Risk sharing with Lambda value at risk under heterogeneous beliefs

In this paper, we study the risk sharing problem among multiple agents using Lambda value at risk as their preferences under heterogenous beliefs, where the beliefs are represented by several probability measures. We obtain semi-explicit formulas for the inf-convolution of multiple Lambda value at risk under heterogenous beliefs and the explicit forms of the corresponding optimal allocations. To show the interplay among the beliefs, we consider three cases: homogeneous beliefs, conditional beliefs and absolutely continuous beliefs. For those cases, we find more explicit expressions for the inf-convolution, showing the influence of the relation of the beliefs on the inf-convolution. Moreover, we consider the inf-convolution of one Lambda value at risk and a general risk measure, including expected utility, distortion risk measures and Lambda value at risk as special cases, with different beliefs. The expression of the inf-convolution and the form of the optimal allocation are obtained. Finally, we discuss the risk sharing for another definition of Lambda value at risk.