# American Institute of Mathematical Sciences

• Previous Article
Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation
• MCRF Home
• This Issue
• Next Article
Generalized homogeneous systems with applications to nonlinear control: A survey
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 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. [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.
 [1] Ludger Overbeck, Jasmin A. L. Röder. Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 4-. doi: 10.1186/s41546-018-0030-2 [2] Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004 [3] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 [4] Tianxiao Wang, Yufeng Shi. Symmetrical solutions of backward stochastic Volterra integral equations and their applications. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 251-274. doi: 10.3934/dcdsb.2010.14.251 [5] Yushi Hamaguchi. Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems. Mathematical Control and Related Fields, 2021, 11 (2) : 433-478. doi: 10.3934/mcrf.2020043 [6] Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035 [7] Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115 [8] Jiongmin Yong. Forward-backward stochastic differential equations: Initiation, development and beyond. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022011 [9] Xiaomin Huang, Yanpei Jiang, Wei Liu. Freidlin-Wentzell's large deviation principle for stochastic integral evolution equations. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022091 [10] Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087 [11] Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 [12] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432 [13] Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043 [14] Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control and Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013 [15] Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7 [16] Anna Karczewska, Carlos Lizama. On stochastic fractional Volterra equations in Hilbert space. Conference Publications, 2007, 2007 (Special) : 541-550. doi: 10.3934/proc.2007.2007.541 [17] Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100 [18] Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002 [19] G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783 [20] M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42

2021 Impact Factor: 1.141