General Mean-field Backward Doubly Stochastic Differential Equations And Their Applications

Pardoux and Peng[82]published their pioneering paper on nonlinear backward stochastic differential equations(BSDEs for short)in 1990.Since then the theory of BSDEs has become a powerful tool in stochastic control(refer,for instance,to[87]),mathematical finance(see,for example,[34]),partial differential equations(refer among others,to[84,89])and other topics.Among its many different applications,one major objective of researchers has been to generalize the original Lipschitz assumption on the driving coefficient in order to guarantee the existence and the uniqueness of a solution.In 1997,Lepeltier and San Martin[59]studied the existence of a solution for BSDEs with only continuous generator.Moreover,Jia[51]studied the uniqueness theorem for BSDEs with continuous coefficient.In 1994,Pardoux and Peng[83]extended the notion of BSDEs to that of backward doubly stochastic differential equations(BDSDEs for short),involving both a standard(forward)stochastic integral driven by a Brownian motion W and a backward stochastic integral governed by a Brownian motion B.For other recent developments on BDSDEs,we refer to the works of[18,19,41,65,95,94]and references therein.The history of mean-field stochastic differential equations(SDEs),known as McKeanVlasov SDEs,can be traced back to the work by Kac[52]in 1956.Since that time,thanks to their numerous applications,mean-field problems have been studied by many authors.In particular,Lasry and Lions[58]extended the application areas for mean-field problems to Economics,Finance and game theory.In recent years,a new impetus to this research was given by the courses given by P.L.Lions at Collège de France[69](also refer to the notes by Cardaliaguet[21]),in which the authors introduced the notion of differentiability with respect to probability measure of a function defined on P2(Rd).Inspired by[69],many works adopt this definition,for example,Buckdahn,Li,Peng and Rainer[17],Li[62],Carmona and Delarue[26],Cardaliaguet[22],Hao and Li[44].In this paper we consider general mean-field backward doubly stochastic differential equations(BDSDEs),i.e.,the generator of our mean-field BDSDEs depends not only on the solution but also on the law of the solution.This paper mainly consider two topics:the first one is to investigate,under suitable assumptions on both of the driving coefficients,existence theorems,uniqueness theorems as well as comparison theorems for general mean-field BDSDEs;and the second one is to study general mean-field BDSDE and the related backward SPDEs of mean-field type.In what follows we will introduce the content and structure of this thesis.In Chapter 1 we give the main problems and their research backgrounds studied in Chapter 2 to Chapter 4.In Chapter 2 we extend BDSDEs introduced by Pardoux and Peng[83]to the meanfield case,and consider the general mean-field BDSDEs.That is the generator of our mean-field BDSDEs depends not only on the solution but also on the joint law of the solution.The first part of Chapter 2 is devoted to the existence and the uniqueness of solutions for general mean-field BDSDEs under the Lipschitz conditions.We emphasise that unlike the pioneering paper on BDSDEs by Pardoux and Peng[83],we handle a driving coefficient in the backward integral of the BDSDE for which the Lipschitz assumption with respect to(w.r.t.)the law of the solution is sufficient,without assuming that this Lipschitz constant is small enough.Furthermore,we also establish higher order moment estimates and obtain a comparison theorem of one dimensional general meanfield BDSDEs.In the second part of Chapter 2,with the help of the comparison result for the Lipschitz case we study an existence theorem and a comparison theorem for general mean-field BDSDEs with a continuous coefficient f.Finally,we investigate uniqueness theorems for general mean-field BDSDEs with a uniformly continuous coefficient f.From[68]we know that when the generator of our BDSDEs does not depend on the law of the solution process,the assumption that f is uniformly continuous w.r.t.y and Lipschitz w.r.t.z does not guarantee the uniqueness of the solution of BDSDE.In particular,for the case when f is not Lipschitz in z but satisfies the condition of uniform continuity w.r.t.z it is difficult to get the uniqueness theorem.With a new method we succeed to prove the uniqueness theorem for our mean-field BDSDE when f is uniformly continuous w.r.t.(?,y,z)The novelty of this chapter:We are the first to give the result without any restriction on the Lipschitz constant for a Z-component in the backward integral of a BDSDE,and to establish higher order moment estimates.Moreover,we study the existence theorem for general mean-field BDSDEs with an only continuous drift coefficient,and based on a new method we obtain uniqueness theorems for general mean-field BDSDEs with a uniformly continuous coefficient.This Chapter is based on:J.Li,C.Xing.General mean-field BDSDEs with continuous coefficients.Journal of Mathematical Analysis and Applications,506(2),125699,2022.Inspired by the work in Chapter 2,the main objective of Chapter 3 concerns the study of the comparison theorem for multi-dimensional mean-field BDSDEs and its application to the study of such BDSDEs with only a continuous drift coefficient.The first part of Chapter 3 is devoted to the study of the comparison theorem for multidimensional mean-field BDSDEs under Lipschitz conditions on the coefficients.As a byproduct of this,a comparison result for multi-dimensional mean-field BSDEs is deduced.Furthermore,we extend the technique first presented by Li,Liang and Zhang[63],for when the generator f is multi-dimensional.In other words,by approximating continuous mean-field coefficients by Lipschitz ones,we use the results of the first part to obtain the existence of a solution and also a comparison theorem for multi-dimensional BDSDEs,and also for multi-dimensional BSDEs with only a continuous drift coefficient satisfying a linear growth condition.The novelty of this chapter:We obtain the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions,and construct a new type of sequence of Lipschitz functions to approximate the function f satisfying continuous condition.With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth,and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.This Chapter is based on:J.Li,C.Xing,Y.Peng.Comparison theorems for multi-dimensional general meanfield BDSDEs.Acta Mathematica Scientia,41(2),535-551,2021.In Chapter 4 we study general mean-field BDSDE associated with a forward diffusion SDE as well as the related nonlocal semi-linear backward stochastic partial differential equations(SPDEs for short)of mean-field type,i.e.,BDSDEs whose driving coefficients also depend on the joint law of the solution process as well as the solution of an associated mean-field forward SDE.Using the splitting method introduced in[17]for mean-field SDEs,we study the unique solutions(Yt,?,Zt,?)and(Yt,x,P?,Zt,x,P?)of our BDSDE.Under suitable regularity assumptions on the coefficients we investigate the first and the second order derivatives of the solution(Yt,x,P?,Zt,x,P?)w.r.t.x,the derivative(??Yt,x,P?(y),??Zt,x,P?(y))of the solution process w.r.t.the measure P?,and the derivative of(??Yt,x,P?(y),??Zt,x,P?(y))w.r.t.y.However,as the parameters(x,P?)and(x,P?,y)run an infinite-dimensional space,unlike Pardoux and Peng,we cannot apply Kolmogorov's continuity criterion to the value function V(t,x,P?):=Ytt,x,P?,while in the classical case studied in[83]the value function V(t,x):=Ytt,x can be shown to be of class C1,2([0,T]×Rd),we have for our value function V(t,x,P?)and its derivative ??V(t,x,P?,y)only the L2-differentiability w.r.t.x and y,respectively.However,we have to apply the(mean-field)It? formula to V(s,Xst,x,P?,PXst,?).To overcome that we have for V only L2-regularity,we introduce the deterministic function?(t,x,P?):=E[V(t,x,P?)·?],for suitable ?? L?(F;R),and for ? we can show that it is a Cb1,2,2-function.Applying the(mean-field)Ito formula to ?(s,Xst,x,P?,PXst,?)will be crucial for the proof that V is a solution of backward SPDE,a classical one but with the derivatives w.r.t.x and y in L2-sense.Using a similar idea,we extend the classical mean-field It? formula to the solutions of BDSDEs.The novelty of this chapter:We extend the classical mean-field It? formula introduced by Buckdahn,Li,Peng and Rainer[17]to smooth functions of solutions of mean-field BDSDEs.Then we give the probabilistic interpretation of the associated nonlocal semi-linear backward SPDEs of mean-field type.This Chapter is based on:R.Buckdahn,J.Li,C.Xing.Mean-field BDSDEs and associated nonlocal semilinear backward stochastic partial differential equations.Preprint.In Chapter 5 we give a summary and put forward the possible research direction in the future.This dissertation include four chapters mentioned above,we now give an outline of the structure and the main conclusion.Chapter 1 Introduction;Chapter 2 General mean-field BDSDEs with continuous coefficients;Chapter 3 Comparison theorems for multi-dimensional mean-field BDSDEs;Chapter 4 Mean-field BDSDEs and associated backward stochastic partial differential equations;Chapter 5 Summary and future works.Chapter 2:We consider the existence and the uniqueness of solutions for general mean-field BDSDEs under the Lipschitz conditions.We also establish higher order moment estimates and obtain a comparison theorem of one dimensional general mean-field BDSDEs.Furthermore,we study an existence theorem and a comparison theorem for general mean-field BDSDEs with a continuous coefficient f.Finally,we investigate uniqueness theorems for general mean-field BDSDEs with a uniformly continuous coefficient f.Given ??L2(?,FT,P;Rk),we consider the following general mean-field BDSDEs:(?)(0.0.7)where the integral with respect to B is the Ito backward one,denoted by dB.Theorem 2.2.1.Under the assumptions(H2.2.1)-(H2.2.6),the general mean-field BDSDE(0.0.7)has a unique solution(Y,Z)?S2(0,T;Rk)×H2(0,T;Rk×d).It should be noticed that in the assumption(H2.2.2)we assume ?1,?2>0 and 00 depending only on the Lipschitz constant of f,g and h,?1 and ?2,such thatProposition 2.2.3.We assume g satisfies(H2.2.1)and(H2.2.2),f satisfies(H2.2.3)and(H2.2.4),and h satisfies(H2.2.5)and(H2.2.6).Moreover,we suppose that,for some p?2,Cp(?1+?2)p/2<1.Here Cp:=2p-1Cp*((p/(p-1)p+1)Cp',Cp*=2-p-23pp3p+2p/2,Cp':=(p/(p-1))p3p-1(2Cp5p-1?(6p3)p5p/2-1),C is the Lipschitz constant in(H2.2.2),(H2.2.4)and(H2.2.6).(Y,Z)is the solution of the mean-field BDSDE(0.0.7).Then there exists Cp?R+ only depending on the Lipschitz constant C of the coefficients and on p,such thatWe also prove another important result,the comparison theorem.From the Examples 3.1 and 3.2 in[16]we know that if the driver f depends on the law of Z or is non-increasing with respect to the law of Y we usually don't have the comparison theorem.On the other hand,if g and h depend on the law of Z or depend on the law of Y we also don't have the comparison theorem.Let k=1,we consider now the general mean-field BDSDE as follows(?)(0.0.9)Theorem 2.2.3.(Comparison Theorem)Let g(s,?,y,z)satisfy the assumption(H2.2.1),(H2.2.2),fi=fi(s,?,?,y,z),i=1,2,satisfy(H2.2,3),one of the fi satisfies(H2.2.4),one of the fi.satisfies(A2.2.2),and let ?1,?2?L2(?,FT,P;R),(Y1,Z1)and(Y2,Z2)be the solutions of mean-field BDSDE(0.0.9)with the data(?1,f1,g)and(?2,f2,g),respectively.Moreover,if ?1??2,P-a.s.,and f1(s,?,y,z)?f2(s,?,y,z),dsdP-a.e.,for all(?,y,z),then Ys1?Ys2,for all s?[0,T],P-a.s.With the help of the comparison result,we obtain an existence theorem and a comparison theorem for one dimensional mean-field BDSDEs(0.0.9)with a continuous coefficient f.Theorem 2.3.1.Assume that f:[0,T]×?×P2(R)×R×Rd?R and g:[0,T]×?×R×Rd?Rl satisfy(H2.2.3),(H2.3.1)-(H2.3.3)and(H2.2.1),(H2.2.2),respectively.Then,for all??L2(?,FT,P;R),the mean-field BDSDE(0.0.9)has a solution(Y,Z)?S2(0,T;R)×H2(0,T;Rd).Moreover,there is a minimal solution(Y,Z)of(0.0.9)and a maximal one(Y,Z).Theorem 2.3.2.(Comparison Theorem)Assume ?1,?2?L2(Q,FT,P),g satisfies(H2.2.1)and(H2.2.2),and f1 satisfies(H2.2.3),(H2.3.1)-(H2.3.3),f2 satisfies(H2.2.3).Let(Y1,Z1)be the minimal solution of mean-field BDSDE(0.0.9)with the data(?1,f1,g);let(Y2,Z2)?S2(0,T;R)× H2(0,T;Rd)be a solution of meanfield BDSDE(0.0.9)with the date(?2,f2,g).If ?1??2,P-a.s.,and f1(s,?,y,z)?f2(s,?,y,z),dsdP-a.e.,for all(?,y,z),then Yt1?Yt2,for all t?[0,T],P-a.s.Furthermore,we prove the uniqueness of the solution even when f depends on the law and is not Lipschitz w.r.t.y.Theorem 2.3.3.Assume(H2.2.1)-(H2.2.3),(H2.3.1)and(H2.3.4).? is concave and satisfies ?0+[?((?))]-2dx=+?.f is Lipschitz in z.Then there exists a unique solution(Y,Z)?S2(0,T;R)×H2(0,T;Rd)which solves mean-field BDSDE(0.0.9).If f is not Lipschitz in z and satisfies only(H2.3.5)w.r.t.z,mean-field BDSDE(0.0.9)may has a unique solution too as follows.Theorem 2.3.4.Let g(s,y,z)satisfy the assumption(H2.2.1)and(H2.2.2),f(s,?,y,z)satisfy the assumption(H2.2.3),(H2.3.1),(H2.3.4)and(H2.3.5).? is concave,and satisfies ?0+[?(x)]-1dx=+?.??D1,?(D[·]=DW[·]is the Malliavin derivative w.r.t.W).Moreover,we suppose(?)g(s,y,z)=g(s)y,where g(s)is Fs,TB-measurable,and |g(s)|?C,s?[0,T].(?)f does not depend on ???.Then there exists a unique solution(Y,Z)?S2(0,T;R)×H2(0,T;Rd)which solves mean-field BDSDE(0.0.9).Chapter 3:We concern the study of the comparison theorem for multidimensional mean-field BDSDEs and its application to the study of such BDSDEs with a continuous coefficient.We first consider the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions.With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth,and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.Given ?1,?2?L2(?,FT,P;Rk),we consider the comparison theorem of the k-dimensiona1 mean-field BDSDEs as follows:(?)(0.0.10)(?)(0.0.11)Theorem 3.2.1.Let ?1,?2 satisfy the assumption(H3.2.1),g(t,?,y,z)the assumptions(H2.2.1)and(H3.2.5),and let fi=fi(t,?,?,y,z),i=1,2,be two drivers satisfying the assumptions(H2.2.3)and(H3.2.2).Moreover,we suppose that one of the fi satisfies assumption(H3.2.3),and one of the fi satisfies assumption(H3.2.4).Denote by(Y1,Z1)and(Y2,Z2)the solution of mean-field BDSDE(0.0.10),(0.0.11)with the data(?1,f1,g)and(?2,f2,g),respectively.Then Ys1?Ys2 for all s?[0,T],P-a.s..As an application of Theorem 3.2.1,we consider the following multi-dimensional mean-field BDSDE with continuous coefficients:(?)(0.0.12)We extend the technique first presented by Li,Liang and Zhang[63],for when the generator f is multi-dimensional,Theorem 3.3.1.Assume that f=f(s,?,?,y,z):[0,T]×?×P2(Rk)×Rk×Rk×d?Rk and g=g(s,?,y,z):[0,T]×?×Rk×Rk×d?Rk×l are measurable and {Ft?adapted.Moreover,g satisfies(H2.2.1)and(H3.2.5),and f(H2.2.3),(H3.3.1)(H3.3.5).Then,for all ??L2(?,FT,P;Rk),the mean-field BDSDE(0.0.12)has an adapted solution(Y,Z)? S2(0,T;Rk)×H2(0,T;Rk×d).Moreover,(Y,Z)is a minimal solution of(0.0.12),in the sense that for any other solution(Y,Z)of(0.0.12),we have Ys?Ys,s?[0,T],P-a.s..Theorem 3.3.2.(Comparison Theorem)Assume that ?1,?2?L2(Q,FT,P),g satisfies(H2.2.1))and(H3.2.5).f1 satisfies(H2.2.3),(H3.3.1)-(H3.3.5),f2 satisfies(H2.2.3),and let(Y1,Z1)be the minimal solution of the mean-field BDSDE(0.0.12)with the data(?1,f1,g)Let(Y2,Z2)?S2(0,T;Rk)×H2(0,T;Rk×d)be a solution of the mean-field BDSDE(0.0.12)with the data(?2,f2,g).If ?1??2,Pa.s.,and f1(s,?,y,z)?f2(s,?,y,z),dsdP-a.e.for all(?,y,z),then Yt1?Yt2 for all t?[0,T],P-a.s..Chapter 4:We study general mean-field BDSDE associated with a forward diffusion SDE as well as the related backward SPDEs of mean-field type.Under suitable regularity assumptions on the coefficients we prove the L2-regularity of the value function V(t,x,P?):=Ytt,x,P?,The characterisation of V=(V(t,x,P?))as the unique solution of the associated mean-field backward stochastic PDE uses the Cb1,2,2-functions ?(t,x,P?):=E[V(t,x,P?)·?]for suitable ??L?(F;R).Besides,we extend the classical mean-field It? formula studied in[17].First,we extend the classical mean-field It? formula studied by Buckdahn,Li,Peng and Rainer[17]to smooth functions of solutions of mean-field BDSDEs.Theorem 4.1.1.(It?'s formula).Let,F?Cb1,2,2([0,T]×Rd×P2(Rd)).Given f?HF2(0,T;Rd),g?HF2(0,T;Rd×l),??L2(FT;Rd),as well as u?HF2(0,T;Rd),v?HF2(0,T;Rd×l),??L2(FT;Rd).We consider the solution(Y,Z),(U,V)?SF2(0,T;Rd)×HF2(0,T;Rd×d)of the following BDSDEs,respectively:(?)t?[0,T],and(?)t?[0,T],Then,for all t?[0,T],we have We consider the following both SDEs:(?)s?[t,T],(?)s?[t,T],and the following both decoupled BDSDEs:where ?st,?:=(Xst,?,Yst,?,Zst,?),?st,x,?:=(Xst,x,?,Yst,x,?,Zst,x,?),(t,x)?[0,T]×Rd and??L2(gt;Rd).We obtain the solutions(Xt,?,Xt,x,P?)of the split forward SDE and those of the split BDSDE(Yt,?Zt,?)and(Yt,x,?Zt,x,?)Moreover,we give the investigation of the existence and the uniqueness of solutions and the corresponding estimates for our split mean-field BDSDEs as well as the fact that(Xt,x,?,Yt,x,?,Zt,x,?)depends on ? only through its law((Xt,x,?,Yt,x,?,Zt,x,?)=(Xst,x,P?,Yst,x,P?,Zst,x,P?)).Now we we investigate the first and the second order derivatives of(Xt,x,P?,Yt,x,P?,Zt,x,P?).Theorem 4.4.1.Suppose Assumption(H4.4.1)holds true.Then the L2-derivative of Xt,x,P? with respect to x exists,it is denoted by ?xXt,x,P?=(?xXt,x,P?,j)1?j?d,and is characterized by a SDE.Theorem 4.4.2.Let(b,?)satisfy Assumption(H4.4.1).Then,for all 0?t?s?T,x?Rd,the lifted process L2(gt;Rd)???Xst,x,??L2(gs;Rd)is Fréchet differentiab le,and the Frechet derivative is given by(?),?=(?1,?2,…,?d)?L2(gt;Rd),where,for y?Rd,??X(t,x,P?,y)?Sg2(t,T;Rd×d)is characterized by a SDE.Theorem 4.5.1.Under the Assumptions(H4.4.1)and(H4.5.1),the L2-derivative of(Yt,x,P?,Zt,x,P?)with respect to x,(?xYt,x,P?,?xZt,x,P?),exists and is characterized by a BDSDE.Theorem 4.5.2.Assume the Assumptions(H4.4.1)and(H4.5.1)hold.Then,for all 0?t?s?T,x?Rd,the lifted processes L2(gt;Rd)???Yst,x,?:=Yst,x,P??L2(Fs;Rd),and L2(gt;Rd)???Zst,x,?:=Zst,x,P??HF2(t,T;Rd)are Fréchet differentiable,with the Frechet derivatives(?),s?[t,T],P-a.s.,(?),dsdP-a.e.,for all ?=(?1,?2,…,?d)?L2(gt;Rd),where for all y?Rd,(??Yt,x,P?(y),??Zt,x,P?(y))?SF2(t,T;Rd)×HF2(t,T;Rd×d)is characterized by a BDSDE.Theorem 4.6.1.Under Assumption(H4.6.1)the first order derivatives x??xXt,x,P?,??Xt,x,P?(y)?Sg2(t,T;Rd)are differentiable w.r.t.x and y in L2-sense,respectively.Theorem 4.7.1.Let the Assumptions(H4.3.2),(H4.6.1),(H4.7.1)and(H4.7.2)hold true.Then,for all t?[0,T],x?Rd,??L2(gt;Rd),the mapping(?xYst,x,P?,?xZst,x,P?)?SF2(t,T;Rd)×HF2(t.T;Rd×d)is differentiable in x,and its derivative(?xx2Yst,x,P?,?xx2Zst,x,P?)is characterized by a BDSDE.Theorem 4.7.2.Let the Assumptions(H4.3.2),(H4.6.1),(H4.7.1)and(H4.7.2)hold true.Then,for all t?[0,T],x?Rd and ??L2(gt;Rd),the process(??Yt,x,P?(y),??Zt,x,P?(y))?SF2(t,T;Rd)×HF2(t,T;Rd×d)is L2-differentiable with respect to y,and its derivative(?y??Yst,x,P?(y),?y??Zst,x,P?(y))is characterized by a BDSDE.Now we introduce the value function V(t,x,P?):=Ytt,x,P?.From above we know the continuity and differentiability of first and second order of V(t,·,·)in the only L2-sense,i.e.,we prove the the L2-regularity of the value function V(t,x,P?):=Ytt,x,P?.Thus,we cannot use the mean-field It? formula(Theorem 4.1.1)to V(s,Xst,x,P?,PXst,?).For this,in order to get the associated semi-linear backward SPDE with ?(s,x,P?):=(V(s,x,P?),?i=1d?xiV(s,x,P?)?i(x,P?)),we introduce the deterministic function ?(t,x,P?):=E[V(t,x,P?)·?],for suitable ??L?(F;R),and for? we can show that it is a Cb1,2,2-function.Applying the mean-field It? formula to?(s,Xst,x,P?,PXst,?)will be crucial for the proof that V is a solution of above SPDE.Proposition 4.8.2.Under assumptions(H4.3.2),(H4.6.1),(H4.7.1)and(H4.7.2),for all ??L?(F;R)with(H4.8.1),??Cb0,2,2([0,T]×Rd×P2(Rd)),and for all??{?,?x?,?x2?,???,?y(???)} it holds|?(t+q,x,P?,y)-?(t,x,P?,y)?(?),0?t?t+q?T,(x,P?,y)?Rd×P2(Rd)×Rd,where ???(t,x,P?,y)=E[??V(t,x,P?,y)·?],and the constant C?'? R+ depends on ?and C?.Now we are able to establish and to prove the main result of this chapter.Theorem 4.8.1.Under the Assumptions(H4.3.2),(H4.6.1),(H4.7.1)and(H4.7.2),V?C0,2,2(?×[0,T]×Rd×P2(Rd))is a classical solution of above backward SPDE,and it is unique in C0,2,2(?×[0,T]×Rd×P2(Rd)).