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

Krystyna Twardowska

GDML_Books, (1993), p.

Résumé

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.............................................................................................................................................50

1991 Mathematics Subject Classification: 34G20, 34K50, 35R15, 60H05, 60H10, 60H15, 60H30.

Détail

Zbl 0777.60051

EUDML-ID : urn:eudml:doc:250986