Probabilistic Interpretation For Sobolev Solution Of A New Type Of McKean-Vlasov Partial Differential Equations

Since 2009,Buckdahn,Djehiche,Li and Peng[1]firstly introduced mean-field back-ward stochastic differential equations,lots of attention have been attracted.Buckdahn,Djehiche,Li and Peng[1]had studied the mean-field backward stochastic differential equations(in short,MFBSDEs)and the relationship between the solution of the MFB-SDEs and the viscosity solution of the related partial differential equations.In this paper,we mainly study the weak solution of a new type of McKean-Vlasov PDEs,i.e.,the Sobolev solution,which is different from the viscousity solution.It does not need to depend on the comparison theorem so that the coefficient of the BSDE we study is able to depend on z.This paper mainly studies the following equations:The McKean-Vlasov SDE:This paper can be divided into three parts:The first part mainly studies the exis-tence and uniqueness of the solution for the new type of McKean-Vlasov PDE(3)and it uses the following assumptions:Assumption 3.1:(A1)(i)function b and a are Lipschitz in x,x.(ii)b(·,0,0)and ?(·,0,0)are F-progressively measurable continuous functions and there exists a constant l>0,such that for all 0? t? T;x,x?Rd,|b(t,x,x)| + |?(t,x,x)|?l(1 +|x|),a.s.(A2)(i)? is a F(?)B(Rd)-medasuralble vdariable,f is a measurable process,such that f(·,x,x,y,y,z,z)is F-adapted,for all(x,x,y,y z,z)?Rd ×Rd×Rn×Rn × Rnxd × Rn×d and f(t,x,x,0,0,0,0)?HF2|(0,T;Rn).(?)f satisfies the Lipschitz condition in x,x,y,y,z,z.(?)f and ? satisfy the linear growth condition,i.e.,there exists a constant c>0,such that a.s.for all x,x?Rd,|f(t,x,x,0,0,0,0)| + |?(x,x)| ?c(1 + |x| + |x|).(?)G,?,?,k:Rd?Rd,A:Rd?Rd,?:Rn×d ?Rn×d are Lipschitz continuous functions.(A3)Given(x,y,z)?Rd × Rn×Rn×d,for all s?[0,T],(x,y,z)?f(s,x,x,y,y,z,z)?Cb3,3,3(Rd×Rn×Rn×d,Rn).(A4)b ? Cb 1,3,3([0,T]×Rd ×Rd,Rd)and ??Cb1,3,3([0,T]×Rd×Rd,Rd×d).At the same time,we can define the value function that u(t,x)= Ytt,x.So under the Assumption 3.1,the classical solution of the McKean-Vlasov PDE(3)is unique,and it has the following respectation:Yst,x=u(s,Xst,x),Zst,x?Dxu(s,Xst,x)?(s,E(?(Xs0,x0)],Xst,x).Using the method of stochastic inverse flow,the result of equivalence of norms and test function,we finally obtain that under the Assumption 3.1-(A2)and(A4),the Sobolev solution of the McKean-Vlasov PDEs(3)is unique.The second part:Firstly,we can study the solution of the MFBSDE(2)with globally monotone coefficient,which is unique under the following assumptions:Assumption 4.1:(H1)for any given(?,t),f(?,t,.,.,.,.)is continuous;(H2)there exists a process ft? HF2{(0,T;R)and a constant L>0,such that|f(t,y,z,y,z)|?ft+ L(|y|+ |z| + |y| + |z|).Secondly,we study that the solution of the new type of MFBSDE(2)with locally monotone coefficients is unique under the following assumptions:Assumption 4.2:Therefore,we can obtain that under the Assumption 4.1-(H1),Assumption 4.2,and the following function is maintained,where ? is any fixed constant,s.t.,0<?<1-2?,the MFBSDE(2)with locally monotone coefficient has unique solution(Y,Z).Thirdly,when we get the above conclusions,we begin to study the existence and uniqueness of the Sobolev solution for the related the McKean-Vlasov PDE(3).At first,we know that under the following assumptions,the value function u(t,x):=Ytt,x is a unique Sobolev solution for the McKean-Vlasov PDE(3)with globally monotone coefficient.Assumption 4.3:(B1)b and ? satisfy the Assumption 3.1-(A1),(A4).(B2)f and ? satisfy the Assumption 3.1-(A2)-(i)(iii)and Assumption 3.1-(A2)-(iv)is satisfied,??L2(Rd,?(x)dx);(B3)for all 0 ?t?T,x1,x2,x1,x2 ? Rn,y,y1,y2,y,y1,y2 ?Rn,,z,z1,z2,z,z1,z2?Rn×d,there exist constants C>0,?1,?2 ? R,such that|?(x1,x1)-?(x2,x2)|2 +|f(t,x1,x1,y,y,z1,z1)-f(t,x2,x2,y,y,z2,z2)|2?C(|x1-x2|2 +|x1-x2|2+|z1-z2|2+ |z1-z2|2).(y1-y2)(f(t,x1,x1,y1,y1,z)-f(t,x2,x2,y2,y2,z,z))??1(y1-y2)(y1-y2)+?2|y1-y2|2.(B4)|f(t1,(?),x,(?),y,(?),z)|?|f(t,(?),x,0,0,0,0)| +K(|y|+|y|+|z|+|z|),f(t,x,x,0,0,0,0)?L2(Rd,?(x)dx)and it satisfies the linear growth.And then we can obtain the existence and uniqueness of the Sobolev solution for the McKean-Vlasov PDE(3)under the locally monotone assumption.The related locally monotone assumption is the following:Assumption 4.4:(B3')for all N ? N,there exists constants LN>0,?N,?N?R,such that for x1,x1,x2,x2 ? Rd,yl,y1,y2,Y2?Rn,z1,z1,z2,z2 ?R×d,it satisfies that |y1|,|y1|,|y2|,|y2|,|z1|,|z1|,|z2|,|z2|? N,we have|?(x1,x1))-?(x2,x2)|2 + f(t,x1,x1,y1,y1,z1,z1)-f(t,x2,x2,y1,y1,z2;z2)|2?LN(|x1-x2|2+|x1-x2|2+|z1-z2|2+|z1-z2|2).(y1-y2)(f(t,x1,x1,y1,y1,z1,z1)-f(t,x1,x1,y2,y2,z1,z1))??N(y1-y2)(y1-y2)+?N|y1-y2|2.(B4')there exists constants K>0 and 0???1,such that|f(t,x,x,y,y,z,z)|? K(1 + |y|? + |z|? + |y|? +|z|?),for all t,x,x,y,y,z,z.Therefore,we finally get that under the Assumption 4.3-(B1),(B2),Assumption 4.4 and the formula(4)is satisfied,the McKean-Vlasov PDEs(3)with the locally monotone coefficient has a unique Sobolev solution.