Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin
Logos-Verl.
1997
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Zugl.: Berlin, Humboldt-Univ., Diss., 1997 |
| Physical Description: | XXII, 264 S. graph. Darst. |
| ISBN: | 3931216942 |
Staff View
MARC
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| 001 | BV011458901 | ||
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| 100 | 1 | |a Schurz, Henri |d 1964- |e Verfasser |0 (DE-588)115479228 |4 aut | |
| 245 | 1 | 0 | |a Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications |c Henri Schurz |
| 264 | 1 | |a Berlin |b Logos-Verl. |c 1997 | |
| 300 | |a XXII, 264 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Zugl.: Berlin, Humboldt-Univ., Diss., 1997 | ||
| 650 | 4 | |a Stochastic differential equations | |
| 650 | 0 | 7 | |a Wiener-Prozess |0 (DE-588)4189870-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
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| _version_ | 1807501559387389952 |
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CONTENTS
GENERAL
NOTATION
0
PREFACE
IX
0.1
ABSTRACT
-
VERBAL
INTRODUCTION
TO
SUBJECT
.
IX
0.2
ABSTRAKT
(SUMMARY
IN
GERMAN
AS
SUPPLEMENT)
.
XVI
0.3
A
PERSONAL
NOTE
TO
READERSHIP
.
XX
0.4
ACKNOWLEDGEMENTS
.
XXII
1
INTRODUCTION
TO
ASYMPTOTICAL
MEAN
SQUARE
STABILITY
OF
TRIVIAL
SOLUTION
OF
IMPLICIT
NUMERICAL
METHODS
FOR
SDES
1
1.1
INTRODUCTION
.
2
1.2
BILINEAR
SYSTEMS
AND
MEAN
SQUARE
STABILITY
.
4
1.3
A
COMPARISON
THEOREM
FOR
MEAN
SQUARE
EVOLUTIONS
.
.
9
1.4
A
COMPLEX-VALUED
EXAMPLE
(KUBO
OSCILLATOR)
.
13
1.4.1
COMPARISON
OF
TEMPORAL
MEAN
SQUARE
EVOLUTIONS
.
14
1.4.2
CONTINUOUS
TIME
KUBO
OSCILLATOR
.
15
1.4.3
DISCRETIZED
KUBO
OSCILLATORS
AND
NUMERICAL
EXPERIMENTS
.
17
1.5
IMPLICIT
METHODS
AND
MEAN
SQUARE
STABILITY
.
19
1.5.1
A
SUFFICIENT
CONDITION
FOR
AUTONOMOUS
IMPLICIT
METHODS
.
19
1.5.2
ON
METHODS
WHICH
ARE
NOT
MEAN
SQUARE
A-STABLE
.
22
1.5.3
A
SUFFICIENT
CONDITION
FOR
NONAUTONOMOUS
METHODS
.
24
1.6
MONOTONIC
NESTING
PRINCIPLES
OF
STABILITY
DOMAINS
.
29
1.6.1
MONOTONIC
NESTING
WITH
RESPECT
TO
INCREASING
IMPLICITNESS
.
29
1.6.2
MONOTONIC
NESTING
WITH
RESPECT
TO
STEP
SIZES
.
31
II
CONTENTS
1.7
AN
ALTERNATIVE
-
BIMS
WITH
SCALAR
CORRECTION
.
34
1.8
AN
APPLICATION
TO
THE
NOISY
BRUSSELATOR
.
38
1.8.1
STABILITY
OF
FIRST
MOMENTS
.
39
1.8.2
STABILITY
OF
SECOND
MOMENTS
.
39
1.8.3
NUMERICAL
EXPERIMENTS
AND
ESTIMATION
OF
SECOND
MOMENTS
.
40
1.9
CONCLUSIONS
AND
REMARKS
.
42
2
PRESERVATION
OF
PROBABILISTIC
LAWS
THROUGH
FAMILY
OF
IMPLICIT
EULER
METH
ODS
FOR
ADDITIVE
NOISE
55
2.1
INTRODUCTION
.
56
2.2
NUMERICAL
SOLUTION
FOR
LINEAR
SDES
.
59
2.3
THE
PRESERVATION
OF
ASYMPTOTICAL
PROPERTIES
.
60
2.4
THE
GENERAL
LAW
OF
LINEAR
EULER
METHODS
.
64
2.5
ASYMPTOTIC
MOMENTS
OF
NONAUTONOMOUS
SYSTEMS
.
68
2.6
TWO
EXAMPLES
.
69
2.6.1
A
STOCHASTICALLY
PERTURBED
ROTATION
.
69
2.6.2
STOCHASTICALLY
PERTURBED
OSCILLATORS
.
70
2.7
CONCLUSIONS
AND
REMARKS
.
71
3
FURTHER
DISCUSSION
ON
AUTONOMOUS
MEAN
SQUARE
STABILITY
ANALYSIS
77
3.1
INTRODUCTION
.
77
3.2
CONTINUOUS
SYSTEMS
AND
MEAN
SQUARE
STABILITY
.
79
3.2.1
MEAN
SQUARE
EQUIVALENT
SYSTEMS
AND
NOISE
REDUCTION
.
80
3.2.2
A
CRITERION
OF
EMS-STABILITY
.
82
3.2.3
MEAN
SQUARE
MAJORANTS
.
82
3.3
CONTINUOUS
TIME
OSCILLATORS
AND
STABILITY
.
84
3.4
DISCRETE
TIME
SYSTEMS
AND
MEAN
SQUARE
STABILITY
.
85
3.5
DISCRETIZED
STOCHASTIC
OSCILLATORS
AND
STABILITY
.
88
3.5.1
STOCHASTIC
0-METHODS
AND
THEIR
CONVERGENCE
.
88
3.5.2
DISCRETIZATIONS
OF
STOCHASTIC
OSCILLATORS
.
89
CONTENTS
III
3.5.3
STABILITY
INVESTIGATION
FOR
0-METHOD
IN
ONE
DIMENSION
.
90
3.5.4
STABILITY
INVESTIGATIONS
FOR
DISCRETIZED
LINEAR
OSCILLATOR
.
94
3.6
CONCLUSIONS
AND
REMARKS
.
99
4
THE
INVARIANCE
OF
ASYMPTOTIC
PROBABILISTIC
LAWS
OF
LINEAR
STOCHASTIC
SYS
TEMS
UNDER
TIME-DISCRETIZATION
103
4.1
INTRODUCTION
.
103
4.2
ASYMPTOTIC
LAWS
OF
LINEAR
STOCHASTIC
SYSTEMS
.
106
4.3
COINCIDENCE
OF
ASYMPTOTIC
LAWS
OF
LINEAR
SYSTEMS
.
108
4.4
NUMERICAL
EXPERIMENTS
FOR
RANDOM
OSCILLATORS
.
ILL
4.5
CONCLUSIONS
AND
REMARKS
.
114
4.6
APPENDIX:
AN
AUXILIARY
LEMMA
.
115
5
CONSTRUCTION
OF
NONNEGATIVE
NUMERICAL
SOLUTIONS
FOR
SDES
119
5.1
INTRODUCTION
.
119
5.2
AUTONOMOUS
SDES
AND
THEIR
NUMERICAL
SOLUTION
.
120
5.3
LIFE
TIME
OF
NUMERICAL
SOLUTIONS
.
122
5.4
CONSTRUCTION
OF
NONNEGATIVE
SOLUTIONS
IN
1R
1
.
124
5.4.1
A
DETERMINISTIC
ONE-DIMENSIONAL
MODEL
(MOTIVATION)
.
124
5.4.2
A
STOCHASTIC
BILINEAR
ONE-DIMENSIONAL
MODEL
.
124
5.4.3
A
NONLINEAR
DIFFUSION
IN
POPULATION
DYNAMICS
.
127
5.4.4
A
DIFFUSION
PROCESS
WITH
LINEAR
DRIFT
.
128
5.4.5
A
MEAN-REVERTING
PROCESS
WITH
NONLINEAR
DIFFUSION
.
129
5.4.6
DIFFUSION
OF
INNOVATION
IN
MARKETING
SCIENCES
.
130
5.5
TWO
THEOREMS
FOR
DIFFUSIONS
WITH
LINEAR
GROWTH
.
131
5.6
AN
EXTENDED
MODEL
OF
COX-INGERSOLL-ROSS
.
133
5.6.1
SOME
INTEREST
RATE
MODELS
.
133
5.6.2
BIMS
FOR
STOCHASTIC
INTEREST
RATES
.
134
5.6.3
AN
EXPLICITLY
SOLVABLE
EXAMPLE
(BESSEL
PROCESS)
.
135
5.6.4
NUMERICAL
EXPERIMENTS:
FAILURE
PROBABILITIES
AND
ERROR
PROCESS
.
.
.
135
IV
CONTENTS
5.6.5
THE
ROLE
OF
IMPLICITNESS
FOR
NONNEGATIVITY
.
137
5.7
CONCLUSIONS
AND
REMARKS
.
138
5.8
APPENDIX:
DISCUSSION
ON
BROWNIAN
BRIDGES
.
140
6
APPLICATION
TO
INNOVATION
DIFFUSION
IN
STOCHASTIC
MARKETING
145
6.1
INTRODUCTION
TO
STOCHASTIC
BASS
MODEL
.
146
6.2
ANALYTICAL
NONREGULARITY
AND
REGULARITY
.
146
6.2.1
NONREGULARITY
WITH
ADDITIVE
NOISE
.
147
6.2.2
REGULARITY
WITH
MULTIPLICATIVE
NOISE
.
148
6.3
NUMERICAL
REGULARIZATION
VIA
IMPLICIT
METHODS
.
149
6.4
SIMULATION
RESULTS
.
151
6.5
SUMMARY
AND
REMARKS
.
152
6.6
APPENDIX:
A
GENERALIZATION
OF
THEOREM
ON
MEAN
SQUARE
CONVERGENCE
.
.
153
7
STABILITY
OF
CONTROLLED
MECHANICAL
STRUCTURES
WITH
DISTRIBUTED
TIME
DELAY
UNDER
RANDOM
VIBRATIONS
157
7.1
INTRODUCTION
.
158
7.2
STOCHASTIC
MODEL
WITH
DELAYS
AND
ACTIVE
CONTROL
.
159
7.2.1
WEAK
DELAY
.
160
7.2.2
STRONG
DELAY
.
161
7.3
A
GENERAL
APPROACH
TO
STOCHASTIC
STABILITY
.
161
7.4
NUMERICAL
RESULTS
FOR
TOP
LYAPUNOV
EXPONENT
.
163
7.5
PHASE
PLANE
ANALYSIS
AND
EXIT
FREQUENCIES
OF
NONLIN
EAR
SYSTEMS
.
165
7.5.1
PHASE
DIAGRAMS
.
165
7.5.2
ESTIMATION
OF
EXIT
FREQUENCY
(FAILURE
PROBABILITIES)
.
167
7.6
SUMMARY,
REMARKS,
AND
OPEN
PROBLEMS
.
167
8
NONLINEAR
STABILITY
AND
CONTRACTIVITY
OF
STOCHASTIC
DYNAMICAL
SYSTEMS
WITH
SOME
DISSIPATIVE
STRUCTURE
173
8.1
INTRODUCTION
.
174
CONTENTS
V
8.2
CONCEPTS
OF
DISSIPATIVITY
ON
TIME
SCALES
.
175
8.3
DISSIPATIVITY
OF
STOCHASTIC
SYSTEMS
.
177
8.3.1
DISSIPATIVITY
OF
CONTINUOUS
TIME
SDES
.
178
8.3.2
DISSIPATIVITY
OF
SOME
DISCRETE
TIME
SYSTEMS
.
180
8.4
FURTHER
CLASSIFICATION
OF
DISSIPATIVE
SYSTEMS
.
186
8.4.1
LINEARLY
GROWTH-BOUNDED
COEFFICIENT
SYSTEMS
.
186
8.4.2
A
AND
AN-DISSIPATIVITY
.
188
8.4.3
MONOTONE
COEFFICIENT
SYSTEMS
.
188
8.4.4
B
AND
BN-DISSIPATIVITY
.
190
8.4.5
ESTIMATES
FOR
GROWTH-BOUNDED
COEFFICIENT
SYSTEMS
.
191
8.4.6
PROPAGATION
OF
INITIAL
PERTURBATIONS
IN
MONOTONE
SYSTEMS
.
192
8.5
P-TH
MEAN
STABILITY
IN
WIDE
AND
NARROW
SENSE
.
194
8.5.1
ASYMPTOTICAL
P-TH
MEAN
STABILITY
ON
TIME
SCALES
.
194
8.5.2
EXPONENTIAL
STABILITY
FOR
NONLINEAR
SDES
IN
WIDE
SENSE
.
195
8.5.3
EXPONENTIAL
STABILITY
FOR
NONLINEAR
NUMERICAL
METHODS
.
196
8.5.4
NONLINEAR
A
AND
AN-STABILITY
.
197
8.6
CONTRACTIVITY
IN
WIDE
AND
NARROW
SENSE
.
198
8.6.1
CONTRACTIVITY
OF
INITIAL
PERTURBATIONS
ON
TIME
SCALES
.
198
8.6.2
CONTRACTIVITY
OF
INITIAL
PERTURBATIONS
OF
SDES
.
199
8.6.3
CONTRACTIVITY
OF
SOME
NUMERICAL
METHODS
.
200
8.6.4
NONLINEAR
B
AND
BN-STABILITY
.
200
8.7
V-DISSIPATIVITY,
V-STABILITY,
V-CONTRACTIVITY
.
201
8.8
SOME
ILLUSTRATIVE
EXAMPLES
.
205
8.8.1
STOCHASTIC
DISK
DYNAMO
MODEL
.
206
8.8.2
UNFORCED
DUFFING
OSCILLATOR
WITH
RANDOM
NOISE
.
207
8.8.3
FORCED
VAN
DER
POL
OSCILLATORS
WITH
RANDOM
NOISE
.
208
8.9
CONCLUSIONS
AND
REMARKS
.
211
8.10
APPENDIX:
ELEMENTARY
LEMMAS
.
212
9
LINEAR-IMPLICIT
NUMERICAL
METHODS
FOR
SOME
NONLINEAR
SDES
219
VI
CONTENTS
9.1
INTRODUCTION
.
219
9.2
NUMERICAL
TREATMENT
OF
NOISY
DUFFING
OSCILLATOR
.
221
9.2.1
DISCRETIZED
STOCHASTIC
DUFFING
OSCILLATORS
.
222
9.2.2
A
COMPARISON
WITH
RESPECT
TO
EFFICIENCY
AND
STATIONARITY
.
224
9.3
LINEAR-IMPLICIT
METHODS
AND
CONVERGENCE
NOTIONS
.
225
9.3.1
GENERAL
(DETERMINISTICALLY)
LINEAR-IMPLICIT
METHODS
.
225
9.3.2
BASIC
CONVERGENCE
NOTIONS
FOR
NUMERICAL
APPROXIMATIONS
.
227
9.4
CONVERGENCE
RATES
OF
LINEAR-IMPLICIT
METHODS
.
228
9.4.1
MEAN
SQUARE
CONVERGENCE
ANALYSIS:
PART
I
.
228
9.4.2
MEAN
SQUARE
CONVERGENCE
ANALYSIS:
PART
II
.
231
9.5
CONCLUSION
AND
REMARKS
.
233
A
SORTED
LIST
OF
REFERENCES
239
A.L
MONOGRAPHS
AND
TEXTBOOKS
ON
THEORY
AND
APPLICATION
OF
STOCHASTIC
DIFFER
ENTIAL
EQUATIONS
.
240
A.2
MONOGRAPHS
AND
FOUNDATIONS
ON
NUMERICAL
ANALYSIS
FOR
STOCHASTIC
DIFFEREN
TIAL
EQUATIONS
.
245
B
CURRICULUM
VITAE
OF
AUTHOR
253
C
LIST
OF
PUBLICATIONS
OF
AUTHOR
255
C.L
MONOGRAPHS
.
255
C.2
PUBLICATIONS
IN
REFEREED
JOURNALS
.
255
C.3
REFEREED
PUBLICATIONS
IN
SPECIAL
VOLUMES
.
256
C.4
PUBLICATIONS
IN
CONFERENCE
PROCEEDINGS
.
256
C.5
SONDERDRUCK
/
TECHNICAL
REPORTS
.
257
INDEX
.
257 |
| any_adam_object | 1 |
| author | Schurz, Henri 1964- |
| author_GND | (DE-588)115479228 |
| author_facet | Schurz, Henri 1964- |
| author_role | aut |
| author_sort | Schurz, Henri 1964- |
| author_variant | h s hs |
| building | Verbundindex |
| bvnumber | BV011458901 |
| 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 920 |
| ctrlnum | (OCoLC)40381682 (DE-599)BVBBV011458901 |
| 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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| genre | (DE-588)4113937-9 Hochschulschrift gnd-content |
| genre_facet | Hochschulschrift |
| id | DE-604.BV011458901 |
| illustrated | Illustrated |
| indexdate | 2024-08-16T00:23:31Z |
| institution | BVB |
| isbn | 3931216942 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007707995 |
| oclc_num | 40381682 |
| open_access_boolean | |
| owner | DE-703 DE-706 |
| owner_facet | DE-703 DE-706 |
| physical | XXII, 264 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Logos-Verl. |
| record_format | marc |
| spelling | Schurz, Henri 1964- Verfasser (DE-588)115479228 aut Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications Henri Schurz Berlin Logos-Verl. 1997 XXII, 264 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Zugl.: Berlin, Humboldt-Univ., Diss., 1997 Stochastic differential equations Wiener-Prozess (DE-588)4189870-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Numerische Integration (DE-588)4172168-8 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Wiener-Prozess (DE-588)4189870-9 s Stochastische Differentialgleichung (DE-588)4057621-8 s Numerische Integration (DE-588)4172168-8 s DE-604 Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007707995&sequence=000001&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 | Schurz, Henri 1964- Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications Stochastic differential equations Wiener-Prozess (DE-588)4189870-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Numerische Integration (DE-588)4172168-8 gnd |
| subject_GND | (DE-588)4189870-9 (DE-588)4128130-5 (DE-588)4057621-8 (DE-588)4172168-8 (DE-588)4113937-9 |
| title | Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications |
| title_auth | Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications |
| title_exact_search | Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications |
| title_full | Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications Henri Schurz |
| title_fullStr | Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications Henri Schurz |
| title_full_unstemmed | Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications Henri Schurz |
| title_short | Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications |
| title_sort | stability stationarity and boundedness of some implicit numerical methods for stochastic differential equations and applications |
| topic | Stochastic differential equations Wiener-Prozess (DE-588)4189870-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Numerische Integration (DE-588)4172168-8 gnd |
| topic_facet | Stochastic differential equations Wiener-Prozess Numerisches Verfahren Stochastische Differentialgleichung Numerische Integration Hochschulschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007707995&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT schurzhenri stabilitystationarityandboundednessofsomeimplicitnumericalmethodsforstochasticdifferentialequationsandapplications |