Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes
Patrick Cheridito, Princeton University, USA
Freddy Delbaen, ETH Zürich, Switzerland
Michael Kupper, ETH Zürich, Switzerland
Abstract
We study dynamic monetary risk measures that
depend on bounded discrete-time processes
describing the evolution of financial values. The time horizon
can be finite or infinite. We call a dynamic risk measure
time-consistent if it assigns to a process of financial
values the same risk irrespective of whether it is calculated
directly or in two steps backwards in time. We show
that this condition translates into a decomposition property for the
corresponding acceptance sets, and we demonstrate
how time-consistent dynamic monetary risk measures can be
constructed by pasting together one-period risk measures.
For conditional coherent and convex monetary risk measures,
we provide dual representations of Legendre--Fenchel type
based on linear functionals induced by adapted
increasing processes of integrable variation.
Then we give dual characterizations of time-consistency for dynamic
coherent and convex monetary risk measures.
To this end, we introduce a concatenation operation for
adapted increasing processes of integrable variation, which generalizes the pasting of
probability measures.
In the coherent case, time-consistency corresponds to
stability under concatenation in the dual.
For dynamic convex monetary risk measures, the dual
characterization of time-consistency generalizes to a
condition on the family of convex conjugates
of the conditional risk measures at different times.
The theoretical results are applied by discussing the time-consistency of various
specific examples of dynamic monetary
risk measures that depend on bounded discrete-time processes.
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