Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes

We study dynamic monetary risk measures thatdepend on bounded discrete-time processesdescribing the evolution of financial values. The time horizoncan be finite or infinite. We call a dynamic risk measuretime-consistent if it assigns to a process of financialvalues the same risk irrespective of whether it is calculateddirectly or in two steps backwards in time. We showthat this condition translates into a decomposition property for thecorresponding acceptance sets, and we demonstratehow time-consistent dynamic monetary risk measures can beconstructed by pasting together one-period risk measures.For conditional coherent and convex monetary risk measures,we provide dual representations of Legendre--Fenchel typebased on linear functionals induced by adaptedincreasing processes of integrable variation.Then we give dual characterizations of time-consistency for dynamiccoherent and convex monetary risk measures.To this end, we introduce a concatenation operation foradapted increasing processes of integrable variation, which generalizes the pasting ofprobability measures.In the coherent case, time-consistency corresponds tostability under concatenation in the dual.For dynamic convex monetary risk measures, the dualcharacterization of time-consistency generalizes to acondition on the family of convex conjugatesof the conditional risk measures at different times.The theoretical results are applied by discussing the time-consistency of variousspecific examples of dynamic monetaryrisk measures that depend on bounded discrete-time processes.