Yosida approximations of stochastic differential equations in infinite dimensions and applications:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cham
Springer
[2016]
|
| Schriftenreihe: | Probability theory and stochastic modelling
volume 79 |
| Schlagworte: | |
| Online-Zugang: | BTU01 FHR01 TUM01 UBM01 UBT01 UBW01 UEI01 URL des Erstveröffentlichers Inhaltsverzeichnis |
| Beschreibung: | 1 Online-Ressource (xix, 407 Seiten) Illustrationen |
| ISBN: | 9783319456843 |
| DOI: | 10.1007/978-3-319-45684-3 |
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Datensatz im Suchindex
| _version_ | 1804176865073561600 |
|---|---|
| adam_text | Titel: Yosida approximations of stochastic differential equations in infinite dimensions and applications
Autor: Govindan, T. E
Jahr: 2016
Contents
1 Introduction and Motivating Examples......................................................................1
1.1 A Heat Equation..............................................................................................................1
1.1.1 Stochastic Evolution Equations..........................................................2
1.2 An Electric Circuit..........................................................................................................3
1.2.1 Stochastic Evolution Equations with Delay ................................4
1.3 An Interacting Particle System................................................................................5
1.3.1 McKean-Vlasov Stochastic Evolution Equations....................5
1.4 A Lumped Control System........................................................................................6
1.4.1 Neutral Stochastic Partial Differential Equations....................6
1.5 A Hyperbolic Equation................................................................................................7
1.5.1 Stochastic Integrodifferential Equations........................................7
1.6 The Stock Price and Option Price Dynamics..................................................8
1.6.1 Stochastic Evolution Equations with Poisson jumps..............10
2 Mathematical Machinery......................................................................................................11
2.1 Semigroup Theory..........................................................................................................11
2.1.1 The Hille-Yosida Theorem....................................................................13
2.1.2 Yosida Approximations of Maximal Monotone Operators 17
2.2 Yosida Approximations and The Central Limit Theorem......................21
2.2.1 Optimal Convergence Rate for Yosida Approximations ... 22
2.2.2 Asymptotic Expansions for Yosida Approximations..............27
2.3 Almost Strong Evolution Operators....................................................................30
2.4 Basics from Analysis and Probability in Banach Spaces........................31
2.4.1 Wiener Processes..........................................................................................38
2.4.2 Poisson Random Measures and Poisson Point Processes .. 40
2.4.3 Lévy Processes..............................................................................................43
2.4.4 Random Operators......................................................................................44
2.4.5 The Gelfand Triple......................................................................................45
2.5 Stochastic Calculus........................................................................................................45
2.5.1 Itô Stochastic Integral with respect to a Q-Wiener process 46
xiii
xiv Contents
2.5.2 Itô Stochastic Integral with respect to a Cylindrical
Wiener Process..............................................................................................50
2.5.3 Stochastic Integral with respect to a Compensated
Poisson Measure..........................................................................................51
2.5.4 Itô s Formula for the case of a Q- Wiener Process....................54
2.5.5 Itô s Formula for the case of a Cylindrical
Wiener Process ............................................................................................55
2.5.6 Itô s Formula for the case of a Compensated
Poisson process ............................................................................................56
2.6 The Stochastic Fubini Theorem ............................................................................58
2.7 Stochastic Convolution Integrals............................................................................59
2.7.1 A Property using Yosida Approximations....................................60
2.8 Burkholder Type Inequalities..................................................................................62
2.9 Bounded Stochastic Integral Contractors..........................................................64
2.9.1 Volterra Series................................................................................................67
3 Yosida Approximations of Stochastic Differential Equations......................69
3.1 Linear Stochastic Evolution Equations..............................................................69
3.2 Semilinear Stochastic Evolution Equations....................................................74
3.3 Stochastic Evolution Equations with Delay....................................................83
3.3.1 Equations with a Constant Delay........................................................83
3.3.2 Strong Solutions by the Variational Method................................86
3.3.3 Equations with a Variable Delay........................................................92
3.4 McKean-Vlasov Stochastic Evolution Equations........................................96
3.4.1 Equations with an Additive Diffusion..............................................97
3.4.2 A Generalization with a Multiplicative Diffusion....................105
3.5 Neutral Stochastic Partial Differential Equations........................................112
3.6 Stochastic Integrodifferential Evolution Equations....................................122
3.6.1 Linear Stochastic Equations..................................................................122
3.6.2 Semilinear Stochastic Equations........................................................129
3.7 Multivalued Stochastic Partial Differential Equations
with a White Noise........................................................................................................135
3.8 Time-Varying Stochastic Evolution Equations..............................................152
3.9 Relaxed Solutions with Polynomial Nonlinearities
for Stochastic Evolution Equations......................................................................159
3.9.1 Radon Nikodym Property and Lifting ............................................160
3.9.2 Topological Compactifications and an
Existence Theorem......................................................................................161
3.9.3 Forward Kolmogorov Equations........................................................167
3.10 Evolution Equations Driven by Stochastic Vector Measures................171
3.10.1 Special Vector Spaces and Generalized Solutions....................171
3.11 Controlled Stochastic Differential Equations ................................................181
3.11.1 Measure-Valued McKean-Vlasov Evolution Equations.... 182
3.11.2 Equations with Partially Observed Relaxed Controls............193
Contents xv
4 Yosida Approximations of Stochastic Differential Equations
with Jumps........................................................................................................................................203
4.1 Stochastic Delay Evolution Equations with Poisson Jumps..................203
4.2 Stochastic Functional Equations with Markovian
Switching Driven by Lévy Martingales..............................................................211
4.3 Switching Diffusion Processes with Poisson Jumps ................................218
4.4 Multivalued Stochastic Partial Differential Equations
with Jumps..........................................................................................................................221
4.4.1 Equations Driven by a Poisson Noise..............................................221
4.4.2 Stochastic Porous Media Equations..................................................233
4.4.3 Equations Driven by a Poisson Noise with a
General Drift Term......................................................................................237
5 Applications to Stochastic Stability................................................................................241
5.1 Stability of Stochastic Evolution Equations....................................................241
5.1.1 Stability of Moments ................................................................................242
5.1.2 Sample Continuity......................................................................................243
5.1.3 Sample Path Stability................................................................................246
5.1.4 Stability in Distribution............................................................................249
5.2 Exponential Stabilizability of Stochastic Evolution Equations............257
5.2.1 Feedback Stabilization with a Constant Decay..........................258
5.2.2 Robust Stabilization with a General Decay..................................261
5.3 Stability of Stochastic Evolution Equations with Delay..........................271
5.3.1 Polynomial Stability and Lyapunov Functionals......................271
5.3.2 Stability in Distribution of Equations with
Poisson Jumps................................................................................................284
5.4 Exponential State Feedback Stabilizability of Stochastic
Evolution Equations with Delay by Razumikhin Type Theorem.... 296
5.5 Stability of McKean-Vlasov Stochastic Evolution Equations..............300
5.5.1 Weak Convergence of Induced Probability Measures............300
5.5.2 Almost Sure Exponential Stability of a General
Equation with a Multiplicative Diffusion......................................301
5.6 Weak Convergence of Probability Measures of Yosida
Approximating Mild Solutions of Neutral SPDEs......................................305
5.7 Stability of Stochastic Integrodifferential Equations..................................308
5.8 Exponential Stability of Stochastic Evolution Equations
with Markovian Switching Driven by Levy Martingales........................311
5.8.1 Equations with a Delay............................................................................312
5.8.2 Equations with Time-Varying Coefficients..................................321
5.9 Exponential Stability of Time-Varying Stochastic
Evolution Equations......................................................................................................330
6 Applications to Stochastic Optimal Control............................................................333
6.1 Optimal Control over a Finite Time Horizon..................................................333
6.2 A Periodic Control Problem under White Noise Perturbations............338
6.2.1 A Deterministic Optimization Problem..........................................341
xvi Contents
6.2.2 A Periodic Stochastic Case....................................................................343
6.2.3 Law of Large Numbers............................................................................345
6.3 Optimal Control for Measure-Valued McKean-Vlasov
Evolution Equations......................................................................................................348
6.4 Necessary Conditions of Optimality for Equations
with Partially Observed Relaxed Controls........................................................356
6.5 Optimal Feedback Control for Equations Driven
by Stochastic Vector Measures................................................................................359
6.5.1 Some Special Cases....................................................................................365
A Nuclear and Hilbert-Schmidt Operators....................................................................369
B Multivalued Maps........................................................................................................................373
C Maximal Monotone Operators..........................................................................................375
D The Duality Mapping................................................................................................................377
E Random Multivalued Operators......................................................................................379
Bibliographical Notes and Remarks......................................................................................383
Bibliography............................................................................................................................................391
Index..............................................................................................................................................................403
|
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-319-45684-3 |
| format | Electronic eBook |
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| indexdate | 2024-07-10T07:38:58Z |
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| isbn | 9783319456843 |
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| publishDateSort | 2016 |
| publisher | Springer |
| record_format | marc |
| series | Probability theory and stochastic modelling |
| series2 | Probability theory and stochastic modelling |
| spelling | Govindan, T. E. Verfasser (DE-588)112093267X aut Yosida approximations of stochastic differential equations in infinite dimensions and applications T.E. Govindan Cham Springer [2016] © 2016 1 Online-Ressource (xix, 407 Seiten) Illustrationen txt rdacontent c rdamedia cr rdacarrier Probability theory and stochastic modelling volume 79 Mathematik Mathematics Partial differential equations Probabilities Control engineering Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Kontrolltheorie (DE-588)4032317-1 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Numerisches Verfahren (DE-588)4128130-5 s Kontrolltheorie (DE-588)4032317-1 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Erscheint auch als Druck-Ausgabe 978-3-319-45682-9 Probability theory and stochastic modelling volume 79 (DE-604)BV041997534 79 https://doi.org/10.1007/978-3-319-45684-3 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029341359&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Govindan, T. E. Yosida approximations of stochastic differential equations in infinite dimensions and applications Probability theory and stochastic modelling Mathematik Mathematics Partial differential equations Probabilities Control engineering Partielle Differentialgleichung (DE-588)4044779-0 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Kontrolltheorie (DE-588)4032317-1 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4057621-8 (DE-588)4079013-7 (DE-588)4128130-5 (DE-588)4032317-1 |
| title | Yosida approximations of stochastic differential equations in infinite dimensions and applications |
| title_auth | Yosida approximations of stochastic differential equations in infinite dimensions and applications |
| title_exact_search | Yosida approximations of stochastic differential equations in infinite dimensions and applications |
| title_full | Yosida approximations of stochastic differential equations in infinite dimensions and applications T.E. Govindan |
| title_fullStr | Yosida approximations of stochastic differential equations in infinite dimensions and applications T.E. Govindan |
| title_full_unstemmed | Yosida approximations of stochastic differential equations in infinite dimensions and applications T.E. Govindan |
| title_short | Yosida approximations of stochastic differential equations in infinite dimensions and applications |
| title_sort | yosida approximations of stochastic differential equations in infinite dimensions and applications |
| topic | Mathematik Mathematics Partial differential equations Probabilities Control engineering Partielle Differentialgleichung (DE-588)4044779-0 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Kontrolltheorie (DE-588)4032317-1 gnd |
| topic_facet | Mathematik Mathematics Partial differential equations Probabilities Control engineering Partielle Differentialgleichung Stochastische Differentialgleichung Wahrscheinlichkeitstheorie Numerisches Verfahren Kontrolltheorie |
| url | https://doi.org/10.1007/978-3-319-45684-3 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029341359&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV041997534 |
| work_keys_str_mv | AT govindante yosidaapproximationsofstochasticdifferentialequationsininfinitedimensionsandapplications |