Dynamic risk measures and related risk-averse decision problems

In this paper, we consider a risk-averse decision problem for systems governed by stochastic differential equations with dynamic risk measures, where the classical average performance criteria may not be sufficient to account for how risks to uncertain outcomes are perceived by a decision maker. In particular, we formulate a risk-averse stochastic optimization problem, involving a family of time-consistent dynamic convex risk measures induced by conditional g-expectations (i.e., filtration-consistent nonlinear expectations) that is associated with the generator function of a certain backward stochastic differential equation. Moreover, under suitable conditions, we establish the existence of optimal risk-averse solutions, in the sense of viscosity solutions, to the associated risk-averse dynamic programming equations. Finally, we briefly remark on the problem of risk-averseness under model uncertainty, when decision maker is allowed to take into account alternative models that are statistically difficult to distinguish.