The Annals of Probability

Global flows for stochastic differential equations without global Lipschitz conditions

Shizan Fang, Peter Imkeller, and Tusheng Zhang

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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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Ann. Probab., Volume 35, Number 1 (2007), 180-205.

First available in Project Euclid: 19 March 2007

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]
Secondary: 60G48: Generalizations of martingales 37C10: Vector fields, flows, ordinary differential equations 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15]

Stochastic differential equation global flow local Lipschitz conditions moment inequalities martingale inequalities approximation by ordinary differential equation uniform convergence


Fang, Shizan; Imkeller, Peter; Zhang, Tusheng. Global flows for stochastic differential equations without global Lipschitz conditions. Ann. Probab. 35 (2007), no. 1, 180--205. doi:10.1214/009117906000000412.

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