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  .
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 new class of dynamic risk measures given in terms of a certain family of g-expectations driven by Wiener and Poisson point processes , which we call dynamic spectral risk measures . 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 . We also define a family of market-consistent dynamic spectral risk measures in the setting of a financial market. We identify the corresponding optimal hedging strategy and show that the negative of the value of such a risk measure is equal to the smallest price under a certain family of equivalent martingale measures. By varying the parameters of the risk measure we show that this price ranges between the subhedging price and the price under the minimal martingale measure .
R
risk-measure
risk measure
A
new
A
given
R
. In analogy with the static case, we show that these risk measures, which we label
, which we call
D
, are locally law-invariant and additive on the set of pathwise increasing random variables
A
. We also define a family of market-consistent dynamic spectral risk measures in the setting of a financial market. We identify the corresponding optimal hedging strategy and show that the negative of the value of such a risk measure is equal to the smallest price under a certain family of equivalent martingale measures. By varying the parameters of the risk measure we show that this price ranges between the subhedging price and the price under the minimal martingale measure