Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

摘要

Recently, Hairer et. al (2012) showed that there exist SDEs with infinitelyoften differentiable and globally bounded coefficient functions whose solutionsfail to be locally Lipschitz continuous in the strong L^p-sense with respect tothe initial value for every p \in [1,\infty). In this article we providesufficient conditions on the coefficient functions of the SDE and on p \in(0,\infty] which ensure local Lipschitz continuity in the strong L^p-sense withrespect to the initial value and we establish explicit estimates for the localLipschitz continuity constants. In particular, we prove local Lipschitzcontinuity in the initial value for several nonlinear SDEs from the literaturesuch as the stochastic van der Pol oscillator, Brownian dynamics, theCox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As anapplication of our estimates, we obtain strong completeness for severalnonlinear SDEs.