We consider stochastic differential equations driven by Wiener processes. The vector fields are supposed to satisfy only local Lipschitz conditions. The Lipschitz constants of the drift vector field, valid on balls of radius R, are supposed to grow not faster than log R, while those of the diffusion vector fields are supposed to grow not faster than $\sqrt{\log R}$ . We regularize the stochastic differential equations by associating with them approximating ordinary differential equations obtained by discretization of the increments of the Wiener process on small intervals. By showing that the flow associated with a regularized equation converges uniformly to the solution of the stochastic differential equation, we simultaneously establish the existence of a global flow for the stochastic equation under local Lipschitz conditions.
Publié le : 2007-01-14
Classification:
Stochastic differential equation,
global flow,
local Lipschitz conditions,
moment inequalities,
martingale inequalities,
approximation by ordinary differential equation,
uniform convergence,
60H10,
34F05,
60G48,
37C10,
37H10
@article{1174324127,
author = {Fang, Shizan and Imkeller, Peter and Zhang, Tusheng},
title = {Global flows for stochastic differential equations without global Lipschitz conditions},
journal = {Ann. Probab.},
volume = {35},
number = {1},
year = {2007},
pages = { 180-205},
language = {en},
url = {http://dml.mathdoc.fr/item/1174324127}
}
Fang, Shizan; Imkeller, Peter; Zhang, Tusheng. Global flows for stochastic differential equations without global Lipschitz conditions. Ann. Probab., Tome 35 (2007) no. 1, pp. 180-205. http://gdmltest.u-ga.fr/item/1174324127/