Existence of invariant manifolds for stochastic equations in infinite dimension
type: Item
instance of: scholarly article
title: Existence of invariant manifolds for stochastic equations in infinite dimension
zbMATH Open document ID: 1013.60035
DOI: 10.1016/S0022-1236(03)00008-9
author: Damir Filipović
author: Josef Teichmann
published in: Journal of Functional Analysis
publication date: 2003-04-09T00:00:00Z
review text: The authors investigate the existence of finite-dimensional invariant manifolds for a stochastic equation of the type \[ dr_t= \bigl(Ar_t+ \alpha(r_t) \bigr)dt+ \sum^d_{j=1} j(r_t)dW^j_t, \quad r_0=h_0, \] on a separable Hilbert space \(H\), in the spirit of \textit{G. Da Prato} and \textit{J. Zabczyk} [``Stochastic equations in infinite dimensions'' (1992; Zbl 0761.60052)]. The operator \(A:D(A) \subset H\to H\) generates a strongly continuous semigroup on \(H\); here \(d\in\mathbb{N}\), and \(W=(W^1,\dots,W^d)\) denotes a standard \(d\)-dimensional Brownian motion, the mappings \(\alpha:H\to H\) and \(\sigma= (\sigma_1, \dots,\sigma_d):\) \(H\to H^d\) satisfy a smoothness condition. The main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application, the authors characterize all finite-dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 37L55
zbMATH DE Number: 1895828
zbMATH Keywords: affine term structure
zbMATH Keywords: analysis on Frechet spaces
zbMATH Keywords: finite-dimensional invariant
zbMATH Keywords: submanifolds
zbMATH Keywords: Frobenius theorem
zbMATH Keywords: interest rate models
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1895828