On the stochastic decision problems with backward stochastic viability property

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.