In the context of multistage stochastic optimization, it is natural to consider nested risk measures, which originate by repeatedly composing risk measures, conditioned on realized observations. Starting from this discrete time setting, we extend the notion of nested risk measures to continuous time by adapting the risk levels in a time dependent manner. This time dependent modification is necessary for a risk measure to be non-degenerate in continuous time. Moreover, the Entropic Value-at-Risk turns out to be the natural and universal choice for a coherent nested risk measure in the context of optimal control. Consequently, we construct the risk-averse analogue of the infinitesimal generator based on risk measures and obtain risk-averse Hamilton–Jacobi–Bellman equations.
Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany