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September  2015, 5(3): 613-649. doi: 10.3934/mcrf.2015.5.613

## Optimal control problems of forward-backward stochastic Volterra integral equations

 1 Institute for Financial Studies and School of Mathematics, Shandong University, Jinan, Shandong 250100 2 School of Mathematics, Sichuan University, Chengdu 610065, China 3 Department of Mathematics, University of Central Florida, Orlando, FL 32816

Received  April 2014 Revised  March 2015 Published  July 2015

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) are formulated and studied. A general duality principle is established for linear backward stochastic integral equation and linear stochastic Fredholm-Volterra integral equation with mean-field. With the help of such a duality principle, together with some other new delicate and subtle skills, Pontryagin type maximum principles are proved for two optimal control problems of FBSVIEs.
Citation: Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
##### References:
 [1] G. Ainslie, Picoeconomics: The Strategic Interaction of Successive Motivational States within the Person,, Cambridge Univ. Press 1992., (1992). [2] D. Duffie and L. G. Epstein, Stochastic differential utility,, Econometrica, 60 (1992), 353. doi: 10.2307/2951600. [3] D. Duffie and C. F. Huang, Stochastic Production-Exchange Equilibria,, Reserach paper No.974, (1986). [4] M. I. Kamien and E. Muller, Optimal control with integral state equations,, Rev. Econ. Stud., 43 (1976), 469. doi: 10.2307/2297225. [5] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer, (1988). doi: 10.1007/978-1-4684-0302-2_2. [6] J. Lin, Adapted solution of backward stochastic nonlinear Volterra integral equation,, Stoch. Anal. Appl., 20 (2002), 165. doi: 10.1081/SAP-120002426. [7] J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications,, Springer-Verlag, (1999). [8] E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients,, Ann. Probab., 18 (1990), 1635. doi: 10.1214/aop/1176990638. [9] S. Peng, A general stochastic maximum principle for optimal control problems,, SIAM J. Control Optim., 28 (1990), 966. doi: 10.1137/0328054. [10] S. Peng, Backward stochastic differential equations and application to optimal control,, Appl. Math. Optim., 27 (1993), 125. doi: 10.1007/BF01195978. [11] Y. Shi and T. Wang, Solvability of general backward stochastic Volterra integral equations,, J. Korean Math. Soc., 49 (2012), 1301. doi: 10.4134/JKMS.2012.49.6.1301. [12] Y. Shi, T. Wang and J. Yong, Mean-field backward stochastic Volterra integral equations,, Discrete. Contin. Dyn. Syst. Ser. B, 18 (2013), 1929. doi: 10.3934/dcdsb.2013.18.1929. [13] T. Wang and Y. Shi, Symmetrical solutions of backward stochastic Volterra integral equations and applications,, Discrete Contin. Dyn. Syst. Ser. B., 14 (2010), 251. doi: 10.3934/dcdsb.2010.14.251. [14] T. Wang and J. Yong, Comparison theorems for backward stochastic Volterra integral equations,, Stoch. Process Appl., 125 (2015), 1756. doi: 10.1016/j.spa.2014.11.013. [15] Z. Wang and X. Zhang, Non-Lipschitz backward stochastic volterra type equations with jumps,, Stoch. Dyn., 7 (2007), 479. doi: 10.1142/S0219493707002128. [16] J. Yong, Backward stochastic Volterra integral equations and some related problems,, Stoch. Process Appl., 116 (2006), 779. doi: 10.1016/j.spa.2006.01.005. [17] J. Yong, Continuous-time dynamic risk measures by backward stochastic Volterra integral equations,, Appl. Anal., 86 (2007), 1429. doi: 10.1080/00036810701697328. [18] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equation,, Probab. Theory Relat. Fields, 142 (2008), 21. doi: 10.1007/s00440-007-0098-6. [19] J. Yong and X. Y. Zhou, Stochstic Control: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3.

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##### References:
 [1] G. Ainslie, Picoeconomics: The Strategic Interaction of Successive Motivational States within the Person,, Cambridge Univ. Press 1992., (1992). [2] D. Duffie and L. G. Epstein, Stochastic differential utility,, Econometrica, 60 (1992), 353. doi: 10.2307/2951600. [3] D. Duffie and C. F. Huang, Stochastic Production-Exchange Equilibria,, Reserach paper No.974, (1986). [4] M. I. Kamien and E. Muller, Optimal control with integral state equations,, Rev. Econ. Stud., 43 (1976), 469. doi: 10.2307/2297225. [5] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer, (1988). doi: 10.1007/978-1-4684-0302-2_2. [6] J. Lin, Adapted solution of backward stochastic nonlinear Volterra integral equation,, Stoch. Anal. Appl., 20 (2002), 165. doi: 10.1081/SAP-120002426. [7] J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications,, Springer-Verlag, (1999). [8] E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients,, Ann. Probab., 18 (1990), 1635. doi: 10.1214/aop/1176990638. [9] S. Peng, A general stochastic maximum principle for optimal control problems,, SIAM J. Control Optim., 28 (1990), 966. doi: 10.1137/0328054. [10] S. Peng, Backward stochastic differential equations and application to optimal control,, Appl. Math. Optim., 27 (1993), 125. doi: 10.1007/BF01195978. [11] Y. Shi and T. Wang, Solvability of general backward stochastic Volterra integral equations,, J. Korean Math. Soc., 49 (2012), 1301. doi: 10.4134/JKMS.2012.49.6.1301. [12] Y. Shi, T. Wang and J. Yong, Mean-field backward stochastic Volterra integral equations,, Discrete. Contin. Dyn. Syst. Ser. B, 18 (2013), 1929. doi: 10.3934/dcdsb.2013.18.1929. [13] T. Wang and Y. Shi, Symmetrical solutions of backward stochastic Volterra integral equations and applications,, Discrete Contin. Dyn. Syst. Ser. B., 14 (2010), 251. doi: 10.3934/dcdsb.2010.14.251. [14] T. Wang and J. Yong, Comparison theorems for backward stochastic Volterra integral equations,, Stoch. Process Appl., 125 (2015), 1756. doi: 10.1016/j.spa.2014.11.013. [15] Z. Wang and X. Zhang, Non-Lipschitz backward stochastic volterra type equations with jumps,, Stoch. Dyn., 7 (2007), 479. doi: 10.1142/S0219493707002128. [16] J. Yong, Backward stochastic Volterra integral equations and some related problems,, Stoch. Process Appl., 116 (2006), 779. doi: 10.1016/j.spa.2006.01.005. [17] J. Yong, Continuous-time dynamic risk measures by backward stochastic Volterra integral equations,, Appl. Anal., 86 (2007), 1429. doi: 10.1080/00036810701697328. [18] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equation,, Probab. Theory Relat. Fields, 142 (2008), 21. doi: 10.1007/s00440-007-0098-6. [19] J. Yong and X. Y. Zhou, Stochstic Control: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3.
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