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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

Yufeng Shi 1, , Tianxiao Wang 2, and  Jiongmin Yong 3,

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.
Keywords: adapted M-solution, Forward-backward stochastic Volterra integral equations, stochastic maximum principle, stochastic Fredholm-Volterra integral equations., duality principle.
Mathematics Subject Classification: Primary: 60H20, 93E20; Secondary: 49K21, 49K4.
Citation: Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control and 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.

[2]

D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600.

[3]

D. Duffie and C. F. Huang, Stochastic Production-Exchange Equilibria, Reserach paper No.974, Graduate School of Business, Stanford University, Stanford, 1986.

[4]

M. I. Kamien and E. Muller, Optimal control with integral state equations, Rev. Econ. Stud., 43 (1976), 469-473. doi: 10.2307/2297225.

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I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, Heidelberg, 1988. doi: 10.1007/978-1-4684-0302-2_2.

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J. Lin, Adapted solution of backward stochastic nonlinear Volterra integral equation, Stoch. Anal. Appl., 20 (2002), 165-183. doi: 10.1081/SAP-120002426.

[7]

J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Springer-Verlag, Berlin, 1999.

[8]

E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab., 18 (1990), 1635-1655. doi: 10.1214/aop/1176990638.

[9]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054.

[10]

S. Peng, Backward stochastic differential equations and application to optimal control, Appl. Math. Optim., 27 (1993), 125-144. doi: 10.1007/BF01195978.

[11]

Y. Shi and T. Wang, Solvability of general backward stochastic Volterra integral equations, J. Korean Math. Soc., 49 (2012), 1301-1321. 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), 1929-1967. 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), 251-274. 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), 1756-1798. 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), 479-496. doi: 10.1142/S0219493707002128.

[16]

J. Yong, Backward stochastic Volterra integral equations and some related problems, Stoch. Process Appl., 116 (2006), 779-795. 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), 1429-1442. doi: 10.1080/00036810701697328.

[18]

J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equation, Probab. Theory Relat. Fields, 142 (2008), 21-77. doi: 10.1007/s00440-007-0098-6.

[19]

J. Yong and X. Y. Zhou, Stochstic Control: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

show all references

References:
[1]

G. Ainslie, Picoeconomics: The Strategic Interaction of Successive Motivational States within the Person, Cambridge Univ. Press 1992.

[2]

D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600.

[3]

D. Duffie and C. F. Huang, Stochastic Production-Exchange Equilibria, Reserach paper No.974, Graduate School of Business, Stanford University, Stanford, 1986.

[4]

M. I. Kamien and E. Muller, Optimal control with integral state equations, Rev. Econ. Stud., 43 (1976), 469-473. doi: 10.2307/2297225.

[5]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, Heidelberg, 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), 165-183. doi: 10.1081/SAP-120002426.

[7]

J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Springer-Verlag, Berlin, 1999.

[8]

E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab., 18 (1990), 1635-1655. doi: 10.1214/aop/1176990638.

[9]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054.

[10]

S. Peng, Backward stochastic differential equations and application to optimal control, Appl. Math. Optim., 27 (1993), 125-144. doi: 10.1007/BF01195978.

[11]

Y. Shi and T. Wang, Solvability of general backward stochastic Volterra integral equations, J. Korean Math. Soc., 49 (2012), 1301-1321. 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), 1929-1967. 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), 251-274. 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), 1756-1798. 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), 479-496. doi: 10.1142/S0219493707002128.

[16]

J. Yong, Backward stochastic Volterra integral equations and some related problems, Stoch. Process Appl., 116 (2006), 779-795. 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), 1429-1442. doi: 10.1080/00036810701697328.

[18]

J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equation, Probab. Theory Relat. Fields, 142 (2008), 21-77. doi: 10.1007/s00440-007-0098-6.

[19]

J. Yong and X. Y. Zhou, Stochstic Control: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

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