## The Annals of Applied Probability

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

Long Jiang

#### 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 (ρtg)t∈[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 ρg 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.

#### Article information

Source
Ann. Appl. Probab., Volume 18, Number 1 (2008), 245-258.

Dates
First available in Project Euclid: 9 January 2008

https://projecteuclid.org/euclid.aoap/1199890022

Digital Object Identifier
doi:10.1214/105051607000000294

Mathematical Reviews number (MathSciNet)
MR2380898

Zentralblatt MATH identifier
1145.60032

#### Citation

Jiang, Long. Convexity, translation invariance and subadditivity for g -expectations and related risk measures. Ann. Appl. Probab. 18 (2008), no. 1, 245--258. doi:10.1214/105051607000000294. https://projecteuclid.org/euclid.aoap/1199890022

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