On the derivability, with respect to initial data, of the solution of a stochastic differential equation with lipschitz coeffcients
Bouleau, Nicolas ; Hirsch, Francis
HAL, hal-00451851 / Harvested from HAL
Text on HAL
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.
Publié le : 1988-07-05
Classification:  stochastic differential equation,  Lipschitz coefficients,  flow,  diffeomorphism,  invertibility,  Dirichlet form,  Kolmogorov's criterion,  MSC 60J35 60J45 60J60,  [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA],  [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] 
@article{hal-00451851,
     author = {Bouleau, Nicolas and Hirsch, Francis},
     title = {On the derivability, with respect to initial data, of the solution of a stochastic differential equation with lipschitz coeffcients},
     journal = {HAL},
     volume = {1988},
     number = {0},
     year = {1988},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-00451851}
}

Bouleau, Nicolas; Hirsch, Francis. On the derivability, with respect to initial data, of the solution of a stochastic differential equation with lipschitz coeffcients. HAL, Tome 1988 (1988) no. 0, . http://gdmltest.u-ga.fr/item/hal-00451851/