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2007.194: Global flows for stochastic differential equations without global Lipschitz conditions

2007.194: Shizan Fang, Peter Imkeller and Tusheng Zhang (2007) Global flows for stochastic differential equations without global Lipschitz conditions. The Annals of Probability, 35 (1). pp. 180-205. ISSN 0091-1798

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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.

Item Type:Article
Uncontrolled Keywords:Stochastic differential equation; global flow; local Lipschitz conditions; moment inequalities; martingale inequalities; approximation by ordinary differential equation; uniform convergence
Subjects:MSC 2000 > 60 Probability theory and stochastic processes
MIMS number:2007.194
Deposited By:Mrs Louise Healey
Deposited On:21 November 2007

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