Introduction to stochastic analysis and Malliavin calculus:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Pisa
Ed. della Normale
2007
|
| Series: | Appunti
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 190 S. |
| ISBN: | 9788876423130 |
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MARC
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| 245 | 1 | 0 | |a Introduction to stochastic analysis and Malliavin calculus |c Giuseppe Da Prato |
| 264 | 1 | |a Pisa |b Ed. della Normale |c 2007 | |
| 300 | |a XVI, 190 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Appunti |v 6 | |
| 650 | 7 | |a Stochastische analyse |2 gtt | |
| 650 | 4 | |a Malliavin calculus | |
| 650 | 4 | |a Stochastic differential equations | |
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Record in the Search Index
| _version_ | 1804137208375934976 |
|---|---|
| adam_text | Contents
Introduction
xi
1
Gaussian
measures in Hubert spaces
1
1.1
General concepts of probability measures in Hubert spaces
1
1.2
Gaussian probability measures
.............. 3
1.2.1
Gaussian measures in
IR
............. 3
1.2.2
Gaussian measures in Hubert spaces
....... 4
1.3
Gaussian random variables
................ 5
1.3.1
Random variables
................. 5
1.4
Gaussian random variables
................ 6
1.4.1
Product measures
................. 7
1.4.2
Independent real variables
............ 8
1.5
Linear functional in L2(H,
μ)
.............. 8
1.5.1
Linear bounded functionals
............ 9
1.5.2
Linear unbounded functional
........... 10
2
L2 and Sobolev spaces with respect to a Gaussian measure
13
2.1
Some useful approximation results
............ 15
2.2
Sobolev spaces
...................... 16
2.2.1
Generalities on closable operators
........ 16
2.2.2
Weak derivatives in L2(H,
μ)
.......... 17
2.3
The Malliavin derivative
................. 19
2.4
Calculus rules for Malliavin derivatives
......... 20
2.5
The basic integration by parts formula
.......... 22
2.6
The adjoint of the Malliavin derivative
.......... 23
2.6.1
Generalities on adjoint operators
......... 23
2.6.2
The adjoint operator of
M
............ 24
Brownian motion
27
3.1
Generalities on stochastic Processes
........... 27
3.1.1
The spaces
<ür([O,
Т];ЩП))
and C([0,
Γ];£Ρ(Ω))
28
3.1.2
The spaces
Jšf
(Ω;
C{[0,
Τ]))
and
ί, (Ω;
С([0,Г]))
28
3.2
Construction of a Brownian motion
............ 29
3.3
The standard Brownian motion
.............. 33
3.4
Quadratic variation of the Brownian motion
....... 34
3.5
Wiener integral
...................... 36
3.6
Multidimensional Brownian motions
........... 40
Markov property of the Brownian motion
43
4.1
Cylindrical sets
...................... 44
4.2
Filtration, zero-one law
.................. 45
4.3
Stopping times
...................... 47
4.4
The Brownian motion B(t
+
r)
-
Β (τ).........
50
4.5
Transition semigroup
................... 51
4.6
Markov property
..................... 51
4.6.1
Strong Markov property
............. 52
4.7
.Reflection Principle
.................... 53
4.8
Application to partial differential equations
....... 56
4.8.1
The Dirichlet problem
.............. 56
4.8.2
The Neumann problem
............. 57
4.8.3
The Ventzell problem
............... 58
The
Ito
integral
59
5.1
Itô s
integral for elementary processes
.......... 60
5.2
General definition of
Itô s
integrals
............ 61
5.3
Some basic spaces of processes
.............. 63
5.3.1
The space L2([0,
Γ]χΩ,^,λχΡ)
...... 63
5.3.2
The space
CßCfO,
Γ]; ¿ (Ω)) ..........
64
5.3.3
The space
¿g
(Ω;
C([0,
Τ]))
........... 64
5.4
Itô s
integral for mean square continuous processes
... 66
5.5
The
Ito
integral as a stochastic process
.......... 68
5.6
Itô s
integral with stopping times
............. 70
5.7
Multidimensional
Itô s
integrals
............. 71
The
Ito
formula
73
6.1
Itô s
formula for real processes
.............. 74
6.1.1
Itô s
formula for functions of a real Brownian
motion
...................... 74
6.1.2
Itô s
formula for a function of an
Itô s
integral
. . 77
6.1.3
Itô s
formula for unbounded functions
...... 82
6.1.4
The Ito
formula for functions of general processes
84
6.2
The
Ito
formula for functions of multi-dimensional pro¬
cesses
........................... 86
6.2.1
The
Ito
formula for Brownian motions
...... 86
6.2.2
The
Ito
formula for functions of multi-dimensional
processes
..................... 87
6.3
The Bismut-Elworthy formula
.............. 89
7
Stochastic differential equations
91
7.1
Existence and uniqueness
................. 92
7.1.1
Differential stochastic equations with random co¬
efficients
..................... 98
7.2
Continuous dependence on data
............. 99
7.3
Differentiability of X(t, s, x) with respect to
χ
...... 101
7.3.1
Differentiability of X(t, s, x) with respect to x.
. 102
7.3.2
Existence of Xxx(t,
s, x
)............. 103
7.4
Ito
Differentiability of
X (t, s, x)
with respect to
tç
.... 106
7.4.1
The deterministic case
.............. 106
7.4.2
The stochastic case
................ 107
7.4.3
Backward
Itô s
formula
.............. 108
8
Transition evolution operators 111
8.1
The deterministic case
..................
Ill
8.1.1
The autonomous case
............... 113
8.2
Stochastic case
...................... 114
8.3
Markov property of X(t, s,x)
.............. 115
8.3.1
The Chapman-Kolmogorov equation
...... 117
8.4
Basic properties of transition operators
.......... 117
8.5
Parabolic equations
.................... 118
8.6
Examples
......................... 121
8.7
The Bismut-Elworthy formula
.............. 122
9
Formulae of Feynman-Kac and Girsanov
125
9.1
The Feynman-Kac formula
................ 126
9.2
The Girsanov formula
................... 129
9.2.1
The autonomous case
............... 133
9.3
The Girsanov transform
.................. 135
9.3.1
Change of probability
.............. 135
9.3.2
An application
.................. 139
10
One dimensional Malliavin calculus
141
10.1
Some further properties of the white noise function
. . . 142
10.2
The Malliavin derivative
................. 143
10.2.1
Malliavin derivative of
Itô s
integrals
...... 144
10.2.2
The Skorohod integral
.............. 145
10.3
Malliavin derivative of a stochastic flows
......... 147
10.4
Regularity of the transition semigroup
.......... 148
10.5
Existence of the density of the law of X(t,x)
...... 150
11
Malliavin calculus in several dimensions
153
11.1
Some properties of the white noise function
....... 153
11.2
The Malliavin derivative
................. 154
11.3
The Clark-Ocone formula
................. 155
11.4
The Skorohod integral
.................. 156
11.5
Malliavin derivative of a stochastic flow
......... 157
11.6
Regularity of the transition semigroup
.......... 158
11.6.1
The case when
det G
Φ
0 ............ 159
11.6.2
The case when
det G
= 0 ............ 159
11.7
Existence of a density
................... 161
12
Asymptotic behaviour of the transition semigroup
163
12.1
Preliminary properties of Pt
............... 163
12.2
Invariant measures
.................... 165
12.2.1
The Prokhorov theorem
............. 166
12.2.2
The Krylov-Bogoliubov theorem
......... 166
12.3
Ergodiciity
........................ 168
12.3.1
Ergodic averages
................. 169
12.3.2
The
Von
Neumann theorem
........... 170
12.3.3
Necessary and sufficient conditions for ergodicity
172
A Conditional expectation
175
A.I Definition
......................... 175
A.2 Basic properties of the conditional expectation
...... 175
В
λ
-systems
and
π
-systems
179
С
Martingales
181
C.I Definitions
......................... 181
C.2 The basic inequality for martingales
........... 182
C.3 Square
integrable
martingales
............... 183
D
Fixed points depending on parameters
185
D.I Introduction
........................ 185
D.2 The main result
...................... 186
|
| adam_txt |
Contents
Introduction
xi
1
Gaussian
measures in Hubert spaces
1
1.1
General concepts of probability measures in Hubert spaces
1
1.2
Gaussian probability measures
. 3
1.2.1
Gaussian measures in
IR
. 3
1.2.2
Gaussian measures in Hubert spaces
. 4
1.3
Gaussian random variables
. 5
1.3.1
Random variables
. 5
1.4
Gaussian random variables
. 6
1.4.1
Product measures
. 7
1.4.2
Independent real variables
. 8
1.5
Linear functional in L2(H,
μ)
. 8
1.5.1
Linear bounded functionals
. 9
1.5.2
Linear unbounded functional
. 10
2
L2 and Sobolev spaces with respect to a Gaussian measure
13
2.1
Some useful approximation results
. 15
2.2
Sobolev spaces
. 16
2.2.1
Generalities on closable operators
. 16
2.2.2
Weak derivatives in L2(H,
μ)
. 17
2.3
The Malliavin derivative
. 19
2.4
Calculus rules for Malliavin derivatives
. 20
2.5
The basic integration by parts formula
. 22
2.6
The adjoint of the Malliavin derivative
. 23
2.6.1
Generalities on adjoint operators
. 23
2.6.2
The adjoint operator of
M
. 24
Brownian motion
27
3.1
Generalities on stochastic Processes
. 27
3.1.1
The spaces
<ür([O,
Т];ЩП))
and C([0,
Γ];£Ρ(Ω))
28
3.1.2
The spaces
Jšf
(Ω;
C{[0,
Τ]))
and
ί,'(Ω;
С([0,Г]))
28
3.2
Construction of a Brownian motion
. 29
3.3
The standard Brownian motion
. 33
3.4
Quadratic variation of the Brownian motion
. 34
3.5
Wiener integral
. 36
3.6
Multidimensional Brownian motions
. 40
Markov property of the Brownian motion
43
4.1
Cylindrical sets
. 44
4.2
Filtration, zero-one law
. 45
4.3
Stopping times
. 47
4.4
The Brownian motion B(t
+
r)
-
Β (τ).
50
4.5
Transition semigroup
. 51
4.6
Markov property
. 51
4.6.1
Strong Markov property
. 52
4.7
.Reflection Principle
. 53
4.8
Application to partial differential equations
. 56
4.8.1
The Dirichlet problem
. 56
4.8.2
The Neumann problem
. 57
4.8.3
The Ventzell problem
. 58
The
Ito
integral
59
5.1
Itô's
integral for elementary processes
. 60
5.2
General definition of
Itô's
integrals
. 61
5.3
Some basic spaces of processes
. 63
5.3.1
The space L2([0,
Γ]χΩ,^,λχΡ)
. 63
5.3.2
The space
CßCfO,
Γ]; ¿'(Ω)) .
64
5.3.3
The space
¿g
(Ω;
C([0,
Τ]))
. 64
5.4
Itô's
integral for mean square continuous processes
. 66
5.5
The
Ito
integral as a stochastic process
. 68
5.6
Itô's
integral with stopping times
. 70
5.7
Multidimensional
Itô's
integrals
. 71
The
Ito
formula
73
6.1
Itô's
formula for real processes
. 74
6.1.1
Itô's
formula for functions of a real Brownian
motion
. 74
6.1.2
Itô's
formula for a function of an
Itô's
integral
. . 77
6.1.3
Itô's
formula for unbounded functions
. 82
6.1.4
The Ito
formula for functions of general processes
84
6.2
The
Ito
formula for functions of multi-dimensional pro¬
cesses
. 86
6.2.1
The
Ito
formula for Brownian motions
. 86
6.2.2
The
Ito
formula for functions of multi-dimensional
processes
. 87
6.3
The Bismut-Elworthy formula
. 89
7
Stochastic differential equations
91
7.1
Existence and uniqueness
. 92
7.1.1
Differential stochastic equations with random co¬
efficients
. 98
7.2
Continuous dependence on data
. 99
7.3
Differentiability of X(t, s, x) with respect to
χ
. 101
7.3.1
Differentiability of X(t, s, x) with respect to x.
. 102
7.3.2
Existence of Xxx(t,
s, x
). 103
7.4
Ito
Differentiability of
X (t, s, x)
with respect to
tç
. 106
7.4.1
The deterministic case
. 106
7.4.2
The stochastic case
. 107
7.4.3
Backward
Itô's
formula
. 108
8
Transition evolution operators 111
8.1
The deterministic case
.
Ill
8.1.1
The autonomous case
. 113
8.2
Stochastic case
. 114
8.3
Markov property of X(t, s,x)
. 115
8.3.1
The Chapman-Kolmogorov equation
. 117
8.4
Basic properties of transition operators
. 117
8.5
Parabolic equations
. 118
8.6
Examples
. 121
8.7
The Bismut-Elworthy formula
. 122
9
Formulae of Feynman-Kac and Girsanov
125
9.1
The Feynman-Kac formula
. 126
9.2
The Girsanov formula
. 129
9.2.1
The autonomous case
. 133
9.3
The Girsanov transform
. 135
9.3.1
Change of probability
. 135
9.3.2
An application
. 139
10
One dimensional Malliavin calculus
141
10.1
Some further properties of the white noise function
. . . 142
10.2
The Malliavin derivative
. 143
10.2.1
Malliavin derivative of
Itô's
integrals
. 144
10.2.2
The Skorohod integral
. 145
10.3
Malliavin derivative of a stochastic flows
. 147
10.4
Regularity of the transition semigroup
. 148
10.5
Existence of the density of the law of X(t,x)
. 150
11
Malliavin calculus in several dimensions
153
11.1
Some properties of the white noise function
. 153
11.2
The Malliavin derivative
. 154
11.3
The Clark-Ocone formula
. 155
11.4
The Skorohod integral
. 156
11.5
Malliavin derivative of a stochastic flow
. 157
11.6
Regularity of the transition semigroup
. 158
11.6.1
The case when
det G
Φ
0 . 159
11.6.2
The case when
det G
= 0 . 159
11.7
Existence of a density
. 161
12
Asymptotic behaviour of the transition semigroup
163
12.1
Preliminary properties of Pt
. 163
12.2
Invariant measures
. 165
12.2.1
The Prokhorov theorem
. 166
12.2.2
The Krylov-Bogoliubov theorem
. 166
12.3
Ergodiciity
. 168
12.3.1
Ergodic averages
. 169
12.3.2
The
Von
Neumann theorem
. 170
12.3.3
Necessary and sufficient conditions for ergodicity
172
A Conditional expectation
175
A.I Definition
. 175
A.2 Basic properties of the conditional expectation
. 175
В
λ
-systems
and
π
-systems
179
С
Martingales
181
C.I Definitions
. 181
C.2 The basic inequality for martingales
. 182
C.3 Square
integrable
martingales
. 183
D
Fixed points depending on parameters
185
D.I Introduction
. 185
D.2 The main result
. 186 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Da Prato, Giuseppe 1936-2023 |
| author_GND | (DE-588)121352641 |
| author_facet | Da Prato, Giuseppe 1936-2023 |
| author_role | aut |
| author_sort | Da Prato, Giuseppe 1936-2023 |
| author_variant | p g d pg pgd |
| building | Verbundindex |
| bvnumber | BV022961363 |
| callnumber-first | Q - Science |
| callnumber-label | QA274 |
| callnumber-raw | QA274.23 |
| callnumber-search | QA274.23 |
| callnumber-sort | QA 3274.23 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)181090561 (DE-599)BVBBV022961363 |
| dewey-full | 519 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519 |
| dewey-search | 519 |
| dewey-sort | 3519 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022961363 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:04:51Z |
| indexdate | 2024-07-09T21:08:38Z |
| institution | BVB |
| isbn | 9788876423130 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016165732 |
| oclc_num | 181090561 |
| open_access_boolean | |
| owner | DE-384 DE-824 DE-19 DE-BY-UBM DE-11 |
| owner_facet | DE-384 DE-824 DE-19 DE-BY-UBM DE-11 |
| physical | XVI, 190 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Ed. della Normale |
| record_format | marc |
| series | Appunti |
| series2 | Appunti |
| spelling | Da Prato, Giuseppe 1936-2023 Verfasser (DE-588)121352641 aut Introduction to stochastic analysis and Malliavin calculus Giuseppe Da Prato Pisa Ed. della Normale 2007 XVI, 190 S. txt rdacontent n rdamedia nc rdacarrier Appunti 6 Stochastische analyse gtt Malliavin calculus Stochastic differential equations Malliavin-Kalkül (DE-588)4242584-0 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 s Malliavin-Kalkül (DE-588)4242584-0 s 1\p DE-604 DE-604 Appunti 6 (DE-604)BV025597622 6 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016165732&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 | Da Prato, Giuseppe 1936-2023 Introduction to stochastic analysis and Malliavin calculus Appunti Stochastische analyse gtt Malliavin calculus Stochastic differential equations Malliavin-Kalkül (DE-588)4242584-0 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| subject_GND | (DE-588)4242584-0 (DE-588)4132272-1 |
| title | Introduction to stochastic analysis and Malliavin calculus |
| title_auth | Introduction to stochastic analysis and Malliavin calculus |
| title_exact_search | Introduction to stochastic analysis and Malliavin calculus |
| title_exact_search_txtP | Introduction to stochastic analysis and Malliavin calculus |
| title_full | Introduction to stochastic analysis and Malliavin calculus Giuseppe Da Prato |
| title_fullStr | Introduction to stochastic analysis and Malliavin calculus Giuseppe Da Prato |
| title_full_unstemmed | Introduction to stochastic analysis and Malliavin calculus Giuseppe Da Prato |
| title_short | Introduction to stochastic analysis and Malliavin calculus |
| title_sort | introduction to stochastic analysis and malliavin calculus |
| topic | Stochastische analyse gtt Malliavin calculus Stochastic differential equations Malliavin-Kalkül (DE-588)4242584-0 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| topic_facet | Stochastische analyse Malliavin calculus Stochastic differential equations Malliavin-Kalkül Stochastische Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016165732&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV025597622 |
| work_keys_str_mv | AT dapratogiuseppe introductiontostochasticanalysisandmalliavincalculus |