In this paper we provide the complete solution to the existence and characterization problem of optimal capital and risk allocations for not necessarily monotone, law-invariant convex risk measures on the model space L p , for any p ∈ [1, ∞]. Our main result says that the capital and risk allocation problem always admits a solution via contracts whose payoffs are defined as increasing Lipschitz continuous functions of the aggregate risk.
In this paper, we establish a one‐to‐one correspondence between law‐invariant convex risk measures on L∞ and L1. This proves that the canonical model space for the predominant class of law‐invariant convex risk measures is L1.
We axiomatically introduce risk-consistent conditional systemic risk measures defined on multidimensional risks. This class consists of those conditional systemic risk measures which can be decomposed into a statewise conditional aggregation and a univariate conditional risk measure. Our studies extend known results for unconditional risk measures on finite state spaces. We argue in favor of a conditional framework on general probability spaces for assessing systemic risk. Mathematically, the problem reduces to selecting a realization of a random field with suitable properties. Moreover, our approach covers many prominent examples of systemic risk measures from the literature and used in practice.
Recently Heyde, Kou and Peng [2] proposed the notion of a natural risk statistic associated with a finite sample that relaxes the subadditivity assumption in the classical coherent risk statistics. In this note we use convex analysis to provide alternate proofs of two of the main results in [2] regarding representation of a natural risk statistic as a supremum over a family of convex combinations of order statistics.
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