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