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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 & 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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[8]

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

[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. Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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

[21]

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

[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. Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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. Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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