Invariant manifolds for weak solutions to stochastic equations
type: Item
instance of: scholarly article
title: Invariant manifolds for weak solutions to stochastic equations
zbMATH Open document ID: 0970.60069
author: Damir Filipović
publication date: 2001-10-06T00:00:00Z
review text: Given stochastic equations \[ d X_t = (A X_t + F(t,X_t)) dt + B(t,X_t) dW_t \] on a separable Hilbert space \(H\) with nonrandom initial conditions \(X_0 = x_0\), where \(W\) denotes a standard Wiener process on some separable Hilbert space \(G\), and \(A\) represents the infinitesimal generator of a strongly continuous semigroup in \(H\). Assume that the random mappings \(F=F(t,\omega,x)\) and \(B=B(t,\omega,x)\) satisfy some appropriate measurability conditions, local Lipschitz and integrability conditions together with some Nagumo type consistency conditions. The author proves a regularity result that any weak solution of these equations, which is viable in a finite-dimensional \(C^2\) submanifold, is a strong solution. The main idea behind the proof steps are the fact that locally any finite-dimensional submanifold \(M\) of \(H\) can be projected diffeomorphically onto a finite-dimensional linear subspace of the domain \(D(A^*)\), where \(A^*\) is the adjoint of \(A\), and an application of the well-known Itô formula. An application of the main theorems to the HJM model for the forward rate curve in mathematical finance is sketched at the end. The main results are related to find finite-dimensional realizations for stochastic equations.
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 93B29
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 1562913
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: viability and invariance problem
zbMATH Keywords: strong and weak solution
zbMATH Keywords: stochastic equations in a Hilbert space
zbMATH Keywords: Nagumo type consistency conditions
MaRDI profile type: Publication
DOI: 10.1007/S004400000085
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 1562913