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

Bixuan Yang; Bixuan Yang

doi:10.3934/era.2026321

Electronic Research Archive

2026, Volume 34, Issue 10: 7448-7476. doi: 10.3934/era.2026321

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Research article

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

Bixuan Yang ^,

School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, China

Received: 30 June 2026 Revised: 21 August 2026 Accepted: 26 August 2026 Published: 28 August 2026

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.
Citation: Bixuan Yang. Mean-field backward stochastic Volterra integral equations with jumps and their applications to stochastic optimal control problems[J]. Electronic Research Archive, 2026, 34(10): 7448-7476. doi: 10.3934/era.2026321

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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.

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