Mean-field Reflected Backward Stochastic Differential Equations With Discontinuous Barriers

Over the last years,researches on reflected backward stochastic differential equations have been very popular.A lot of researchers have been attracted by the numerous applications,e.g.,in optimal stopping time problems,American option pricing and mixed control problems.In 1997,El Karoui et al.[1]introduced in their pioneering work reflected backward stochastic differential equation with a continuous lower barrier.Cvitanic and Karatzas[2]extended the results of[1]to the case with two continuous barriers.Inspired by these works,many researchers have worked to relax the continuity assumption made on the barriers.In 2002,Hamadene[3]dealt with the situation of a lower barrier,which is right-continuous with left limits;in 2005,Peng and Xu[4]studied related problems with square integrable barriers.On the basis of previous work,and motivated by recent works in the mean-field theory which has found large applications after the seminal paper by Lasry and Lions[5],this paper mainly generalizes reflected backward stochastic differential equations with discontinuous barriers to the mean-field case,and studies the properties and comparison theorems for the solution when the generator satisfies a Lipschitz condition or,more generally,a continuity condition.The studies in this thesis can be divided into the following two parts.In the first part,we consider the following mean-field reflected backward stochastic differential equation with a single square integrable barrier:(?)To study this equation we made the following assumptions on the generator and the terminal value and the barrier:(A3.1)f(ω,t,δ0,0,0)∈HF2(0,T;R),where δ0 is the Dirac measure with mass at 0 ∈R1+d.(A3.2)f is Lipschitz in(μ,y,z),that is,there exists a constant L>0,such that for allμ,μ’∈P2(R1+d),y,y’∈R,z,z’∈Rd,|f(t,μ,y,z)-f(t,μ’,y’,z’)|≤L(W2(μ,μ’)+|y-y’|+|z-z’|),dtdP-a.e.(A3.3)The terminal value ζ belongs to L2(Ω,TF,P;R).(A3.4)S ∈HF2(0,T;R)satisfies E[(?)(St+)2]<∞,and ST≤ζ,P-a.s.Under the above assumptions,the existence and the uniqueness of the solution of equation(0-1)are proved by using a fixed point theorem.Moreover,we give the relevant estimates of the solution.Interested in a comparison result for solutions of(0-1)we give a counterexample to illustrate that the comparison theorem can not hold when the generator depends on the law of Z or is non-increasing with respect to the law of Y.Therefore,the comparison theorem for the solutions of the following equation is studied:(?)Our comparison result is obtained by using the Meyer-It(?) formula.In the following our comparison theorem plays a key role in proving the existence of solutions under only continuity conditions.Generalizing the frame of the study of(0-2)we replace the Lipschitz condition by a continuity one.More precisely,we suppose for f:(A4.1)Linear growth:There exists a constant L>0,such that,for all(μ,y,z)∈P2(R)×R× Rd，|f(t,μ,y,z)|≤L(1+W2(μ,δ0)+|y|+|z|),dtdP-a.e.(A4.2)Monotonicity in μ:For all θ1,θ2∈L2(Q,F,P,R)and(y,z)∈R×Rd,whenθ2≤θ1,P-a.s.,f(t,Pθ2,y,z)≤f)(t,Pθ1,y,z),didP-a.e.(A4.3)For a.s.ω∈Ω,f(ω,·,·,·,·)is continuous and there is an increasing continuity modulus ρ:R+→R+,ρ(0+)=0,such that,for all μ1,μ2∈P2(R),(y,z)∈R×Rd,|f(ω,tμ1,y,z)-f(ω,t,μ2,y,z)|≤ρ(W2(μ1,μ2)),dtdP-a.e.Under the above assumptions,the method of approximating the continuous function f by a sequence of Lipschitz coefficients constructed by inf-convolution proves the existence of a solution of equation(0-2),using the prior estimates,as well as the monotonic limit theorem in the mean-field case.Moreover,we consider the corresponding comparison theorem.In particular,under the above continuity conditions,another way of proving the existence of maximal and minimal solutions of equation(0-3)is given when the barrier is right-continuous with left limits.(?)We observe that in(0-3),the reflection condition has changed.This has as consequence that the existence of a solution of equation(0-3)can be obtained by using the properties of the Snell envelope.Further,by using the comparison theorem for the case of Lipschitz driving coefficient obtained by sup-convolution and inf-convolution,the existence of maximal and minimal solutions can be proved.In the second part,we consider the following mean-field reflected backward stochastic differential equation with two square integrable barriers,an upper and lower one:(?)The following assumptions are made about both barriers:(?)(A5.1)S,U∈HF2(0,T;R)satisfy E[(?)(St+)2]+E[(?)(Ut-)2]<∞,ST≤ζ≤UT,P-a.s.(A5.2)There exists a progress Qt=Q0+Kto-Dt0+∫0tZs0dBs,0≤t≤T,such that St≤Qt≤Ut,dtdP-a.e.,where Z0∈HF0(0,T;Rd):K0,D0∈AF2(0,T;R).Firstly,under the conditions(A3.1)-(A3.2)satisfied by the generator and(A5.1)(A5.2)satisfied by two barriers,we prove that for any terminal value ζ belongs to L2(Ω,FT,P;R),there exists a unique adapted solution of equation(0-4).We give the continuous dependence theorem of the solution and the comparison theorem for the solutions of the following equation:(?)(0-5)As an application,under the suitable conditions,we study the relationship between the solutions of mean-field reflected backward stochastic differential equations with two barriers which are right-continuous with left limits and Dynkin games as follows:(?)Secondly,we relax the Lipschitz assumption of the generator in the above problem to the continuity condition,and use indirect and direct penalization methods combining the generalized monotonic limit theorem under the mean-field case to prove that under the assumptions(A4.1)-(A4.3)and(A5.1)-(A5.2),for any terminal value ζ belongs to L2(Ω,FT,P;R),there exists a solution of equation(0-5).