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

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

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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.  Google Scholar

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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.  Google Scholar

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

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

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

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. Yong, Continuous-time dynamic risk measures by backward stochastic Volterra integral equations, Appl. Anal., 86 (2007), 1429-1442. doi: 10.1080/00036810701697328.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. Yong, Continuous-time dynamic risk measures by backward stochastic Volterra integral equations, Appl. Anal., 86 (2007), 1429-1442. doi: 10.1080/00036810701697328.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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