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Séminaire de Théorie du Potentiel n°9, France (1988)

On the derivability, with respect to initial data, of the solution of a stochastic differential equation with lipschitz coeffcients

Nicolas Bouleau 1, Francis Hirsch 2

(1988)

We consider a stochastic differential equation, driven by a Brownian motion, with Lipschitz coefficients. We prove that the corresponding flow is, almost surely, almost everywhere derivable with respect to the initial data for any time, and the process defined by the Jacobian matrices is a GLn(R)-valued continuous solution of a linear stochastic differential equation. In dimension one, this process is given by an explicit formula. These results partially extend those which are known when the coefficients are C-1-alpha-Holder continuous. Dirichlet forms are involved in the proofs.

1:	Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
	INRIA – Ecole des Ponts ParisTech
2:	Département de Mathématiques (DP)
	Université d'Evry-Val d'Essonne

Subject

:

Mathematics/Functional Analysis Mathematics/Probability

Keyword(s): stochastic differential equation – Lipschitz coefficients – flow – diffeomorphism – invertibility – Dirichlet form – Kolmogorov's criterion

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From: Nicolas Bouleau <>

Submitted on: Monday, 1 February 2010 08:24:11

Updated on: Monday, 1 February 2010 09:11:20