  Recently, Hairer et. al (2012) showed that there exist SDEs with infinitely
often differentiable and globally bounded coefficient functions whose solutions
fail to be locally Lipschitz continuous in the strong L^p-sense with respect to
the initial value for every p \in [1,\infty). In this article we provide
sufficient 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 with
respect to the initial value and we establish explicit estimates for the local
Lipschitz continuity constants. In particular, we prove local Lipschitz
continuity in the initial value for several nonlinear SDEs from the literature
such as the stochastic van der Pol oscillator, Brownian dynamics, the
Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an
application of our estimates, we obtain strong completeness for several
nonlinear SDEs.

Probability