Risk sharing with Lambda value at risk

In this paper, we study the risk sharing problem among multiple agents using Lambda value at risk (ΛVaR) as their preferences via the tool of inf-convolution, where ΛVaR is an extension of Value-at-Risk (VaR). We obtain explicit formulas of the inf-convolution of multiple ΛVaR with monotone Λ, and explicit forms of the corresponding optimal allocations, extending the results of the inf-convolution of VaR. It turns out that the inf-convolution of several ΛVaR is still a ΛVaR under some mild condition. Moreover, we investigate the inf-convolution of one ΛVaR and a general monotone risk measure without cash-additivity, including ΛVaR, expected utility and rank-dependent expected utility as special cases. The expression of the inf-convolution and the explicit forms of the optimal allocation are derived, leading to some partial solution of the risk sharing problem with multiple ΛVaR for general Λ functions. Finally, we discuss the risk sharing problem with ΛVaR+, another definition of Lambda value at risk. We focus on the inf-convolution of ΛVaR+ and a risk measure that is consistent with the second-order stochastic dominance, deriving very different expression of the inf-convolution and the forms of the optimal allocations.