L~p-Solutions For Mean-field Reflected Backward Stochastic Differential Equations With Stochastic Lipschitz Coefficients

This paper is devoted to the Lp-solutions of three different types of reflected backward stochastic differential equations under a stochastic Lipschitz assumption,where p ∈(1,2].In the first part,this paper studies the existence and the uniqueness of Lp-solution of reflected backward stochastic differential equations in a classical form.We prove namely a comparison theorem under a stochastic Lipschitz condition.The second part extends the above studies to the general case in which the generator depends not only on the solution processes but also on the joint law of the solution processes.We prove in particular the existence and the uniqueness of the Lp-solution of such general mean-field reflected backward stochastic differential equation and the associated comparison theorem when the generator does not depend on the law of Z.Finally,we consider the properties of the Lp-solution when the reflecting barrier depends on the expectation of the solution component Y,that is,the properties of the Lp-solution of mean-field backward stochastic differential equation with mean reflection.Let us be a bit more precise.Part Ⅰ:In this part we consider the following type of the classical reflected backward stochastic differential equation but with a generator f satisfying a stochastic Lipschitz condition,i.e.,there are two nonnegative F-adapted processes μ=(μt)t∈[0,T],γ=(γt)t∈[0,T]such that,for all t ∈[0,T],y1,y2 ∈ R,z1,z2 ∈ Rd,|f(t,y1,z1)-f(t,y2,z2)| ≤ μt|y1-y2|+γt|z1-z2|,P-a.s.The terminal condition ξ and the barrier process L satisfy suitable assumptions of p-integrability.The existence and the uniqueness of the Lp-solution of the above equation are proved by using the penalization method.As illustration an example in a special case is given.Finally,a comparison theorem of this type of equation is obtained for one-dimensional case.Part Ⅱ:Here we consider the generator not only depends on the solution processes but also their law,that is,we study the mean-field reflected backward stochastic differential equation Under the condition that the generator f satisfies a stochastic Lipschitz condition and the terminal ξ as well as barrier process L satisfy a suitable p-integrability assumption,we give several a priori estimates of the solution.The existence and the uniqueness of the Lp-solution of the equation are proved for the both cases p=2 and p ∈(1,2),separately.Finally,we give a comparison theorem when the generator does not depend on the law of Z.Part Ⅲ:In this part we investigate the following mean-field backward stochastic differential equation with mean reflection where we suppose again that the generator f satisfies a suitable stochastic Lipschitz condition,and the terminal ξ and barrier process L are supposed to satisfy a suitable p-integrability condition.We study the existence and the uniqueness of the deterministic flat Lp-solution of the equation when the continuous increasing process K is deterministic and the following condition holds true:f0T E[l(t,Yt)]dKt=0.Last but not least,in Chapter 6 we study an application to the pricing of American options and to the mean-field BSDEs with risk measure reflection.