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

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Seria
Rozprawy Matematyczne tom/nr w serii: 325 wydano: 1993
Zawartość
Warianty tytułu
Abstrakty
EN
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.
EN

CONTENTS
1. 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..........................................................9
2. 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.........................................................................................................................................24
3. 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..........................................................................................................45
4. Comparison of the results...................................................................................................................46
4.1. Finite-dimensional case..................................................................................................................46
4.2. Stochastic delay equations.............................................................................................................47
5. On the relation between the Itô and Stratonovich integrals in Hilbert spaces.....................................47
6. Conclusions........................................................................................................................................49
References.............................................................................................................................................50
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 325
Liczba stron
54
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, CCCXXV
Daty
wydano
1993
otrzymano
1992-03-27
poprawiono
1992-07-13
Twórcy
autor
• Institute of Mathematics, Warsaw Technical University, Pl. Politechniki 1, 00-661 Warszawa, Poland
Bibliografia
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Języki publikacji
 EN
Uwagi
1991 Mathematics Subject Classification: 34G20, 34K50, 35R15, 60H05, 60H10, 60H15, 60H30.