Global flows for stochastic differential equations without global Lipschitz conditions

By S. Fang, Peter Imkeller, Tusheng Zhang. 2007

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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.

Subfields: Demography, Finance

View Paper: https://www.semanticscholar.org/paper/c2975444d06a8f23e4c52255f6a0115801c67fa3

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Citations in the corpus

Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations (2013, background): “Strong completeness for the SDE (165) follows from Theorem 2.4 in Zhang [54] under the stronger assumption that condition (164) holds with η2 = 0 and that condition (171) holds with ρ1 = 0.”
Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients (2010, background): “1) under various non-Lipschitz assumptions on coe fficients (see [3, 16, 5] and references therein).”
Flow of diffeomorphisms for SDEs with unbounded H (2009, background): “For (at least) locally Lipschitz coefficients, there are indeed positive results of strong completeness (see [7,21,23]).”
Well-posedness of the transport equation by stochastic perturbation (2008, background): “For papers that tackle existence of global ows of homeomorphisms for SDEs without global Lipschitz coecients see [37, 11 ] and the references therein.”
Stochastic flows of SDEs with irregular coefficients and stochastic transport equations (2009, background): “Recently, there are increasing interests for studying the stochastic homeomorphism flow property associated with SDE (1.1) under variousnon-Lipschitz assumptions onb andσ (cf. [21, 1, 20, 25, 8, 9, 10, 11, 28], etc.).”
Loss of regularity for Kolmogorov equations (2012, background): “, also Elworthy [15], Kunita [48] and Fang, Imkeller & Zhan [19] for further results on strong completeness of SDEs.”