Global flows for stochastic differential equations without global Lipschitz conditions

Details Global flows for stochastic differential equations without global Lipschitz conditions Global flows for stochastic differential equations without global Lipschitz conditions

2007-03-27

arxiv.org/abs/math/0703791v1

Annals of Probability 2007, Vol. 35, No. 1, 180-205

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000412 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000412

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.

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