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In this paper, we consider a stochastic decision problem for a systemgoverned by a stochastic differential equation, in which an optimal decision ismade in such a way to minimize a vector-valued accumulated cost over afinite-time horizon that is associated with the solution of a certainmulti-dimensional backward stochastic differential equation (BSDE). Here, wealso assume that the solution for such a multi-dimensional BSDE {\it almostsurely} satisfies a backward stochastic viability property w.r.t. a givenclosed convex set. Moreover, under suitable conditions, we establish theexistence of an optimal solution, in the sense of viscosity solutions, to theassociated system of semilinear parabolic PDEs. Finally, we briefly comment onthe implication of our results.

On the stochastic decision problems with backward stochastic viability property