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

• 成果类型: Article
• 署名作者: Jiang, Long
• 署名单位: China University of Mining & Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051607000000294
• 发表日期: 2008
• 页码: 245-258
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; THEOREM

摘要

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 (epsilon(g)[.vertical bar F-t])(t is an element of[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))(tE[0,T]). In this paper we prove that, without the additional continuous assumption on g, a g-expectation epsilon(g) satisfies translation invariance if and only if g is independent of y, and epsilon(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 Og 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 (2606) 19-34] on these subjects.