Abstract
We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.
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Acknowledgements
The authors would like to thank the referees for their constructive comments and suggestions which helped us to improve this paper. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11871309, 11671229, 71871129, 11371226, 11301298), the National Key R&D Program of China (Grant No. 2018 YFA0703900), the Natural Science Foundation of Shandong Province (No. ZR2019MA013), the Special Funds of Taishan Scholar Project (No. tsqn20161041), and the Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions.
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Zhu, Q., Su, L., Liu, F. et al. Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games. Front. Math. China 15, 1307–1326 (2020). https://doi.org/10.1007/s11464-020-0889-y
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DOI: https://doi.org/10.1007/s11464-020-0889-y
Keywords
- Non-zero sum stochastic differential game
- mean-field
- backward doubly stochastic differential equation (BDSDE)
- Nash equilibrium point
- maximum principle