  Scalar dynamic risk measures for univariate positions in continuous time are
commonly represented as backward stochastic differential equations. In the
multivariate setting, dynamic risk measures have been defined and studied as
families of set-valued functionals in the recent literature. There are two
possible extensions of scalar backward stochastic differential equations for
the set-valued framework: (1) backward stochastic differential inclusions,
which evaluate the risk dynamics on the selectors of acceptable capital
allocations; or (2) set-valued backward stochastic differential equations,
which evaluate the risk dynamics on the full set of acceptable capital
allocations as a singular object. In this work, the discrete time setting is
investigated with difference inclusions and difference equations in order to
provide insights for such differential representations for set-valued dynamic
risk measures in continuous time.
