  In this paper we prove the existence of strong solutions to a SDE with a
generalized drift driven by a multidimensional fractional Brownian motion for
small Hurst parameters H<1/2. Here the generalized drift is given as the local
time of the unknown solution process, which can be considered an extension of
the concept of a skew Brownian motion to the case of fractional Brownian
motion. Our approach for the construction of strong solutions is new and relies
on techniques from Malliavin calculus combined with a "local time variational
calculus" argument.
