Before revision:  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 .
After revision:  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, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse dynamic portfolio optimisation problems under DSR, providing verification theorems for the optimal (equilibrium) strategies and the corresponding value functions in terms of associated (extended) HJB equations. In the case of a long-only investor we explicitly identify optimal portfolio allocation strategies .