# American Institute of Mathematical Sciences

September  2018, 8(3&4): 753-775. doi: 10.3934/mcrf.2018033

## Recursive utility optimization with concave coefficients

 1 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, China 2 School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, China

* Corresponding authorr: Xiaomin Shi

Received  March 2017 Revised  February 2018 Published  September 2018

Fund Project: The first author is supported by NSF (No. 11571203); Supported by the Programme of Introducing Talents of Discipline to Universities of China (No. B12023). The second author is supported by NSF (No. 11801315, 11401091, 11701330); Supported by Natural Science Foundation of Shandong Province (No. ZR2018QA001).

This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our problem, we employ the Fenchel-Legendre transform and derive the corresponding variational formulation. By the convex duality method, the primal "sup-inf" problem is translated to a dual minimization problem and the saddle point of our problem is derived. Finally, we obtain the optimal terminal wealth. To illustrate our results, three cases for investors with ambiguity aversion are explicitly worked out under some special assumptions.

Citation: Shaolin Ji, Xiaomin Shi. Recursive utility optimization with concave coefficients. Mathematical Control & Related Fields, 2018, 8 (3&4) : 753-775. doi: 10.3934/mcrf.2018033
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