Convex Risk Measures: Lebesgue Property on one Period and Multi Period Risk Measures and Application in Capital Allocation Problem

In this work we study the Lebesgue property for convex risk measures on the space of bounded c\`adl\`ag random processes ($\mathcal{R}^\infty$). Lebesgue property has been defined for one period convex risk measures in \cite{Jo} and earlier had been studied in \cite{De} for coherent risk measures. We introduce and study the Lebesgue property for convex risk measures in the multi period framework. We give presentation of all convex risk measures with Lebesgue property on bounded c\`adl\`ag processes. To do that we need to have a complete description of compact sets of $\mathcal{A}^1$. The main mathematical contribution of this paper is the characterization of the compact sets of $\mathcal{A}^p$ (including $\mathcal{A}^1$). At the final part of this paper, we will solve the Capital Allocation Problem when we work with coherent risk measures.