Pareto optimal allocations and optimal risk sharing for quasiconvex risk measures

Abstract

The main goal of this paper is to generalize the characterization of Pareto optimal allocations known for convex risk measures (see, among others, Jouini et al., in Math Financ 18(2):269–292, 2008 and Filipovic and Kupper, in Int J Theor Appl Financ, 11:325–343, 2008) to the wider class of quasiconvex risk measures. Following the approach of Jouini et al., in Math Financ 18(2):269–292, 2008 for convex risk measures, in the quasiconvex case we provide sufficient conditions for allocations to be (weakly) Pareto optimal in terms of exactness of the so-called quasiconvex inf-convolution as well as an existence result for weakly Pareto optimal allocations. Moreover, we give a necessary condition for weakly optimal risk sharing that is also sufficient under cash-additivity of at least one between the risk measures.

Appendix

In the following, we recall some basic definitions and results on different notions of subdifferentiability for quasiconvex functions. We refer to Penot [21] and to Penot and Zalinescu [23] for a deep and wide treatment.

Definition 17

(see Penot [21], Penot and Zalinescu [23]) Let $\mathcal {X}$ be a locally convex topological vector space and $\mathcal {X}^*$ be its topological dual space. Let $f: \mathcal {X}\, \rightarrow \, \overline{\mathbb {R}}$ be a quasiconvex function and let $x_0 \in \mathcal {X}$ be such that $f(x_0)$ is finite.

The Greenberg-Pierskalla subdifferential $\partial ^{GP} f(x_0)$, the star subdifferential $\partial ^{(*)} f(x_0)$ and the lower subdifferential of Plastria $\partial ^< f(x_0)$ of $f$ at $x_0 \in \mathcal {X}$ are defined, respectively, as

$$\begin{aligned}&\partial ^{GP} f(x_0) \triangleq \left\{ x^* \in \mathcal {X}^*: \langle x^* , x-x_0\rangle <0 \ \text{ for } \text{ any } \ x \in \{f<f(x_0)\} \right\} \\&\partial ^{(*)} f(x_0) \triangleq \left\{ x^* \in \mathcal {X}^*: \langle x^* , x-x_0\rangle \le 0 \ \text{ for } \text{ any } x \in \{f<f(x_0)\}\right\} \\&\partial ^< f(x_0)\triangleq \{x^*\in \mathcal {X}^*: \langle x^*, x-x_0\rangle \le f(x)-f(x_0) \ \text{ for } \text{ any } x\in \{f <f(x_0)\} \}. \end{aligned}$$

Notice that the last definition is similar but weaker than the one of Fenchel-Moreau subdifferential. Other relations among the different notions above can be found in Penot [21] and Penot and Zalinescu [23], among others.

Proposition 18

(see Prop. 2.8 of Penot and Zalinescu [23]) Let $f: \mathcal {X}\rightarrow \mathbb R$ and $g: \mathcal {X}\rightarrow \mathbb R$ be two given functions and let $z_0 \in \mathcal {X}$. If the qco-convolution $(f \nabla g) (z_0)$ is exact with $(f\nabla g)(z_0) = f(x_0) \vee g(y_0)$ for $x_0, y_0 \in \mathcal {X}$ such that $x_0+y_0=z_0$ and $f(x_0)=g(y_0)$, then

$$\begin{aligned} (\partial ^{(*)} f(x_0) \cap \partial ^{GP}g(y_0)) \cup (\partial ^{GP} f(x_0) \cap \partial ^{(*)}g(y_0)) \subseteq \partial ^{GP}\left( f\nabla g \right) (z_0), \end{aligned}$$

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where equality holds if $x_0,y_0$ are not local minimizers of $f,g$, respectively.

Given two functionals $f,g: \mathcal {X}\rightarrow \bar{\mathbb {R}}$, given $x_0,y_0\in \mathcal {X}$ and the subdifferentials $\partial ^< f(x_0)$ and $\partial ^< g(y_0)$, the set $(\partial ^< f(x_0)) \nabla (\partial ^< g(y_0))$ is defined as

$$\begin{aligned} (\partial ^< f(x_0)) \, \nabla \, (\partial ^< g(y_0))&\triangleq \left( \bigcup \limits _{\lambda \in (0,1)} [\lambda \partial ^<f(x_0) \cap (1-\lambda )\partial ^< g(y_0)] \right) \nonumber \\&\quad \cup \,(\partial ^{(*)} f(x_0) \cap \partial ^< g(y_0)) \cup ( \partial ^< f(x_0)\cap \partial ^{(*)} g(y_0)).\qquad \end{aligned}$$

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See Penot and Zalinescu [23] for a general definition and for further explanations.

Proposition 19

(see Prop. 3.23 of Penot and Zalinescu [23]) Let $f,g: \mathcal {X}\rightarrow \bar{\mathbb {R}}$ be two quasiconvex functions, $x_0\in \mathrm{dom}f$ and $y_0 \in \mathrm{dom}g$.

If $(\partial ^< f(x_0))\, \nabla \, (\partial ^< g(y_0))$ is nonempty and $f(x_0)=g(y_0)$, then $(f \nabla g) (x_0+y_0)$ is exact with $(f \nabla g) (x_0+y_0) = f(x_0)\vee g(y_0)$, and $(\partial ^< f(x_0))\, \nabla \, (\partial ^< g(y_0)) \subseteq \partial ^< (f \nabla g )(x_0+y_0)$.