# Global flows for stochastic differential equations without global Lipschitz conditions

Fang, Shizan and Imkeller, Peter and Zhang, Tusheng (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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## Abstract

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 Stochastic differential equation; global flow; local Lipschitz conditions; moment inequalities; martingale inequalities; approximation by ordinary differential equation; uniform convergence MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes Ms Lucy van Russelt 21 Nov 2007 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/938

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