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 preference functional, under heterogeneous beliefs, where beliefs are represented by several probability measures. We obtain semi-explicit formulas for the inf-convolution of multiple Lambda Value-at-Risk measures under heterogeneous beliefs and the explicit forms of the corresponding optimal allocations. To show the impact of belief heterogeneity, 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, in a two-agent setting, 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 differing beliefs. The expression of the inf-convolution and the form of the optimal allocation are obtained. In all above cases we demonstrate that trivial outcomes arise when both belief inconsistency and risk tolerance are high. Finally, we discuss risk sharing for an alternative definition of Lambda Value-at-Risk.