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数学进展 - 2020, Vol. 49(1): 115-127
研究论文
等级依赖效用下最优投资问题的扩展研究
Extended Research on the Optimal Investment Problem with Rank-dependent Utility

罗马
LUO Ma

北京航空航天大学数学与系统科学学院, 北京, 100191
School of Mathematics and System Sciences, Beijing University of Aeronautics and Astronautics, Beijing, 100191, P. R. China

收稿日期: 2018-12-05
出版日期: 2020-03-25
2020, Vol. 49(1): 115-127
DOI: 10.11845/sxjz.2018101b


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摘要 本文在文献[https://ssrn.com/abstract=3135695]的基础上, 去掉了等级依赖效用投资者的概率加权函数的连续性和单调增加的严格性, 以及其原始效用函数的连续可微性. 通过引入一般单调函数的广义逆函数以及凹函数的超微分, 克服了分析上所带来的新的困难, 证明了新模型最优解的存在性并给出其显式表达.
关键词 等级依赖效用风险约束概率加权函数非光滑广义逆函数超微分    
Abstract:Based on the literature [https://ssrn.com/abstract=3135695], this paper removes the continuity and the strictness of monotonous increase of the probability weighting function in the rank-dependent utility theory, as well as the continuous differentiability of the original utility function. By introducing the generalized inverse function of the general monotone function and the super-differential of the concave function, we will overcome the new difficulties in analysis, prove the existence of the optimal solution of the new model and give its explicit expression.
Key wordsrank-dependent utility    risk constraint    probability weighting function    non-smooth    generalized inverse function    super-differential
PACS:  O224  
  F224  
  F830  
基金资助:国家自然科学基金(No. 11801032)和中国科学院数学与系统科学研究院随机复杂结构与数据科学重点实验室资助(No. 2008DP173182).
通讯作者: E-mail: lm_910622@126.com   
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[2] Föllmer, H. and Schied, A., Stochastic Finance --- An Introduction in Discrete Time, Second Revised and Extended Edition, De Gruyter Studies in Mathematics, Vol. 27, Berlin: Walter de Gruyter, 2008.
[3] Gao Y.,Non-smooth Optimization, Beijing: Science Press, 2008 (in Chinese).
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