  This paper is concerned with the strong solution to the Cauchy-Dirichlet
problem for backward stochastic partial differential equations of parabolic
type. Existence and uniqueness theorems are obtained, due to an application of
the continuation method under fairly weak conditions on variable coefficients
and $C^2$ domains. The problem is also considered in weighted Sobolev spaces
which allow the derivatives of the solutions to blow up near the boundary. As
applications, a comparison theorem is obtained and the semi-linear equation is
discussed in the $C^2$ domain.
