Mean-field Backward Doubly Stochastic Differential Equations And Its Applications

Since Pardoux and Peng [22] introduced nonlinear backward stochastic differential equations(BSDEs)in 1990,many researchers have been attracted.In the paper [22], they proved the existence and the uniqueness of the solutions of nonlinear BSDEs under the assumption that the coefficient (or generator) is Lipschitz,the terminal conditionξis a square integrable random variable and (g(t,O.O))t∈[O.T] is a square integrable stochastic process.They also proved the important comparison theorem.Many researchers also work on how to reduce the Lipschitz assumption for the existence and the uniqueness of the solutions of BSDEs.There are lots of papers on the theory of BSDEs and its appli-cations,In 1994, Pardoux and Peng[23] introduced a new class of backward stochastic differential equations, which is called backward doubly stochastic differential equations (in short.BDSDE). They proved that if the coefficients f and g are Lipschitz, the terminalξis a square integrable random variable and (f(t,O,O))t∈[O,T], (g(t,O,O))t∈[O,T] are both square integrable stochastic process-es, the BDSDE has a unique solution. Shi Yufeng [29] obtained the comparison theorem with the above assumptions, and got the minimal solution of the BDS-DE when the coefficient f is continuous. In 2009, Buckdahn,Djehiche.Li and Peng[1]introduced a new kind of BSDE.that is.mean-field Backward Stochas-tic differential equations(in short.mean-field BSDE). After that, they obtained the existence and the uniqueness of the solutions of the mean-field BSDEs un-der the assumption that the coefficient f is Lipschitz, the terminal conditionξis a square integrable random variable and (f(t,O,O,O,O))t∈[O,T] is a square integrable stochastic process.and also the according comparison theorem and converse comparison theorem in[2]. Inspired by these works, we will study the mean-field BDSDEs in this pa-per. Fistly,we will study the one dimensional mean-field BDSDEs, when the co-efficients f, g are Lipschitz, the terminal conditionξis a square integrable ran-dom variable and (f(t.,O,O,O,O))t∈[O,T],(g(t,O,O,O,O))t∈[O,T] are both square integrable stochastic processes.We will prove the existence and the uniqueness of the solution, and also a generalized comparison theorem. Later,we will s-tudy the mean-field BDSDEs with non-Lipschitz coefficients f and continuous coefficients f respectively.In the end,we will work on the multi-dimensional mean-field BDSDEs, prove the comparison theorem, and the existence of the solution under the continuous coefficients.