| Abstract: |
| Backward stochastic
Volterra integral equations are used in Mathematical Finance and Risk Theory as a tool to define dynamic risk measures. We will adress the corresponding topic of Capital allocation, which requires some differentiablility of BSVIE. Capital allocations have been studied in conjunction with static risk measures in various
papers. The dynamic case has been studied only in a discrete-time setting. We address the
problem of allocating risk capital to subportfolios in a continuous-time dynamic context.
For this purpose we introduce a classical differentiability result for backward stochastic
Volterra integral equations and apply this result to derive continuous-time dynamic capital
allocations. Moreover, we study a dynamic capital allocation principle that is based on
backward stochastic differential equations and derive the dynamic gradient allocation for
the dynamic entropic risk measure. As a consequence we finally provide a representation result for
dynamic risk measures that is based on the full allocation property of the Aumann-Shapley
allocation, which is also new in the static case. |
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