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Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds
Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems
Department of Mathematics, Kyoto University, Kyoto 606–8502, Japan |
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.
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A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Math. Control Relat. Fields, 9 (2019), 541-570.
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Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.
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Y. Hamaguchi, Small-time solvability of a flow of forward-backward stochastic differential equations, Appl. Math. Optim., 2020.
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Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2 (2017), 1-20.
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Time-inconsistent stochastic linear-quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.
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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.
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J. Lin,
Adapted solutions of a backward stochastic nonlinear Volterra integral equation, Stoch. Anal. Appl., 20 (2002), 165-183.
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Solvability of general backward stochastic Volterra integral equations, J. Korean Math. Soc., 49 (2012), 1301-1321.
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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.
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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 |
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H. Wang and J. Yong, Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations, preprint, 2019, arXiv: 1911.04995. Google Scholar |
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T. Wang,
Characterization of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I, Math. Control Relat. Fields, 9 (2019), 385-409.
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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.
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[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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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