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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 |
References:
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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. |
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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