal" filtration
93
C. The
Blumenthal
zero-one law
94
2.8.
Computations Based on Passage Times
94
A. Brownian motion and its running maximum
95
B. Brownian motion on a half-line
97
C. Brownian motion on a finite interval
97
D. Distributions involving last exit times
100
2.9.
The Brownian Sample Paths
103
A. Elementary properties
103
B. The zero set and the quadratic variation
104
C. Local maxima and points of increase
106
D. Nowhere differentiability
109
E. Law of the iterated logarithm 111
F. Modulus of continuity
114
2.10.
Solutions to Selected Problems
116
2.11.
Notes
126
Chapter
3
Stochastic Integration
128
3.1.
Introduction
128
3.2.
Construction of the Stochastic Integral
129
A. Simple processes and approximations
132
B. Construction and elementary properties of the integral
137
C. A
characterization of the integral
141
D. Integration with respect to continuous, local martingales
145
3.3.
The Change-of-Variable Formula
148
A. The
Ito
rule
149
B. Martingale characterization of Brownian motion
156
C. Bessel processes, questions of recurrence
158
D. Martingale moment inequalities
163
E. Supplementary exercises
167
3.4.
Representations of Continuous Martingales in Terms of
Brownian Motion
169
A. Continuous local martingales as stochastic integrals with
respect to Brownian motion
170
В.
Continuous local martingales as time-changed Brownian motions
С
A theorem of
F.
В.
Knight
D.
Brownian martingales as stochastic integrals
E. Brownian functional as stochastic integrals
3.5.
The Girsanov Theorem
A. The basic result
B. Proof and ramifications
C. Brownian motion with drift
D. The Novikov condition
3.6.
Local Time and a Generalized
Ito
Rule for Brownian Motion
A. Definition of local time and the Tanaka formula
B. The Trotter existence theorem
C. Reflected Brownian motion and the Skorohod equation
D. A generalized
Ito
rule for convex functions
E. The
Engelbert-Schmidt
zero-one law
3.7.
Local Time for Continuous Semimartingales
3.8.
Solutions to Selected Problems
3.9.
Notes
Chapter
4
Brownian Motion and Partial Differential Equations
4.1.
Introduction
4.2.
Harmonic Functions and the Dirichlet Problem
A. The mean-value property
B. The Dirichlet problem
С
Conditions for regularity
D. Integral formulas of
Poisson
E.
Supplementary exercises
4.3.
The One-Dimensional Heat Equation
A. The Tychonoff uniqueness theorem
B.
Nonnegative
solutions of the heat equation
С
Boundary crossing probabilities for Brownian motion
D. Mixed initial/boundary value problems
4.4.
The Formulas of Feynman and
Кас
A. The multidimensional formula
B. The one-dimensional formula
4.5.
Solutions to selected problems
4.6.
Notes
Chapter
5
Stochastic Differential Equations
5.1.
Introduction
5.2.
Strong Solutions
A. Definitions
B. The
Ito
theory
C. Comparison results and other refinements
D. Approximations of stochastic differential equations
E. Supplementary exercises
5.3.
Weak
Solutions
300
A. Two notions of uniqueness
301
B. Weak solutions by means of the Girsanov theorem
302
С
A digression on regular conditional probabilities
306
D. Results of Yamada and Watanabe on weak and strong solutions
308
5.4.
The Martingale Problem of Stroock and
Varadhan
311
A. Some fundamental martingales
312
B. Weak solutions and martingale problems
314
C. Well-posedness
and the strong Markov property
319
D. Questions of existence
323
E. Questions of uniqueness
325
F. Supplementary exercises
328
5.5.
A Study of the One-Dimensional Case
329
A. The method of time change
330
B. The method of removal of drift
339
С
Feller's test for explosions
342
D. Supplementary exercises
351
5.6.
Linear Equations
354
A. Gauss-Markov processes
355
B. Brownian bridge
358
C. The general, one-dimensional, linear equation
360
D. Supplementary exercises
361
5.7.
Connections with Partial Differential Equations
363
A. The Dirichlet problem
364
B. The Cauchy problem and a Feynman-Kac representation
366
C. Supplementary exercises
369
5.8.
Applications to Economics
371
A. Portfolio and consumption processes
371
B. Option pricing
376
C. Optimal consumption and investment (general theory)
379
D. Optimal consumption and investment (constant
coefficients)
381
5.9.
Solutions to Selected Problems
387
5.10.
Notes
394
Chapter
6
P. Levy's Theory of Brownian Local Time
399
6.1.
Introduction
399
6.2.
Alternate Representations of Brownian Local Time
400
A. The process of passage times
400
B. Poisson
random measures
403
C. Subordinators
405
D. The process of passage times revisited
411
E. The excursion and downcrossing representations of local time
414
6.3.
Two Independent Reflected Brownian Motions
418
A. The positive and negative parts of a Brownian motion
418
B. The first formula of D. Williams
421
С
The joint density of
(W(t), L(t),
Г+
(t))
423
6.4.
Elastic Brownian
Motion 425
A. The Feynman-Kac formulas for elastic Brownian motion
426
B. The Ray-Knight description of local time
430
С
The second formula of
D.
Williams
434
6.5.
An Application: Transition Probabilities of Brownian Motion with
Two-Valued Drift
437
6.6.
Solutions to Selected Problems
442
6.7.
Notes
445
Bibliography
447
Index
459 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Karatzas, Ioannis Shreve, Steven E. |
| author_facet | Karatzas, Ioannis Shreve, Steven E. |
| author_role | aut aut |
| author_sort | Karatzas, Ioannis |
| author_variant | i k ik s e s se ses |
| building | Verbundindex |
| bvnumber | BV022875549 |
| classification_rvk | SK 920 |
| classification_tum | MAT 606f PHY 015f MAT 607f |
| ctrlnum | (OCoLC)263658576 (DE-599)BVBBV022875549 |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| edition | 2. ed., 9. print. [Nachdr.] |
| format | Book |
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| id | DE-604.BV022875549 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:48:51Z |
| indexdate | 2024-07-09T21:07:30Z |
| institution | BVB |
| isbn | 9780387976556 0387976558 3540976558 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016080620 |
| oclc_num | 263658576 |
| open_access_boolean | |
| owner | DE-473 DE-BY-UBG DE-29T DE-19 DE-BY-UBM DE-739 DE-91G DE-BY-TUM |
| owner_facet | DE-473 DE-BY-UBG DE-29T DE-19 DE-BY-UBM DE-739 DE-91G DE-BY-TUM |
| physical | XXIII, 470 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Karatzas, Ioannis Verfasser aut Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve 2. ed., 9. print. [Nachdr.] New York Springer [2007] XXIII, 470 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 113 Stochastik (DE-588)4121729-9 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stetigkeit (DE-588)4183167-6 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 s Stochastischer Prozess (DE-588)4057630-9 s DE-604 Stochastische Analysis (DE-588)4132272-1 s Stetigkeit (DE-588)4183167-6 s Stochastik (DE-588)4121729-9 s Shreve, Steven E. Verfasser aut Graduate texts in mathematics 113 (DE-604)BV000000067 113 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016080620&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Karatzas, Ioannis Shreve, Steven E. Brownian motion and stochastic calculus Graduate texts in mathematics Stochastik (DE-588)4121729-9 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Stochastische Analysis (DE-588)4132272-1 gnd Stetigkeit (DE-588)4183167-6 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| subject_GND | (DE-588)4121729-9 (DE-588)4057630-9 (DE-588)4132272-1 (DE-588)4183167-6 (DE-588)4128328-4 |
| title | Brownian motion and stochastic calculus |
| title_auth | Brownian motion and stochastic calculus |
| title_exact_search | Brownian motion and stochastic calculus |
| title_exact_search_txtP | Brownian motion and stochastic calculus |
| title_full | Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve |
| title_fullStr | Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve |
| title_full_unstemmed | Brownian motion and stochastic calculus Ioannis Karatzas ; Steven E. Shreve |
| title_short | Brownian motion and stochastic calculus |
| title_sort | brownian motion and stochastic calculus |
| topic | Stochastik (DE-588)4121729-9 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Stochastische Analysis (DE-588)4132272-1 gnd Stetigkeit (DE-588)4183167-6 gnd Brownsche Bewegung (DE-588)4128328-4 gnd |
| topic_facet | Stochastik Stochastischer Prozess Stochastische Analysis Stetigkeit Brownsche Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016080620&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT karatzasioannis brownianmotionandstochasticcalculus AT shrevestevene brownianmotionandstochasticcalculus |