Mean-field Reflected Backward Stochastic Differential Equations With Stochastic Conditions

In this paper,we mainly study general mean-field reflected backward stochastic differential equations with stochastic conditions.We study these equations under different assumptions on their generators with stochastic Lipschitz condition,stochastic linear growth condition and stochastic monotone condition,respectively.Our research is divided into the following three parts.In the first part,we study the mean-field reflected backward stochastic differential equation with nonlinear resistance under stochastic Lipschitz condition.Firstly,we consider the following mean-field reflected backward stochastic differential equation with nonlinear resistance and a single barrier:We use the fixed point theorem to prove the existence and the uniqueness of the solution and give the corresponding comparison theorem and a prior estimates.Furthermore,the following mean-field doubly reflected backward stochastic differential equation with nonlinear resistance is considered:We prove the existence and the uniqueness of the solution of this equation and give the comparison theorem,in which the comparison theorem not only gives the comparison result of the solution Y,but also gives the comparison result of the continuous increasing processes K and R.Notice that the comparison theorem may not hold when the generator f depends on the distribution of Z or is non-increasing with respect to the law of Y.We give a counter-example to explain the comparison theorem.The results what we get in this part play an important role in the following ones.In the second part,based on the above results,we study the mean-field reflected backward stochastic differential equation with nonlinear resistance under stochastic linear growth condition.Due to the comparison theorem under stochastic Lipschitz will be used in the proof,the following mean-field reflected backward stochastic differential equation with nonlinear resistance with a single barrier is considered:By approximating the generator f by constructing convolution function sequences satisfying stochastic Lipschitz condition,we prove the existence of the maximum solution and the minimum solution of the above equation,and give the corresponding comparison theorem.Moreover,we study the following mean-field doubly reflected backward stochastic differential equation with nonlinear resistance:In the third part,we consider the case where the generator f does not depend on K and R,and satisfies the general growth condition and the stochastic monotone condition on y.Firstly,we consider the following mean-field reflected backward stochastic differential equation with a single barrier:We obtain the existence and the uniqueness of the solution of the reflected backward stochastic differential equation via using the penalization method and the fixed point theorem,and then use the fixed point theorem to extend it to equation(0-5),and give the corresponding comparison theorem.Furthermore,we derive the relevant results of the mean-field doubly reflected backward stochastic differential equation as follows;Finally,as applications,in Chapter 6,we give the applications of mean-field reflected backward stochastic differential equation in finance,that is,the connection with superhedging,American options and Dynkin games.