On the dynamic consistency of hierarchical risk-averse decision problems

In this paper, we consider a risk-averse decision problem for controlled-diffusion processes, with dynamic risk measures, in which there are two risk-averse decision makers (i.e., {\it leader} and {\it follower}) with different risk-averse related responsibilities and information. Moreover, we assume that there are two objectives that these decision makers are expected to achieve. That is, the first objective being of {\it stochastic controllability} type that describes an acceptable risk-exposure set vis-\'a-vis some uncertain future payoff, and while the {\it second one} is making sure the solution of a certain risk-related system equation has to stay always above a given continuous stochastic process, namely {\it obstacle}. In particular, we introduce multi-structure, time-consistent, dynamic risk measures induced from conditional $g$-expectations, where the latter are associated with the generator functionals of two backward-SDEs that implicitly take into account the above two objectives along with the given continuous obstacle process. Moreover, under certain conditions, we establish the existence of optimal hierarchical risk-averse solutions, in the sense of viscosity solutions, to the associated risk-averse dynamic programming equations that formalize the way in which both the {\it leader} and {\it follower} consistently choose their respective risk-averse decisions. Finally, we remark on the implication of our result in assessing the influence of the {\it leader'}s decisions on the risk-averseness of the {\it follower} in relation to the direction of {\it leader-follower} information flow.