An extension of Itô's formula for elliptic diffusion processes
type: Item
instance of: scholarly article
title: An extension of Itô's formula for elliptic diffusion processes
zbMATH Open document ID: 0911.60035
DOI: 10.1016/S0304-4149(97)00026-4
author: Xavier Bardina
author: Maria Jolis
published in: Stochastic Processes and their Applications
publication date: 1999-01-14T00:00:00Z
review text: Recently, \textit{H. Föllmer, P. Protter} and \textit{A. N. Shiryayev} extended Itô's formula to processes of the form \(F(X_{t})\), where \(F\) is an absolutely continuous function on \(\mathbb R\) with locally square integrable derivative and \(X\) is a standard one-dimensional Wiener process [see Bernoulli 1, No. 1/2, 149-169 (1995; Zbl 0851.60048)]. In the present paper, their result is further generalized, \(X_{t}\) being assumed now to be a one-dimensional diffusion process solving a stochastic differential equation \[ dX_{t} = b(t,X_{t}) dt + \sigma (t,X_{t}) dW_{t} \] with Lipschitz continuous coefficients of a linear growth and such that \(X_{t}\) has a density \(p_{t}(x)\) satisfying some integrability conditions. Functions \(F\) depending on \(t\) are allowed as well. In the second part of the paper the authors find, using the Malliavin calculus, sufficient conditions (in terms of the non-degeneracy of \(\sigma \) and regularity of \(b\) and \(\sigma \)) for the assumptions on \(p_{t}\) to hold.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1240028
zbMATH Keywords: Itô's formula
zbMATH Keywords: diffusion processes
zbMATH Keywords: Malliavin calculus
reviewed by: Jan Seidler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1240028