General mean-field BDSDEs with continuous coefficients

In this paper we consider general mean-field backward doubly stochastic differential equations (BDSDEs), i.e., the generator of our mean-field BDSDEs depends not only on the solution but also on the law of the solution. The first part of the paper is devoted to the existence and the uniqueness of solutions for general mean-field BDSDEs under the Lipschitz conditions. We emphasize that in the mean-field BDSDEs we study, from Pardoux-Peng's pioneering work [17] the expected condition that the Lipschitz constant of the coefficient for the backward integral with respect to z is strictly less than 1 is considerably weakened for the Z-component of the law variable of the coefficient. We also establish higher order moment estimates and obtain a comparison theorem of one dimensional general mean-field BDSDEs. Furthermore, we study an existence theorem and a comparison theorem for general mean-field BDSDEs with a continuous coefficient f. Finally, we investigate uniqueness theorems for general mean-field BDSDEs with a uniformly continuous coefficient f.