Diffusion processes and stochastic calculus:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
European Mathematical Society
[2014]
|
| Series: | EMS Textbooks in Mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | IX, 276 Seiten |
| ISBN: | 9783037191330 |
Staff View
MARC
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| 100 | 1 | |a Baudoin, Fabrice |d 1975- |e Verfasser |0 (DE-588)1076992641 |4 aut | |
| 245 | 1 | 0 | |a Diffusion processes and stochastic calculus |c Fabrice Baudoin |
| 264 | 1 | |a Zürich |b European Mathematical Society |c [2014] | |
| 264 | 4 | |c © 2014 | |
| 300 | |a IX, 276 Seiten | ||
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Record in the Search Index
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|---|---|
| adam_text |
Contents
Preface
v
Conventions
and frequently used notations
xi
Introduction
1
1
Stochastic processes
6
1.1
Measure theory in function spaces
. 6
1.2
Stochastic processes
. 8
1.3
Filtrations
. 9
1.4
The
Danieli—
Kolmogorov extension theorem
. 11
1.5
The Kolmogorov continuity theorem
. 16
1.6
Stopping times
. 18
1.7
Martingales
. 20
1.8
Martingale inequalities
. 32
Notes and comments
. 34
2
Brownian motion
35
2.1
Definition and basic properties
. 35
2.2
Basic properties
. 39
2.3
The law of iterated logarithm
. 42
2.4
Symmetric random walks
. 45
2.5
Donsker theorem
. 55
Notes and comments
. 62
3
Markov processes
63
3.1
Markov processes
. 63
3.2
Strong Markov processes
. 72
3.3
Feller-Dynkin diffusions
. 75
3.4
Levy processes
. 87
Notes and comments
. 96
4
Symmetric diffusion semigroups
97
4.1
Essential self-
adj
ointness, spectral theorem
. 97
4.2
Existence and regularity of the heat kernel
. 106
4.3
The sub-Markov property
. 109
4.4
/^-theory: The interpolation method
. 112
viii Contents
4.5 L^-theory: The Hille-Yosida
method
. 114
4.6 Diffusion
semigroups as solutions of a parabolic Cauchy problem
125
4.7
The Dirichlet semigroup
. 127
4.8
The Neumann semigroup
. 131
4.9
Symmetric diffusion processes
. 132
Notes and comments
. 135
5
Ito
calculus
138
5.1
Variation of the Brownian paths
. 138
5.2
Ito
integral
. 140
5.3
Square
integrable
martingales and quadratic variations
. 146
5.4
Local martingales, semimartingales and integrators
. 154
5.5
Doblin-Itô
formula
. 161
5.6
Recurrence and transience of the Brownian motion
in higher dimensions
. 166
5.7
Ito
representation theorem
. 168
5.8
Time changed martingales and planar Brownian motion
. 171
5.9
Burkholder-Davis-Gundy inequalities
. 175
5.10
Girsanov theorem
. 178
Notes and comments
. 183
6
Stochastic differential equations and Malliavin calculus
185
6.1
Existence and uniqueness of solutions
. 185
6.2
Continuity and differentiability of stochastic flows
. 190
6.3
The Feynman-Kac formula
. 193
6.4
The strong Markov property of solutions
. 199
6.5
Stratonovitch stochastic differential equations and the language
of vector fields
. 201
6.6
Malliavin calculus
. 205
6.7
Existence of a smooth density
. 216
Notes and comments
. 219
7
An introduction to Lyons'rough paths theory
221
7.1
Continuous paths with finite
ρ
-vaii ation
. 221
7.2
The signature of a bounded variation path
. 226
7.3
Estimating iterated integrals
. 228
7.4
Rough differential equations
. 239
7.5
The Brownian motion as a rough path
. 245
Notes and comments
. 254
Appendix A Unbounded operators
257
Contents ix
Appendix
В
Regularity theory
262
References
269
Index
275
The main purpose of the book is to present at a graduate level and in a self-
contained way the most important aspects of the theory of continuous stochastic
processes in continuous time and to introduce to some of its ramifications like
the theory of semigroups, the Malliavin calculus and the Lyons' rough paths. It is
intended for students, or even researchers, who wish to learn the basics in a con¬
cise but complete and rigorous manner. Several exercises are distributed through¬
out the text to test the understanding of the reader and each chapter ends up
with bibliographic comments aimed to those interested in exploring
further the materials.
The stochastic calculus has been developed in the
1950s
and the range of
its applications is huge and still growing today. Besides being a fundamental
component of modern probability theory, domains of applications include but are
not limited to: mathematical finance, biology, physics, and engineering sciences.
The first part of the text is devoted the general theory of stochastic processes,
we focus on existence and regularity results for processes and on the theory of
martingales. This allows to quickly introduce the Brownian motion and to study
its most fundamental properties. The second part deals with the study of Markov
processes, in particular diffusions. Our goal is to stress the connections between
these processes and the theory of evolution semigroups. The third part deals
with stochastic integrals, stochastic differential equations and Malliavin calculus.
Finally, in the fourth part we present an introduction to the very new theory of
rough paths by Terry Lyons. |
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| discipline | Mathematik |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-09-20T04:19:25Z |
| institution | BVB |
| isbn | 9783037191330 |
| language | English |
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| physical | IX, 276 Seiten |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | European Mathematical Society |
| record_format | marc |
| series2 | EMS Textbooks in Mathematics |
| spelling | Baudoin, Fabrice 1975- Verfasser (DE-588)1076992641 aut Diffusion processes and stochastic calculus Fabrice Baudoin Zürich European Mathematical Society [2014] © 2014 IX, 276 Seiten txt rdacontent n rdamedia nc rdacarrier EMS Textbooks in Mathematics Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Diffusionsprozess (DE-588)4274463-5 gnd rswk-swf Diffusionsprozess (DE-588)4274463-5 s Stochastische Analysis (DE-588)4132272-1 s DE-604 Erscheint auch als Online-Ausgabe 978-3-03719-633-5 (DE-604)BV042019257 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027456341&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027456341&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Baudoin, Fabrice 1975- Diffusion processes and stochastic calculus Stochastische Analysis (DE-588)4132272-1 gnd Diffusionsprozess (DE-588)4274463-5 gnd |
| subject_GND | (DE-588)4132272-1 (DE-588)4274463-5 |
| title | Diffusion processes and stochastic calculus |
| title_auth | Diffusion processes and stochastic calculus |
| title_exact_search | Diffusion processes and stochastic calculus |
| title_full | Diffusion processes and stochastic calculus Fabrice Baudoin |
| title_fullStr | Diffusion processes and stochastic calculus Fabrice Baudoin |
| title_full_unstemmed | Diffusion processes and stochastic calculus Fabrice Baudoin |
| title_short | Diffusion processes and stochastic calculus |
| title_sort | diffusion processes and stochastic calculus |
| topic | Stochastische Analysis (DE-588)4132272-1 gnd Diffusionsprozess (DE-588)4274463-5 gnd |
| topic_facet | Stochastische Analysis Diffusionsprozess |
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