Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions

Krystyna Twardowska. Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1993. <http://eudml.org/doc/250986>.

@book{KrystynaTwardowska1993,
abstract = {Some generalizations of the approximation theorem of Wong-Zakai type for stochastic differential equations are examined. One of them deals with functional stochastic differential equations defined on some spaces of continuous functions. The second one concerns the situations when the state space and the Wiener process have values in some Hilbert spaces. The comparison of these results as well as some examples are also included. The correction terms computed here are then applied to the derivation of the relation between the Itô and Stratonovich integrals. Other important applications of the above theorems are indicated.CONTENTS1. Introduction...........................................................................................................................................5 1.1. The Wong-Zakai theorem and its generalizations.............................................................................5 1.2. Approximation methods for stochastic differential equations.............................................................7 1.3. Extensions of the Wong-Zakai theorem and their applications..........................................................92. Approximation theorem of Wong-Zakai type for functional stochastic differential equations................10 2.1. Introductory remarks........................................................................................................................10 2.2. Definitions and notation...................................................................................................................10 2.3. Description of the model..................................................................................................................11 2.4. Approximation theorem....................................................................................................................15 2.5. Examples.........................................................................................................................................243. An extension of the Wong-Zakai theorem to stochastic evolution equations in Hilbert spaces............26 3.1. Introductory remarks.......................................................................................................................26 3.2. Definitions and notation..................................................................................................................26 3.3. Description of the model.................................................................................................................27 3.4. The main theorem...........................................................................................................................31 3.5. Examples.........................................................................................................................................41  3.5.1. Equations satisfying the assumptions of Theorem 3.4.1.............................................................41  3.5.2. Stochastic delay equations.........................................................................................................43  3.5.3. Stochastic wave equations..........................................................................................................454. Comparison of the results...................................................................................................................46 4.1. Finite-dimensional case..................................................................................................................46 4.2. Stochastic delay equations.............................................................................................................475. On the relation between the Itô and Stratonovich integrals in Hilbert spaces.....................................476. Conclusions........................................................................................................................................49References.............................................................................................................................................501991 Mathematics Subject Classification: 34G20, 34K50, 35R15, 60H05, 60H10, 60H15, 60H30.},
author = {Krystyna Twardowska},
keywords = {theorem of Wong-Zakai type; stochastic delay equations; Stratonovich integrals in Hilbert spaces; stochastic evolution equation in Hilbert spaces; strongly continuous semigroup},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions},
url = {http://eudml.org/doc/250986},
year = {1993},
}

TY - BOOK
AU - Krystyna Twardowska
TI - Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions
PY - 1993
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - Some generalizations of the approximation theorem of Wong-Zakai type for stochastic differential equations are examined. One of them deals with functional stochastic differential equations defined on some spaces of continuous functions. The second one concerns the situations when the state space and the Wiener process have values in some Hilbert spaces. The comparison of these results as well as some examples are also included. The correction terms computed here are then applied to the derivation of the relation between the Itô and Stratonovich integrals. Other important applications of the above theorems are indicated.CONTENTS1. Introduction...........................................................................................................................................5 1.1. The Wong-Zakai theorem and its generalizations.............................................................................5 1.2. Approximation methods for stochastic differential equations.............................................................7 1.3. Extensions of the Wong-Zakai theorem and their applications..........................................................92. Approximation theorem of Wong-Zakai type for functional stochastic differential equations................10 2.1. Introductory remarks........................................................................................................................10 2.2. Definitions and notation...................................................................................................................10 2.3. Description of the model..................................................................................................................11 2.4. Approximation theorem....................................................................................................................15 2.5. Examples.........................................................................................................................................243. An extension of the Wong-Zakai theorem to stochastic evolution equations in Hilbert spaces............26 3.1. Introductory remarks.......................................................................................................................26 3.2. Definitions and notation..................................................................................................................26 3.3. Description of the model.................................................................................................................27 3.4. The main theorem...........................................................................................................................31 3.5. Examples.........................................................................................................................................41  3.5.1. Equations satisfying the assumptions of Theorem 3.4.1.............................................................41  3.5.2. Stochastic delay equations.........................................................................................................43  3.5.3. Stochastic wave equations..........................................................................................................454. Comparison of the results...................................................................................................................46 4.1. Finite-dimensional case..................................................................................................................46 4.2. Stochastic delay equations.............................................................................................................475. On the relation between the Itô and Stratonovich integrals in Hilbert spaces.....................................476. Conclusions........................................................................................................................................49References.............................................................................................................................................501991 Mathematics Subject Classification: 34G20, 34K50, 35R15, 60H05, 60H10, 60H15, 60H30.
LA - eng
KW - theorem of Wong-Zakai type; stochastic delay equations; Stratonovich integrals in Hilbert spaces; stochastic evolution equation in Hilbert spaces; strongly continuous semigroup
UR - http://eudml.org/doc/250986
ER -