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.Comment: 20 pages (Additional Note: This work is, in some sense, a continuation of our previous papers arXiv:1610.07201, arXiv:1603.03359 and arXiv:1611.03405

On the hierarchical risk-averse control problems for diffusion processes

In this paper, we consider a risk-averse control problem for diffusion processes, in which there is a partition of the admissible control strategy into two decision-making groups (namely, the {\it leader} and {\it follower}) with different cost functionals and risk-averse satisfactions. Our approach, based on a hierarchical optimization framework, requires that a certain level of risk-averse satisfaction be achieved for the {\it leader} as a priority over that of the {\it follower's} risk-averseness. In particular, we formulate such a risk-averse control problem involving a family of time-consistent dynamic convex risk measures induced by conditional $g$-expectations (i.e., filtration-consistent nonlinear expectations associated with the generators of certain backward stochastic differential equations). Moreover, under suitable conditions, we establish the existence of optimal risk-averse solutions, in the sense of viscosity solutions, for the corresponding risk-averse dynamic programming equations. Finally, we briefly comment on the implication of our results.Comment: 25 Pages - Version 4.