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 Published  March 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) Google Scholar

[2]

Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403-1443(2002) Google Scholar

[3]

Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143-165 (1999) Google Scholar

[4]

Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992) Google Scholar

[5]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[9]

Lim, A, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450-474 (2001) Google Scholar

[10]

Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 55-61 (1990) Google Scholar

[11]

Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966-979 (1990) Google Scholar

[12]

Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125-144 (1993) Google Scholar

[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) Google Scholar

[14]

Shi, J, Wu, Z:The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161-169 (2006) Google Scholar

[15]

Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447-1475 (1994) Google Scholar

[16]

Wu, Z:Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11, 249-259 (1998) Google Scholar

[17]

Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013) Google Scholar

[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) Google Scholar

[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) Google Scholar

[20]

Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999) Google Scholar

show all references

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) Google Scholar

[2]

Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403-1443(2002) Google Scholar

[3]

Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143-165 (1999) Google Scholar

[4]

Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992) Google Scholar

[5]

El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[9]

Lim, A, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450-474 (2001) Google Scholar

[10]

Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 55-61 (1990) Google Scholar

[11]

Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966-979 (1990) Google Scholar

[12]

Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125-144 (1993) Google Scholar

[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) Google Scholar

[14]

Shi, J, Wu, Z:The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161-169 (2006) Google Scholar

[15]

Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447-1475 (1994) Google Scholar

[16]

Wu, Z:Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11, 249-259 (1998) Google Scholar

[17]

Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013) Google Scholar

[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) Google Scholar

[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) Google Scholar

[20]

Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999) Google Scholar

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