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

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 (Eg[.|Ft])t.[0,T] and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19.34] showed some connections between g and the corresponding dynamic risk measure (.tg)t.[0, T]. In this paper we prove that, without the additional continuous assumption on g, a g-expectation Eg satisfies translation invariance if and only if g is independent of y, and Eg 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 .g induced by a g-expectation Eg 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.