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Pareto optimal allocations and optimal risk sharing for quasiconvex risk measures. (English) Zbl 1320.91087

Consider a risk premium \(\pi: L \to \bar{\mathbb R}\) for some space of random variables \(L\). Then \(\rho(X) = \pi(-X)\) is a risk measure. The premium is called quasiconvex if \(\pi(\alpha X + (1-\alpha) Y) \leq \pi(X) \vee \pi(Y)\) for all \(\alpha \in [0,1]\) and \(X,Y \in L\) and convex if \(\pi(\alpha X + (1-\alpha) Y) \leq \alpha \pi(X) +(1-\alpha) \pi(Y)\). In [E. Jouini et al., Math. Finance 18, No. 2, 269–292 (2008; Zbl 1133.91360)] and [D. Filipović and M. Kupper, Int. J. Theor. Appl. Finance 11, No. 3, 325–343 (2008; Zbl 1151.91608)], properties of convex risk measures are proved. In this paper, these results are generalised to quasiconvex risk measures.
The quasiconvex inf-convolution is defined as \[ (\pi_1 \nabla \pi_2)(X) = \inf_{Y \in L} (\pi_1(X-Y) \vee \pi_2(Y))\;. \] If two risk measures are quasiconvex and monotone, then \(\pi_1 \nabla \pi_2\) will be quasiconvex and monotone.
Let \(X\) be a joint risk and two (or more) agents have to share the risk. That is, the risk is split into two risks \(\xi_1\) and \(\xi_2\) for agent one and agent two, respectively, such that \(\xi_1 + \xi_2 = X\). A risk allocation is called Pareto optimal if whenever \(\pi_1(\eta_1) \leq \pi_1(\xi_1)\) and \(\pi_2(\eta_2) \leq \pi_2(\xi_2)\) for some allocation \((\eta_1, \eta_2)\), then \(\pi_1(\eta_1) = \pi_1(\xi_1)\) and \(\pi_2(\eta_2) = \pi_2(\xi_2)\). It is called weakly Pareto optimal, if there does not exist an allocation \((\eta_1, \eta_2)\) such that \(\pi_1(\eta_1) < \pi_1(\xi_1)\) and \(\pi_2(\eta_2) < \pi_2(\xi_2)\).
Sufficient conditions for weak Pareto optimality are given. Further, several counterexamples show that these conditions are not necessary.

MSC:

91B30 Risk theory, insurance (MSC2010)
26A51 Convexity of real functions in one variable, generalizations
91B32 Resource and cost allocation (including fair division, apportionment, etc.)
91B16 Utility theory
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References:

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