Hilbert-valued anticipating stochastic differential equations
type: Item
instance of: scholarly article
title: Hilbert-valued anticipating stochastic differential equations
zbMATH Open document ID: 0796.60061
author: Marta Sanz-Solé
author: A. Grorud
author: David Nualart
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1994-09-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77472
review text: The authors study the Hilbert space \(H\)-valued solution \(X_ t(x)\) of the stochastic differential equation with Stratonovich integral \[ X_ t(x)=x+\int^ t_ 0\sigma(X_ s(x))\circ dW_ s+\int^ t_ 0 b(X_ s(x))ds,\quad 0\leq t\leq 1,\quad x\in H,\tag{1} \] where \(W\) is a Hilbert space \(K\)-valued Brownian motion with a nuclear covariance operator. Using a generalization of the Kolmogorov continuity criterion for processes indexed by a metric space, as presented in \textit{X. Fernique's} paper [École d'été de probabilités de Saint-Flour XI -- 1981, Lect. Notes Math. 976, 1-74 (1983; Zbl 0507.60027)], they derive the continuity of \(x\mapsto X_ t(x)\) in the compact sets \(B\) of elements \(x\) whose Fourier coefficients in some fixed basis of \(H\) converge to zero fast enough. This result is then applied to prove the existence of the solution of an anticipating version of (1), where the initial value \(x\in H\) is replaced by a \(B\)-valued random variable \(X_ 0\) which may depend on the whole path of \(W\). This is done by using an infinite- dimensional version of Nualart's and Pardoux's formula for Stratonovich integrals depending on a parameter.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 522701
zbMATH Keywords: Fernique's entropy criterion
zbMATH Keywords: stochastic differential equation with Stratonovich integral
zbMATH Keywords: Kolmogorov continuity criterion
zbMATH Keywords: Pardoux's formula for Stratonovich integrals
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 522701