In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.
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The work has been supported in part by the NSF of P.R.China (11871037; 11222110), Shandong Province (JQ201202), NSFC-RS (11661130148; NA150344), 111 Project (B12023).
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Li, J., Xing, C. & Peng, Y. Comparison Theorems for Multi-Dimensional General Mean-Field BDSDES. Acta Math Sci 41, 535–551 (2021). https://doi.org/10.1007/s10473-021-0216-z
- Backward doubly stochastic differential equations
- multi-dimensional comparison theorem
- continuous condition
2010 MR Subject Classification