Mean-field backward stochastic Volterra integral equations with jumps and their applications to stochastic optimal control problems

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Abstract

Motivated by the stochastic optimal control problem, in this paper we systematically study a class of mean-field backward stochastic Volterra integral equations with jumps (MF-BSVIEs with jumps, for short). More precisely, by a special kind of BSVIEs with jumps, we present the well-posedness of MF-BSVIEs with jumps in the sense of adapted M-solutions. For completeness, both the well-posedness of mean-field forward stochastic Volterra integral equations with jumps (MF-FSVIEs with jumps, for short) and the comparison theorem for MF-BSVIEs with jumps are directly proved. Based on this, by virtue of the duality principle, we are able to establish Pontryagin’s type maximum principle for optimal control problems of MF-FSVIEs with jumps. We illustrate our results with an application to the linear-quadratic optimal control problem of MF-FSVIEs with jumps.

Keywords

mean-field backward stochastic Volterra integral equation with jump / mean-field forward stochastic Volterra integral equation with jump / comparison theorem / stochastic optimal control / maximum principle

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References

[1] F. Yılmaz, H. Özbakan, G. Weber, Weak second-order conditions of Runge-Kutta method for stochastic optimal control problems, J. Optim. Theory Appl., 202 (2024), 497-517. https://doi.org/10.1007/s10957-023-02324-y

[2] B. Øksendal, A. Sulem, Applied Stochastic Control of Jump Diffusions , Springer, Berlin, 2009. https://doi.org/10.1007/978-3-030-02781-0

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

[4] J. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444. https://doi.org/10.1137/0314028

[5] E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61. https://doi.org/10.1016/0167-6911(90)90082-6

[6] Y. Hamaguchi, Global maximum principle for optimal control of stochastic Volterra equations with singular kernels: An infinite dimensional approach, J. Differ. Equations, 446 (2025), 113618. https://doi.org/10.1016/j.jde.2025.113618

[7] W. Ji, Y. Lei, Y. Wei, Y. Zhang, Open loop equilibrium strategy of time-inconsistent deterministic linear-quadratic problems, Electron. Res. Arch., 34 (2026), 2419-2432. https://doi.org/10.3934/era.2026111

[8] H. Liu, X. Lv, Hybrid control strategy for input-to-state stability of nonlinear systems with mixed impulses, Trans. Inst. Meas. Control, 48 (2026), 2796-2807. https://doi.org/10.1177/01423312251359367

[9] S. Luo, X. Li, Q. Wei, Infinite time horizon stochastic recursive control problems with jumps: Dynamic programming and stochastic verification theorems, SIAM J. Control Optim., 63 (2025), 796-821. https://doi.org/10.1137/24M1685080

[10] Z. Meng, S. Lyu, M. Zhang, X. Li, Q. Zhang, Sufficient and necessary conditions of near-optimal controls for a stochastic listeriosis model with spatial diffusion, Electron. Res. Arch., 32 (2024), 3059-3091. https://doi.org/10.3934/era.2024140

[11] Y. Youssri, M. Muttardi, A mingled tau-finite difference method for stochastic first-order partial differential equations, Int. J. Appl. Comput. Math., 9 (2023), 14. https://doi.org/10.1007/s40819-023-01489-4

[12] S. Tang, X. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475. https://doi.org/10.1137/S0363012992233858

[13] M. Quenez, A. Sulem, BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Processes Appl., 123 (2013), 3328-3357. https://doi.org/10.1016/j.spa.2013.02.016

[14] K. Andersson, A. Gnoatto, M. Patacca, A. Picarelli, A deep solver for BSDEs with jumps, SIAM J. Financ. Math., 16 (2025), 875-911. https://doi.org/10.1137/23M1615048

[15] R. Hafez, M. Abdelkawy, Y. Youssri, A. Biswas, Spectral Bernoulli-Gauss collocation scheme for pantograph-type Volterra integro-differential equations, Contemp. Math., 7 (2026), 1553-1571. https://doi.org/10.37256/cm.7220268911

[16] F. Wu, X. Li, X. Zhang, Stochastic linear quadratic optimal control problems with regime-switching jumps in infinite horizon, SIAM J. Control Optim., 63 (2025), 852-891. https://doi.org/10.1137/24M1645334

[17] Y. Youssri, R. Hafez, Chebyshev collocation treatment of Volterra-Fredholm integral equation with error analysis, Arab. J. Math., 9 (2020), 471-480. https://doi.org/10.1007/s40065-019-0243-y

[18] F. Zhang, Y. Dong, Q. Meng, Backward stochastic Riccati equation with jumps associated with stochastic linear quadratic optimal control with jumps and random coefficients, SIAM J. Control Optim., 58 (2020), 393-424. https://doi.org/10.1137/18M1209684

[19] M. Kac, Foundations of kinetic theory, in Proceedings of 3rd Berkeley Symposium on Mathematical Statistics and Probability , 3 (1956), 171-197.

[20] R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 37 (2009), 1524-1565. https://doi.org/10.1214/08-AOP442

[21] R. Buckdahn, J. Li, S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes Appl., 119 (2009), 3133-3154. https://doi.org/10.1016/j.spa.2009.05.002

[22] Y. Shen, T. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Anal., 86 (2013), 58-73. https://doi.org/10.1016/j.na.2013.02.029

[23] J. Li, Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs, Stochastic Processes Appl., 128 (2018), 3118-3180. https://doi.org/10.1016/j.spa.2017.10.011

[24] N. Li, Y. Wei, Q. Zhu, Stochastic linear-quadratic mean-field games of controls for delayed systems with jump diffusion, J. Optim. Theory Appl., 206 (2025), 66. https://doi.org/10.1007/s10957-025-02730-4

[25] J. Lin, Adapted solution of a backward stochastic nonlinear Volterra integral equation, Stochastic Anal. Appl., 20 (2002), 165-183. https://doi.org/10.1081/SAP-120002426

[26] T. Wang, M. Zheng, Singular backward stochastic Volterra integral equations in infinite dimensional spaces, J. Differ. Equations, 407 (2024), 1-56. https://doi.org/10.1016/j.jde.2024.06.002

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

[28] T. Wang, J. Yong, Comparison theorems for some backward stochastic Volterra integral equations, Stochastic Processes Appl., 125 (2015), 1756-1798. https://doi.org/10.1016/j.spa.2014.11.013

[29] T. Wang, Backward stochastic Volterra integro-differential equations and applications in optimal control problems, SIAM J. Control Optim., 60 (2022), 2393-2419. https://doi.org/10.1137/20M1371464

[30] T. Wang, J. Yong, Spike variations for stochastic Volterra integral equations, SIAM J. Control Optim., 61 (2023), 3608-3634. https://doi.org/10.1137/22M1522097

[31] B. Yang, T. Guo, Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games, J. Math. Anal. Appl., 565 (2027), 130992. https://doi.org/10.1016/j.jmaa.2026.130992

[32] T. Wang, J. Yong, Backward stochastic Volterra integral equations-Representation of adapted solutions, Stochastic Processes Appl., 129 (2019), 4926-4964. https://doi.org/10.1016/j.spa.2018.12.016

[33] H. Wang, J. Sun, J. Yong, Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations, Appl. Math. Optim., 84 (2021), 145-190. https://doi.org/10.1007/s00245-019-09641-7

[34] B. Yang, J. Wu, T. Guo, Well-posedness and regularity of mean-field backward doubly stochastic Volterra integral equations and applications to dynamic risk measures, J. Math. Anal. Appl., 535 (2024), 128089. https://doi.org/10.1016/j.jmaa.2024.128089

[35] Y. Shi, T. Wang, J. Yong, Mean-field backward stochastic Volterra integral equations, Discrete Contin. Dyn. Syst.-B, 18 (2013), 1929-1967. https://doi.org/10.3934/DCDSB.2013.18.1929

[36] A. Popier, Backward stochastic Volterra integral equations with jumps in a general filltration, ESAIM Probab. Stat., 25 (2021), 133-203. https://doi.org/10.1051/ps/2021006

[37] Z. Fu, S. Shen, J. Wu, Backward stochastic Volterra integral equations with jumps and some related problems, J. Differ. Equations, 435 (2025), 113240. https://doi.org/10.1016/j.jde.2025.113240

[38] J. Gong, T. Wang, Causal feedback strategies for controlled stochastic Volterra systems: A unified treatment, SIAM J. Control Optim., 63 (2025), 1950-1980. https://doi.org/10.1137/24M1668901

[39] Y. Shen, Q. Meng, P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica, 50 (2014), 1565-1579. https://doi.org/10.1016/j.automatica.2014.03.021

[40] L. Miao, Y. Chen, X. Xiao, Y. Hu, Anticipated backward stochastic Volterra integral equations with jumps and applications to dynamic risk measures, Acta Math. Sci., 43 (2023), 1365-1381. https://doi.org/10.1007/s10473-023-0321-2

[41] H. Chen, P. Shi, C. Lim, Stability analysis for nonautonomous impulsive hybrid stochastic delay systems, Syst. Control Lett., 187 (2024), 105785. https://doi.org/10.1016/j.sysconle.2024.105785

[42] M. De la Sen, A study of the stability of integro-differential Volterra-type systems of equations with impulsive effects and point delay dynamics, Mathematics , 12 (2024), 960. https://doi.org/10.3390/math12070960

[43] H. Hammad, M. De la Sen, Stability and controllability study for mixed integral fractional delay dynamic systems endowed with impulsive effects on time scales, Fractal Fractional, 7 (2023), 92. https://doi.org/10.3390/fractalfract7010092