# 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
##### References:
 [1] B. Bian, S. Miao and H. Zheng, Smooth value functions for a class of nonsmooth utility maximization problems, SIAM Journal on Financial Mathematics, 2 (2011), 727-747. doi: 10.1137/100793396. [2] Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337. [3] D. Cuoco and J. Cvitanic, Optimal consumption choices for a 'large' investor, J. Econom. Dynam.Control, 22 (1998), 401-436. doi: 10.1016/S0165-1889(97)00065-1. [4] J. Cvitanic and I. Karatzas, Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2 (1992), 767-818. doi: 10.1214/aoap/1177005576. [5] J. Cvitanic and I. Karatzas, Generalized Neyman-Pearson lemma via convex duality, Bernoulli, 7 (2001), 79-97. doi: 10.2307/3318603. [6] D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600. [7] N. El Karoui, Les aspects probabilistes du controle stochastique, in Lecture Notes in Math., Springer, Berlin-New York, 876 (1981), 73–238. [8] N. El Karoui, S. Peng and M. Quenez, A dynamic maximum principle for the optimization of recursive utilities under constraints, Ann. Appl. Probab., 11 (2001), 664-693. doi: 10.1214/aoap/1015345345. [9] N. El Karoui, S. Peng and M. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [10] L. Epstein and S. Ji, Ambiguous volatility and asset pricing in continuous time, The Review of Financial Study, 26 (2013), 1740-1786. [11] L. Epstein and S. Ji, Ambiguous volatility, possibility and utility in continuous time, J. Math. Econom, 50 (2014), 269-282. doi: 10.1016/j.jmateco.2013.09.005. [12] W. Faidi, A. Matoussi and M. Mnif, Maximization of recursive utilities: A dynamic maximum principle approach, SIAM J. Financial Math., 2 (2011), 1014-1041. doi: 10.1137/100814354. [13] Y. Hu, P. Imkeller and M. Muller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188. [14] S. Ji and S. Peng, Terminal perturbation method for the backward approach to continuous-time mean-variance portfolio selection, Stochastic Process. Appl., 118 (2008), 952-967. doi: 10.1016/j.spa.2007.07.005. [15] S. Ji and X. Zhou, A maximum principle for stochastic optimal control with terminal state constraints, and its applications, Commun. Inf. and Syst, 6 (2006), 321-337. doi: 10.4310/CIS.2006.v6.n4.a4. [16] S. Ji and X. Zhou, A generalized Neyman-Pearson lemma for g-probabilities, Probab. Theory Related Fields, 148 (2010), 645-669. doi: 10.1007/s00440-009-0244-4. [17] H. Jin and X. Zhou, Continuous-Time Portfolio Selection under Ambiguity, Math. Control Relat. Fields, 5 (2015), 475-488. doi: 10.3934/mcrf.2015.5.475. [18] E. Jouini and H. Kallal, Arbitrage in Securities Markets with Short-Sales Constraints, Math. Finance, 5 (1995), 197-232. doi: 10.1111/j.1467-9965.1995.tb00065.x. [19] I. Karatzas, J. Lehoczky and S. Shreve, Optimal portfolio and consumption decisions for a "small investor" on a finite horizon, SIAM J. Control Optim., 25 (1987), 1557-1586. doi: 10.1137/0325086. [20] I. Karatzas, J. Lehoczky, S. Shreve and G. Xu, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim., 29 (1991), 702-730. doi: 10.1137/0329039. [21] I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840. [22] M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. doi: 10.1214/aop/1019160253. [23] A. Matoussi and H. Xing, Convex duality for stochastic differential utility, (2016), 25pp, arXiv: 1601.03562 doi: 10.2139/ssrn.2715425. [24] S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics Stochastics Rep., 38 (1992), 119-134. doi: 10.1080/17442509208833749. [25] S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144. doi: 10.1007/BF01195978. [26] M. Quenez, Optimal portfolio in a multiple-priors model, in Seminar on Stochastic Analysis, Random Fields and Applications IV, Birkhauser Basel, 58 (2004), 291–321. doi: 10.1007/978-3-0348-7943-9_18. [27] A. Schied, Optimal investments for robust utility functionals in complete market models, Math. Oper. Res., 30 (2005), 750-764. doi: 10.1287/moor.1040.0138. [28] N. Westray and H. Zheng, Constrained nonsmooth utility maximization without quadratic inf convolution, Stochastic Process. Appl., 119 (2009), 1561-1579. doi: 10.1016/j.spa.2008.08.002. [29] N. Westray and H. Zheng, Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization, Finance Stoch., 15 (2011), 501-512. doi: 10.1007/s00780-010-0128-6.

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##### References:
 [1] B. Bian, S. Miao and H. Zheng, Smooth value functions for a class of nonsmooth utility maximization problems, SIAM Journal on Financial Mathematics, 2 (2011), 727-747. doi: 10.1137/100793396. [2] Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337. [3] D. Cuoco and J. Cvitanic, Optimal consumption choices for a 'large' investor, J. Econom. Dynam.Control, 22 (1998), 401-436. doi: 10.1016/S0165-1889(97)00065-1. [4] J. Cvitanic and I. Karatzas, Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2 (1992), 767-818. doi: 10.1214/aoap/1177005576. [5] J. Cvitanic and I. Karatzas, Generalized Neyman-Pearson lemma via convex duality, Bernoulli, 7 (2001), 79-97. doi: 10.2307/3318603. [6] D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600. [7] N. El Karoui, Les aspects probabilistes du controle stochastique, in Lecture Notes in Math., Springer, Berlin-New York, 876 (1981), 73–238. [8] N. El Karoui, S. Peng and M. Quenez, A dynamic maximum principle for the optimization of recursive utilities under constraints, Ann. Appl. Probab., 11 (2001), 664-693. doi: 10.1214/aoap/1015345345. [9] N. El Karoui, S. Peng and M. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [10] L. Epstein and S. Ji, Ambiguous volatility and asset pricing in continuous time, The Review of Financial Study, 26 (2013), 1740-1786. [11] L. Epstein and S. Ji, Ambiguous volatility, possibility and utility in continuous time, J. Math. Econom, 50 (2014), 269-282. doi: 10.1016/j.jmateco.2013.09.005. [12] W. Faidi, A. Matoussi and M. Mnif, Maximization of recursive utilities: A dynamic maximum principle approach, SIAM J. Financial Math., 2 (2011), 1014-1041. doi: 10.1137/100814354. [13] Y. Hu, P. Imkeller and M. Muller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188. [14] S. Ji and S. Peng, Terminal perturbation method for the backward approach to continuous-time mean-variance portfolio selection, Stochastic Process. Appl., 118 (2008), 952-967. doi: 10.1016/j.spa.2007.07.005. [15] S. Ji and X. Zhou, A maximum principle for stochastic optimal control with terminal state constraints, and its applications, Commun. Inf. and Syst, 6 (2006), 321-337. doi: 10.4310/CIS.2006.v6.n4.a4. [16] S. Ji and X. Zhou, A generalized Neyman-Pearson lemma for g-probabilities, Probab. Theory Related Fields, 148 (2010), 645-669. doi: 10.1007/s00440-009-0244-4. [17] H. Jin and X. Zhou, Continuous-Time Portfolio Selection under Ambiguity, Math. Control Relat. Fields, 5 (2015), 475-488. doi: 10.3934/mcrf.2015.5.475. [18] E. Jouini and H. Kallal, Arbitrage in Securities Markets with Short-Sales Constraints, Math. Finance, 5 (1995), 197-232. doi: 10.1111/j.1467-9965.1995.tb00065.x. [19] I. Karatzas, J. Lehoczky and S. Shreve, Optimal portfolio and consumption decisions for a "small investor" on a finite horizon, SIAM J. Control Optim., 25 (1987), 1557-1586. doi: 10.1137/0325086. [20] I. Karatzas, J. Lehoczky, S. Shreve and G. Xu, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim., 29 (1991), 702-730. doi: 10.1137/0329039. [21] I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840. [22] M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. doi: 10.1214/aop/1019160253. [23] A. Matoussi and H. Xing, Convex duality for stochastic differential utility, (2016), 25pp, arXiv: 1601.03562 doi: 10.2139/ssrn.2715425. [24] S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics Stochastics Rep., 38 (1992), 119-134. doi: 10.1080/17442509208833749. [25] S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144. doi: 10.1007/BF01195978. [26] M. Quenez, Optimal portfolio in a multiple-priors model, in Seminar on Stochastic Analysis, Random Fields and Applications IV, Birkhauser Basel, 58 (2004), 291–321. doi: 10.1007/978-3-0348-7943-9_18. [27] A. Schied, Optimal investments for robust utility functionals in complete market models, Math. Oper. Res., 30 (2005), 750-764. doi: 10.1287/moor.1040.0138. [28] N. Westray and H. Zheng, Constrained nonsmooth utility maximization without quadratic inf convolution, Stochastic Process. Appl., 119 (2009), 1561-1579. doi: 10.1016/j.spa.2008.08.002. [29] N. Westray and H. Zheng, Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization, Finance Stoch., 15 (2011), 501-512. doi: 10.1007/s00780-010-0128-6.
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