  In this paper, we consider a stochastic decision problem for a system
governed by a stochastic differential equation, in which an optimal decision is
made in such a way to minimize a vector-valued accumulated cost over a
finite-time horizon that is associated with the solution of a certain
multi-dimensional backward stochastic differential equation (BSDE). Here, we
also assume that the solution for such a multi-dimensional BSDE {\it almost
surely} satisfies a backward stochastic viability property w.r.t. a given
closed convex set. Moreover, under suitable conditions, we establish the
existence of an optimal solution, in the sense of viscosity solutions, to the
associated system of semilinear parabolic PDEs. Finally, we briefly comment on
the implication of our results.
