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January 2007 Global flows for stochastic differential equations without global Lipschitz conditions
Shizan Fang, Peter Imkeller, Tusheng Zhang
Ann. Probab. 35(1): 180-205 (January 2007). DOI: 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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Shizan Fang. Peter Imkeller. Tusheng Zhang. "Global flows for stochastic differential equations without global Lipschitz conditions." Ann. Probab. 35 (1) 180 - 205, January 2007.


Published: January 2007
First available in Project Euclid: 19 March 2007

zbMATH: 1128.60046
MathSciNet: MR2303947
Digital Object Identifier: 10.1214/009117906000000412

Primary: 34F05 , 60H10
Secondary: 37C10 , 37H10 , 60G48

Keywords: approximation by ordinary differential equation , global flow , local Lipschitz conditions , martingale inequalities , Moment inequalities , Stochastic differential equation , Uniform convergence

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.35 • No. 1 • January 2007
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