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

Simple item page
dc.creatorJiang, Long
dc.date2008-01-22
dc.date.accessioned2026-07-07T08:56:25Z
dc.date.available2026-07-07T08:56:25Z
dc.descriptionUnder 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 $(ρ^g_t)_{t\in[0,T]}$. 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 $ρ^g$ 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.
dc.descriptionPublished in at http://dx.doi.org/10.1214/105051607000000294 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/0801.3340
dc.identifierhttp://arxiv.org/abs/0801.3340
dc.identifierAnnals of Applied Probability 2008, Vol. 18, No. 1, 245-258
dc.identifierdoi:10.1214/105051607000000294
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/146603
dc.subjectProbability
dc.subject60H10 (Primary); 60H30, 91B30 (Secondary)
dc.titleConvexity, translation invariance and subadditivity for $g$-expectations and related risk measures
dc.typetext

Files

Collections

arXiv