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June  2021, 11(2): 433-478. doi: 10.3934/mcrf.2020043

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

Yushi Hamaguchi

Department of Mathematics, Kyoto University, Kyoto 606–8502, Japan

Received  April 2020 Revised  August 2020 Published  June 2021 Early access  October 2020

In this paper, we study extended backward stochastic Volterra integral equations (EBSVIEs, for short). We establish the well-posedness under weaker assumptions than the literature, and prove a new kind of regularity property for the solutions. As an application, we investigate, in the open-loop framework, a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short). We show that the corresponding adjoint equations become EBSVIEs, and provide a necessary and sufficient condition for an open-loop equilibrium control via variational methods.

Keywords: Extended backward stochastic Volterra integral equation, backward stochastic Volterra integral equation, time-inconsistent stochastic recursive control problem, open-loop equilibrium control.
Mathematics Subject Classification: Primary: 93E20, 60H20; Secondary: 49K45, 47J30.
Citation: 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
References:
[1]

I. Alia, A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Math. Control Relat. Fields, 9 (2019), 541-570.  doi: 10.3934/mcrf.2019025.

[2]

I. Alia, F. Chighoub, N. Khelfallah and J. Vives, Time-consistent investment and consumption strategies under a general discount function, preprint, 2020, arXiv: 1705.10602.

[3]

T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.

[4]

B. Djehiche and M. Huang, A characterization of sub-game perfect equilibria for SDEs of mean-field type, Dyn. Games Appl., 6 (2016), 55-81.  doi: 10.1007/s13235-015-0140-8.

[5]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Math. Financ. Econ., 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.

[6]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[7]

Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Appl. Math. Optim., 2020. doi: 10.1007/s00245-020-09654-7.

[8]

Y. Hamaguchi, Time-inconsistent consumption-investment problems in incomplete markets under general discount functions, preprint, 2020, arXiv: 1912.01281.

[9]

C. Hernández and D. Possamaï, A unified approach to well-posedness of Type-Ⅰ backward stochastic Volterra integral equations, preprint, 2020, arXiv: 2007.12258.

[10]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, AMS, Providence, RI, 1957.

[11]

M. Hu, Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2 (2017), 1-20.  doi: 10.1186/s41546-017-0014-7.

[12]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear-quadratic control under constraint, preprint, 2020, arXiv: 1703.09415. doi: 10.1137/15M1019040.

[13]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.

[14]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279.  doi: 10.1137/15M1019040.

[15]

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

[16]

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

[17]

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.

[18]

Y. Shi, T. Wang and J. Yong, Optimal control problems of forward-backward stochastic Volterra integral equations, Math. Control Relat. Fields, 5 (2015), 613-649.  doi: 10.3934/mcrf.2015.5.613.

[19]

Y. Shi, J. Wen and J. Xiong, Backward doubly stochastic Volterra integral equations and their applications, J. Differential Equations, 269 (2020), 6492-6528.  doi: 10.1016/j.jde.2020.05.006.

[20]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, 23 (1973), 128-143. 

[21]

H. Wang, Extended backward stochastic Volterra integral equations, quasilinear parabolic equations, and Feynman–Kac formula, Stoch. Dyn., 2020. doi: 10.1142/S0219493721500040.

[22]

H. Wang, J. Sun and J. Yong, Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations, Appl. Math. Optim., 2019.

[23]

H. Wang and J. Yong, Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations, preprint, 2019, arXiv: 1911.04995.

[24]

T. Wang, Characterization of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I, Math. Control Relat. Fields, 9 (2019), 385-409.  doi: 10.3934/mcrf.2019018.

[25]

T. Wang, Equilibrium controls in time inconsistent stochastic linear quadratic problems, Appl. Math. Optim., 81 (2020), 591-619.  doi: 10.1007/s00245-018-9513-x.

[26]

T. Wang and J. Yong, Backward stochastic Volterra integral equations–representation of adapted solutions, Stochastic Process. Appl., 129 (2019), 4926-4964.  doi: 10.1016/j.spa.2018.12.016.

[27]

T. Wang and H. Zhang, Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions, SIAM J. Control Optim., 55 (2017), 2574-2602.  doi: 10.1137/16M1059801.

[28]

Q. Wei, J. Yong and Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM J. Control Optim., 55 (2017), 4156-4201.  doi: 10.1137/16M1079415.

[29]

W. Yan and J. Yong, Time-inconsistent optimal control problems and related issues, in Modeling, Stochastic Control, Optimization, and Applications, IMA Vol. Math. Appl., Springer, Cham, 2019,533–569. doi: 10.1007/978-3-030-25498-8_22.

[30]

J. Yong, Backward stochastic Volterra integral equations and some related problems, Stochastic Process. Appl., 116 (2006), 779-795.  doi: 10.1016/j.spa.2006.01.005.

[31]

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

[32]

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

[33]

J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271-329.  doi: 10.3934/mcrf.2012.2.271.

[34]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.

[35]

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

[36]

J. Zhang, Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2.

show all references

References:
[1]

I. Alia, A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Math. Control Relat. Fields, 9 (2019), 541-570.  doi: 10.3934/mcrf.2019025.

[2]

I. Alia, F. Chighoub, N. Khelfallah and J. Vives, Time-consistent investment and consumption strategies under a general discount function, preprint, 2020, arXiv: 1705.10602.

[3]

T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance Stoch., 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.

[4]

B. Djehiche and M. Huang, A characterization of sub-game perfect equilibria for SDEs of mean-field type, Dyn. Games Appl., 6 (2016), 55-81.  doi: 10.1007/s13235-015-0140-8.

[5]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Math. Financ. Econ., 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.

[6]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[7]

Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Appl. Math. Optim., 2020. doi: 10.1007/s00245-020-09654-7.

[8]

Y. Hamaguchi, Time-inconsistent consumption-investment problems in incomplete markets under general discount functions, preprint, 2020, arXiv: 1912.01281.

[9]

C. Hernández and D. Possamaï, A unified approach to well-posedness of Type-Ⅰ backward stochastic Volterra integral equations, preprint, 2020, arXiv: 2007.12258.

[10]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, AMS, Providence, RI, 1957.

[11]

M. Hu, Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2 (2017), 1-20.  doi: 10.1186/s41546-017-0014-7.

[12]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear-quadratic control under constraint, preprint, 2020, arXiv: 1703.09415. doi: 10.1137/15M1019040.

[13]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.

[14]

Y. Hu, H. Jin and X. Y. Zhou, Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279.  doi: 10.1137/15M1019040.

[15]

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

[16]

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

[17]

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.

[18]

Y. Shi, T. Wang and J. Yong, Optimal control problems of forward-backward stochastic Volterra integral equations, Math. Control Relat. Fields, 5 (2015), 613-649.  doi: 10.3934/mcrf.2015.5.613.

[19]

Y. Shi, J. Wen and J. Xiong, Backward doubly stochastic Volterra integral equations and their applications, J. Differential Equations, 269 (2020), 6492-6528.  doi: 10.1016/j.jde.2020.05.006.

[20]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, 23 (1973), 128-143. 

[21]

H. Wang, Extended backward stochastic Volterra integral equations, quasilinear parabolic equations, and Feynman–Kac formula, Stoch. Dyn., 2020. doi: 10.1142/S0219493721500040.

[22]

H. Wang, J. Sun and J. Yong, Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations, Appl. Math. Optim., 2019.

[23]

H. Wang and J. Yong, Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations, preprint, 2019, arXiv: 1911.04995.

[24]

T. Wang, Characterization of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I, Math. Control Relat. Fields, 9 (2019), 385-409.  doi: 10.3934/mcrf.2019018.

[25]

T. Wang, Equilibrium controls in time inconsistent stochastic linear quadratic problems, Appl. Math. Optim., 81 (2020), 591-619.  doi: 10.1007/s00245-018-9513-x.

[26]

T. Wang and J. Yong, Backward stochastic Volterra integral equations–representation of adapted solutions, Stochastic Process. Appl., 129 (2019), 4926-4964.  doi: 10.1016/j.spa.2018.12.016.

[27]

T. Wang and H. Zhang, Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions, SIAM J. Control Optim., 55 (2017), 2574-2602.  doi: 10.1137/16M1059801.

[28]

Q. Wei, J. Yong and Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM J. Control Optim., 55 (2017), 4156-4201.  doi: 10.1137/16M1079415.

[29]

W. Yan and J. Yong, Time-inconsistent optimal control problems and related issues, in Modeling, Stochastic Control, Optimization, and Applications, IMA Vol. Math. Appl., Springer, Cham, 2019,533–569. doi: 10.1007/978-3-030-25498-8_22.

[30]

J. Yong, Backward stochastic Volterra integral equations and some related problems, Stochastic Process. Appl., 116 (2006), 779-795.  doi: 10.1016/j.spa.2006.01.005.

[31]

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

[32]

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

[33]

J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields, 2 (2012), 271-329.  doi: 10.3934/mcrf.2012.2.271.

[34]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions, Trans. Amer. Math. Soc., 369 (2017), 5467-5523.  doi: 10.1090/tran/6502.

[35]

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

[36]

J. Zhang, Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2.

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