  In this paper, we introduce a new kind of "variant" reflected backward doubly
stochastic differential equations (VRBDSDEs in short), where the drift is the
nonlinear function of the barrier process. In the one stochastic case, this
type of equations have been already studied by Ma and Wang. They called it as
"variant" reflected BSDEs (VRBSDEs in short) based on the general version of
the Skorohod problem recently studied by Bank and El Karoui. Among others, Ma
and Wang showed that VRBSDEs is a novel tool for some problems in finance and
optimal stopping problems where no existing methods can be easily applicable.
Since more of those models have their stochastic counterpart, it is very useful
to transpose the work of Ma and Wang to doubly stochastic version. In doing so,
we firstly establish the stochastic variant Skorohod problem based on the
stochastic representation theorem, which extends the work of Bank and El
Karoui. We prove the existence and uniqueness of the solution for VRBDSDEs by
means of the contraction mapping theorem. By the way, we show the comparison
theorem and stability result for the solutions of VRBDSDEs.
