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Implied fractional hazard rates and default risk distributions
Stochastic global maximum principle for optimization with recursive utilities
Zhongtai Institute of Finance, Shandong University, Jinan, Shandong 250100, People's Republic of China 
References:
[1] 
Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:L^{p} solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109129 (2003), 
[2] 
Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 14031443(2002), 
[3] 
Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143165 (1999), 
[4] 
Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353394 (1992), 
[5] 
El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 171 (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, 664693 (2001), 
[7] 
Ji, S, Zhou, XY:A maximum principle for stochastic optimal control with terminal state constrains, and its applications. Comm. Inf. Syst. 6, 321337 (2006), 
[8] 
Kohlmann, M, Zhou, XY:Relationship between backward stochastic differential equations and stochastic controls:a linearquadratic approach. SIAM J. Control Optim. 38, 13921407 (2000), 
[9] 
Lim, A, Zhou, XY:Linearquadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450474 (2001), 
[10] 
Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 5561 (1990), 
[11] 
Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966979 (1990), 
[12] 
Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125144 (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. 265273, Boston:Kluwer Acad.Pub. (1998), 
[14] 
Shi, J, Wu, Z:The maximum principle for fully coupled forwardbackward stochastic control system. Acta Automat. Sinica 32, 161169 (2006), 
[15] 
Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 14471475 (1994), 
[16] 
Wu, Z:Maximum principle for optimal control problem of fully coupled forwardbackward stochastic systems. Syst. Sci. Math. Sci. 11, 249259 (1998), 
[17] 
Wu, Z:A general maximum principle for optimal control of forwardbackward stochastic systems.Automatica 49, 14731480 (2013), 
[18] 
Xu, W:Stochastic maximum principle for optimal control problem of forward and backward system. J.Austral. Math. Soc. Ser. B 37, 172185 (1995), 
[19] 
Yong, J:Optimality variational principle for controlled forwardbackward stochastic differential equations with mixed initialterminal conditions. SIAM J. Control Optim. 48, 41194156 (2010), 
[20] 
Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. SpringerVerlag, New York (1999), 
show all references
References:
[1] 
Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:L^{p} solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109129 (2003), 
[2] 
Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 14031443(2002), 
[3] 
Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143165 (1999), 
[4] 
Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353394 (1992), 
[5] 
El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 171 (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, 664693 (2001), 
[7] 
Ji, S, Zhou, XY:A maximum principle for stochastic optimal control with terminal state constrains, and its applications. Comm. Inf. Syst. 6, 321337 (2006), 
[8] 
Kohlmann, M, Zhou, XY:Relationship between backward stochastic differential equations and stochastic controls:a linearquadratic approach. SIAM J. Control Optim. 38, 13921407 (2000), 
[9] 
Lim, A, Zhou, XY:Linearquadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450474 (2001), 
[10] 
Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 5561 (1990), 
[11] 
Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966979 (1990), 
[12] 
Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125144 (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. 265273, Boston:Kluwer Acad.Pub. (1998), 
[14] 
Shi, J, Wu, Z:The maximum principle for fully coupled forwardbackward stochastic control system. Acta Automat. Sinica 32, 161169 (2006), 
[15] 
Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 14471475 (1994), 
[16] 
Wu, Z:Maximum principle for optimal control problem of fully coupled forwardbackward stochastic systems. Syst. Sci. Math. Sci. 11, 249259 (1998), 
[17] 
Wu, Z:A general maximum principle for optimal control of forwardbackward stochastic systems.Automatica 49, 14731480 (2013), 
[18] 
Xu, W:Stochastic maximum principle for optimal control problem of forward and backward system. J.Austral. Math. Soc. Ser. B 37, 172185 (1995), 
[19] 
Yong, J:Optimality variational principle for controlled forwardbackward stochastic differential equations with mixed initialterminal conditions. SIAM J. Control Optim. 48, 41194156 (2010), 
[20] 
Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. SpringerVerlag, New York (1999), 
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