Stochastic Decision Problems with Multiple Risk-Averse Agents

We consider a stochastic decision problem, with dynamic risk measures, in which multiple risk-averse agents make their decisions to minimize their individual accumulated risk-costs over a finite-time horizon. Specifically, we introduce multi-structure dynamic risk measures induced from conditional g-expectations, where the latter are associated with the generator functionals of certain BSDEs that implicitly take into account the risk-cost functionals of the risk-averse agents. Here, we also assume that the solutions for such BSDEs almost surely satisfy a stochastic viability property w.r.t. a certain given closed convex set. Using a result similar to that of the Arrow–Barankin–Blackwell theorem, we establish the existence of consistent optimal decisions for the risk-averse agents, when the set of all Pareto optimal solutions, in the sense of viscosity solutions, for the associated dynamic programming equations is dense in the given closed convex set. Finally, we comment on the characteristics of acceptable risks w.r.t. some uncertain future outcomes or costs, where results from the dynamic risk analysis are part of the information used in the risk-averse decision criteria.