On Dynamic Spectral Risk Measures and a Limit Theorem

In this paper we explore a novel way to combine the dynamic notion of time-consistency with the static notion of quantile-based coherent risk-measure or spectral risk measure, of which Expected Shortfall is a prime example. We introduce a class of dynamic risk measures in terms of a certain family of g-expectations driven by Wiener and Poisson point processes. In analogy with the static case, we show that these risk measures, which we label dynamic spectral risk measures, are locally law-invariant and additive on the set of pathwise increasing random variables. We substantiate the link between dynamic spectral risk measures and their static counterparts by establishing a limit theorem for general path-functionals which shows that such dynamic risk measures arise as limits under vanishing time-step of iterated spectral risk measures driven by approximating lattice random walks. This involves a certain non-standard scaling of the corresponding spectral weight-measures that we identify explicitly.