# American Institute of Mathematical Sciences

January  2017, 2: 1 doi: 10.1186/s41546-017-0014-7

## Stochastic global maximum principle for optimization with recursive utilities

 Zhongtai Institute of Finance, Shandong University, Jinan, Shandong 250100, People's Republic of China

Received  September 19, 2016 Revised  January 05, 2017

Fund Project: supported by NSF (No. 11671231, 11201262 and 10921101), Shandong Province (No.BS2013SF020 and ZR2014AP005), Young Scholars Program of Shandong University and the 111 Project (No. B12023).

In this paper, we study the recursive stochastic optimal control problems. The control domain does not need to be convex, and the generator of the backward stochastic differential equation can contain z. We obtain the variational equations for backward stochastic differential equations, and then obtain the maximum principle which solves completely Peng's open problem.
Citation: Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7
##### References:
 [1] Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109-129 (2003), [2] Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403-1443(2002), [3] Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143-165 (1999), [4] Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992), [5] El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997), [6] El Karoui, N, Peng, S, Quenez, MC:A dynamic maximum priciple for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11, 664-693 (2001), [7] Ji, S, Zhou, XY:A maximum principle for stochastic optimal control with terminal state constrains, and its applications. Comm. Inf. Syst. 6, 321-337 (2006), [8] Kohlmann, M, Zhou, XY:Relationship between backward stochastic differential equations and stochastic controls:a linear-quadratic approach. SIAM J. Control Optim. 38, 1392-1407 (2000), [9] Lim, A, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450-474 (2001), [10] Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 55-61 (1990), [11] Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966-979 (1990), [12] Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125-144 (1993), [13] Peng, S:Open problems on backward stochastic differential equations. In:Chen, S, Li, X, Yong, J, Zhou, XY (eds.) Control of distributed parameter and stocastic systems, pp. 265-273, Boston:Kluwer Acad.Pub. (1998), [14] Shi, J, Wu, Z:The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161-169 (2006), [15] Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447-1475 (1994), [16] Wu, Z:Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11, 249-259 (1998), [17] Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013), [18] Xu, W:Stochastic maximum principle for optimal control problem of forward and backward system. J.Austral. Math. Soc. Ser. B 37, 172-185 (1995), [19] Yong, J:Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48, 4119-4156 (2010), [20] Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999),

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##### References:
 [1] Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109-129 (2003), [2] Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403-1443(2002), [3] Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143-165 (1999), [4] Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992), [5] El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997), [6] El Karoui, N, Peng, S, Quenez, MC:A dynamic maximum priciple for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11, 664-693 (2001), [7] Ji, S, Zhou, XY:A maximum principle for stochastic optimal control with terminal state constrains, and its applications. Comm. Inf. Syst. 6, 321-337 (2006), [8] Kohlmann, M, Zhou, XY:Relationship between backward stochastic differential equations and stochastic controls:a linear-quadratic approach. SIAM J. Control Optim. 38, 1392-1407 (2000), [9] Lim, A, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450-474 (2001), [10] Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 55-61 (1990), [11] Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966-979 (1990), [12] Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125-144 (1993), [13] Peng, S:Open problems on backward stochastic differential equations. In:Chen, S, Li, X, Yong, J, Zhou, XY (eds.) Control of distributed parameter and stocastic systems, pp. 265-273, Boston:Kluwer Acad.Pub. (1998), [14] Shi, J, Wu, Z:The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161-169 (2006), [15] Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447-1475 (1994), [16] Wu, Z:Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11, 249-259 (1998), [17] Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013), [18] Xu, W:Stochastic maximum principle for optimal control problem of forward and backward system. J.Austral. Math. Soc. Ser. B 37, 172-185 (1995), [19] Yong, J:Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48, 4119-4156 (2010), [20] Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999),
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