On the risk measures representation and capital allocation in the Backward Stochastic Differential Equation framework

Abstract

In this thesis, we study the representation of dynamic risk measures based on backward stochastic differential equations (BSDEs) and ergodic-BSDEs, and capital allocation. We consider the equations driven by the Brownian motion and the compensated Poisson process. We obtain four results. Firstly, we consider the representation of dynamic risk measures defined under BSDE, with generators that have quadratic-exponential growth in the control variables. Under this setting, the dynamic capital allocation of the risk measure is obtained via the differentiability of BSDEs with jumps. In this case, we introduce the Malliavin directional derivative that generalises the classical Gˆateaux-derivative. Using the capital allocation results and the full allocation property of the Aumann-Shapley, we obtain the representation of the dynamic convex and coherent risk measures. The results are illustrated for the dynamic entropic risk and static coherent risk measures. Secondly, we consider the representation of dynamic convex risk measure based on the ergodic-BSDEs in the diffusion framework. The maturityindependent risk measure is defined as the first component to the solution of a BSDE whose generator depends on the second component of the solution to the ergodic-BSDE. Using the differentiability results of BSDEs, we determine the capital allocation. Furthermore, we give an example in the form of the forward entropic risk measure and the capital allocation. Thirdly, we investigate the representation of capital allocation for dynamic risk measures based on BSVIEs from Kromer and Overbeck 2017 and extend it to risk measures based on BSVIEs with jumps. The extension of dynamic risk measure based on BSVIEs with jumps is studied by Agram 2019. In our case, we study capital allocation for dynamic risk measures based on BSVIEs with jumps. In particular, we determine the capital allocation of the dynamic risk measures based on BSVIEs with jumps. Finally, we study the representation for a forward entropic risk measure using ergodic BSDEs under the jump-diffusion framework. In this case, we notice that when the ergodic BSDE includes jump term the forward entropic risk measure does not satisfy the translation property.