Convexity, translation invariance and subadditivity for $g$-expectations and related risk measures

TL;DR

This paper characterizes when $g$-expectations and related risk measures exhibit properties like translation invariance, convexity, and subadditivity, based on the independence and convexity of the generator $g$, extending previous results.

Contribution

It proves that without the continuous assumption on $g$, properties like translation invariance and convexity of $g$-expectations are characterized by specific independence and convexity conditions of the generator $g$, extending prior work.

Findings

$g$-expectation is translation invariant iff $g$ is independent of $y$.

Convexity and subadditivity of $g$-expectation correspond to $g$ being convex/sublinear in $z$ and independent of $y$.

Static risk measures derived from $g$-expectations are convex or coherent under these conditions.

Abstract

Under the continuous assumption on the generator $g$, Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] showed some connections between $g$ and the conditional $g$-expectation $({\mathcal{E}}_g[\cdot|{\mathcal{F}}_t])_{t\in[0,T]}$ and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] showed some connections between $g$ and the corresponding dynamic risk measure $(\rho^g_t)_{t\in[0,T]}$. In this paper we prove that, without the additional continuous assumption on $g$, a $g$-expectation ${\mathcal{E}}_g$ satisfies translation invariance if and only if $g$ is independent of $y$, and ${\mathcal{E}}_g$ satisfies convexity (resp. subadditivity) if and only if $g$ is independent of $y$ and $g$ is convex (resp. subadditive) with respect to $z$. By these conclusions we deduce that the static risk measure $\rho^g$ induced by a $g$-expectation ${\mathcal{E}}_g$ is a convex (resp. coherent) risk measure if and only if $g$ is independent of $y$ and $g$ is convex (resp. sublinear) with respect to $z$. Our results extend the results in Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] on these subjects.