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)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 Département de Mathématiques (Evry)

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.

Mots clés: stochastic differential equation; Lipschitz coefficients; flow; diffeomorphism; invertibility; Dirichlet form; Kolmogorov's criterion

Domaines: Analyse fonctionnelle [math.FA]; Probabilités [math.PR]

Nicolas Bouleau, Francis Hirsch. On the derivability, with respect to initial data, of the solution of a stochastic differential equation with lipschitz coeffcients. Séminaire de Théorie du Potentiel n°9, 1988, France. pp.39-57. ⟨hal-00451851⟩