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Controlled mean-field backward stochastic differential equations with jumps involving the value function

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Abstract

This paper discusses mean-field backward stochastic differential equations (mean-field BSDEs) with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled with the value function of the associated control problem. The authors first prove the existence and the uniqueness as well as a comparison theorem for the above two types of BSDEs. For this the authors use an approximation method. Then, with the help of the notion of stochastic backward semigroups introduced by Peng in 1997, the authors get the dynamic programming principle (DPP) for the value functions. Furthermore, the authors prove that the value function is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman (HJB) integro-partial differential equation, which is unique in an adequate space of continuous functions introduced by Barles, et al. in 1997.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University, Weihai, Weihai, 264209, China

    Juan Li & Hui Min

Authors
  1. Juan Li
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  2. Hui Min
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Correspondence to Hui Min.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 11171187, 11222110, Shandong Province under Grant No. JQ201202, Program for New Century Excellent Talents in University under Grant No. NCET-12-0331, 111 Project under Grant No. B12023.

This paper was recommended for publication by Editor HONG Yiguang.

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Li, J., Min, H. Controlled mean-field backward stochastic differential equations with jumps involving the value function. J Syst Sci Complex 29, 1238–1268 (2016). https://doi.org/10.1007/s11424-016-4275-5

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  • DOI: https://doi.org/10.1007/s11424-016-4275-5

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