A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations
type: Item
instance of: scholarly article
title: A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations
zbMATH Open document ID: 0674.60057
author: Daniel L. Ocone
author: Etienne Pardoux
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77339
review text: The first part of this paper is devoted to establish an extended version of the Itô-Ventzell formula for anticipating processes that can be written as the sum of an absolutely continuous term plus an indefinite Skorokhod integral. The proof of this result is based on an Itô formula for nonadapted Hilbert-valued processes which generalizes the formula obtained by the reviewer and the second author [Probab. Theory Relat. Fields 78, 535-581 (1988; Zbl 0629.60061)] in the finite-dimensional case. A similar Itô-Ventzell formula for Stratonovich stochastic integrals is also presented. In the second part, these results are applied to show the existence and uniqueness of a solution for a Stratonovich stochastic differential equation of the type \[ X_ t=X_ 0+\int^{t}_{0}b(s,X_ s)ds+\int^{t}_{0}\sum^{k}_{i=1}\sigma_ i(s,X_ s)\circ dW^ i_ s, \] where \(X_ 0\) and \(\{\) b(t,x); \(t\geq 0\), \(x\in R^ d\}\) are random and may depend on the whole \(\sigma\)-field generated by the Brownian motion. The solution of the above equation is of the form \(X_ t=\phi_ t(Y_ t)\), where \(\phi_ t(x)\) denotes the flow associated to the same equation with \(b=0\) and \(Y_ t\) solves a certain ordinary differential equation with random coefficients.
reviewed by: David Nualart
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 4104121
zbMATH Keywords: Itô-Ventzell formula for anticipating processes
zbMATH Keywords: Stratonovich stochastic integrals
zbMATH Keywords: ordinary differential equation with random coefficients
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4104121