Study On Numerical Methods For MFBSDEs

Mean-field stochastic differential equations are also called Mckean-Vlasov equa-tions,which have found important applications in many fields as kinetic gas theory,quantum mechanics,and quantum chemistry.Buckdahn.Djehiche.Li and Peng[10]recently studied a special mean-field problem in a purely stochastic approach and deduced new kinds of BSDEs which they called mean-field backward stochastic differ-ential equations(MBSDEs).Then.Buckdahn,Li and Peng[15]obtained the existence and uniqueness of the solutions of mean-field forward backward stochastic differential equations(MFBSDEs),and gave a stochastic interpretation to the related semilinear nonlocal partial differential equations(PDEs).After that,the theory of mean-field forward backward stochastic differential equations are studied by many authors and found important,applications in stochastic optimal control problems,stochastic differ-ential games and nonlocal integral-partial differential equations,and so on.However,it is usually difficult to get the analytic solutions in an explicit closed form.Therefore,the study on numerical solutions of MFBSDEs is of vital importance for the theory of MFBSDEs and their applications in practice.In this thesis,we mainly focus on high accurate numerical methods for mean-field forward stochastic differential equations(SDEs).mean-field forward backward stochastic differential equations(MFBSDEs)and mean-field forward backward stochas-tic differential equations with jumps(MFBSDEJs).Specifically speaking,based on Ito's formula,we deduce the mean-field Ito's formula and mean-field Ito-Taylor expan-sion,and then propose mean-field Ito-Taylor schemes of strong order ? and weak order? for solving MSDEs:we also rigorously prove the error estimate of the strong order ?Ito-Taylor scheme.Based on the mean-field Feynman-Kac formula,the structure of the solution to MBSDEs and the theory of BSDEs and deterministic numerical method,we propose the general explicit ?-scheme for solving MBSDEs,and also analyze the stability and accuracy of scheme theoretically and numerically;propose the explicit second-order and multistep schemes for solving MFBSDEs,and also rigorously analyze their stability and accuracy theoretically and numerically.Besides,we also propose the explicit second-order scheme for solving MFBSDEJs,and analyze the scheme nu-merically.The main contribution and innovation:(1)Develop the mean-field Ito's formula,which is the basis of studying the munerical schemes for MSDEs and MFBSDEs.The study,result has been published in Nurner.Math.Theor.Meth.Appl.[81].(2)By using the mean-field Ito's formula,we propose the Ito-Taylor expansion,and based on which propose Ito-Taylor schemes for solving MSDEs.We rigorously prove the convergence rate of the strong Ito-Taylor schemes and numerically analyze the stability and convergence of Ito-Taylor schemes.The study result has been published in Numer.Math.Theor.Meth.Appl.[81].(3)Propose a general explicit ?-scheme for solving MBSDEs and rigorously analyze the stability and convergence of the scheme theoretically and numerically.The study result has been published in SIAM.J.Numer.Anal.[86].(4)Propose an explicit prediction correction scheme for solving decoupled MFBS-DEs,and rigorously analyze the stability and second-order convergence of the scheme theoretically and numerically.The study result has been finished[85].(5)Propose an explicit multistep scheme for solving decoupled MFBSDEs.The nu-merical scheme is proved to be high order accurate theoretically and numerically.The study rcesult has been fmnished[83].(6)Propose an explicit second order scheme for solving decoupled FBSDEJs.The numerical experiments confirm the second-order of convergence.The study result has been finished[82].The framework:The thesis is divided into seven chaptersChapter I.Introduction In Chapter 1,we make a brief introduction of the background,research motiva-tion,and development of our topic in the following chapters.Chapter 2.Preliminaries In Chapter 2.we introduce the existing theoretical results for our topic,includ-ing that of stochastic differential equations(SDEs),mean-field stochastic differ-ential equations(MSDEs)and mean-field stochastic differential equations with jumps(MSDEJs):the relation between the solutions of FBSDEs,MFBSDEs,FBSDEJs.and their corresponding kind of parabolic PDEs,i.e.,three types of Feynmanp-Kac formula.Chapter 3.Ito-Taylor Schemes for MSDEs In Chapter 3,we focus on numerical methods for mean-field stochastic differential equations(MSDEs).We first develop the mean-field Ito formula and mean-field Ito-Taylor expansion.Then based on the new formula and expansion,we propose the Ito-Taylor schemes of strong order ? and weak order ? for MSDEs,and theoretically obtain the convergence rate ? of the strong Ito-Taylor scheme.This chapter is mainly based on the paper:·YABING SUN,JIE YANG AND WIDONG ZHAO,Ito-Taylor schemes for solving mean-field stochastic differential equations,Numer.Math.Theor.Meth.Appl.,10(2017,pp.798-828.(SCI)Chapter 4.General ?-Schemes for MBSDEs In Chapter 4.based on the mean-field Feynman-Kac formula,we propose a class of explicit 0-schemes for solving MBSDEs.We first prove a rigorous stability result,based on wchich sharp error estimates are presented-showing that the proposed ?-schemes yield a second order rate of convergence.Several numerical experiments are carried out to verify the theoretical results.This chapter is mainly based on the paper:·YABING SUN,WEIDONG ZHAO AND TAO ZHOU,Explicit ?-scheme for solv-ing mean-field backward stochastic differential equations,SIAM J.Numer.Anal.,56(2018),pp.2672-2697.(SCI)Capter 5.The Prediction-Correction Scheme for MFBSDEs In Chapter 5,we mainly study on the explicit prediction correction scheme.for solving the decoupled MFBSDEs.In the first place,a rigorous stability result is discussed in detail,then based on which,we present the error estimates-showing that,the proposed scheme is second-order accurate when the weak order 2.0 Taylor scheme is used to solve MSDEs.Several numerical experiments are also carried out to verify the theoretical results.This chapter is mainly based on the paper:·YABING SUN AND WEIDONG ZHAO,A Numerical Method for Decoupled Mean-field Forward Backward stochastic differential equations,Finished.Chapter 6.Explicit Multistep Numerical Scheme for MFBSDEs In Chapter 6,based on the Lagrange interpolation method and the backward orthogonal polynomials,we propose an explicit multistep numerical scheme for solving the decoupled MFBSDEs and analyze its error estimate rigorously.Sev-eral numerical experiments also verify our theoretical conclusions.This chapter is mainly based on the paper:·YABING SUN,JIE YANG,WEIDONG ZHAO AND TAO ZHOU,An explicit multistep scheme for solving mean-field forward backward stochastic differ-ential equations,Finished.Chapter 7.Second-Order Numerical Scheme for MFBSDEs In Chapter 7.based on two constructed martingales,we propose an explicit second-order scheme for solving the decoupled FBSDEJs.Numerical experi-ments verify the stability and second-order convergence of the proposed scheme.This chapter is mainly based on the paper:·YABING SUN,JIE YANG AND WEIDONG ZHAO,Numerical methods for decoupled mean-field forward backward stochastic differential equations with jumps.Finished.The main results:Chapter 3:we propose the Ito-Taylor schemes for solving MSDEs,and theoretically obtain the convergence rate ? of the strong Ito-Taylor scheme.Consider the following MSDEs:dXt=E[b(t,Xt,?)]|?=Xt ds + E[?(t,Xt,?)]|?=Xt dWt,0?t0?t?T,(0.45)where t0 is the initial time,and Xt0 is Ft0 measurable.To obtain the Ito-Taylor scheme for MSDEs,we first give the following mean-field Ito formula and Ito-Taylor expansion.To this end,consider MSDEs:dXt?b?(t,Xt)dt+??(t,Xt)dWt,t?0,(0.46)where b?(t,Xt)=E[b(t,?t,x)]|x=Xt,??(t,Xt)=E[?(t,?t,x)]|x=Xt,(0.47)with d?t=?tdt+?tdWt,(0.48)where ?t and ?t are progressively measurable processes and satisfy:?0T|?t|dt<+?,?0T|?s|2dt<+?Define f?(t,x)=E[f(t,?t,x)],(0.49)where f:[0,?)×Rd×Rd?R.By using Ito's formula,we deduce the following mean-field Ito formula.Theorem 0.1(Mean-field Ite's formula).Let Xt and?t be d-dimensional lto processes satisfying(0.46)and(0.48),respectively,and f be a C1,2,2 function mapping from[0,?)×Rd×Rd to R.Then it holds that f?(t,Xt)= f3(0,X0)+?0t L0f3(s,Xs)ds+?i=1m?0tLjf?(s,Xs)dWs,(0.50)where operators L0 and L? are defined as(?)with(?)where(?)xf =((?)f/(?)x1,…,(?)f/(?)xd)are d dimensional vector,fxx=((?)2f/)(?)xi(?)xj)d×d is d×d dimen-sional matrix,(?)x'f and fx'x' are defined in the same way,and Tr(A)is the trace of the matrix A.We deduce the Ito-Taylor expansion for MSDEs by iterated application of the mean-field Ito formula(0.50).Theorem 0.2.Let ? and ? be two stopping times with t0??(w)?(?)?T,and let A(?)M be a hierarchical set,and f:R+×Rd×Rd?R.Then the mean-field Ito-Taylor expansion f?(?,X?)=? ??A I?[f??(?,X?)]?,?[f??(·,X.)]??(0.52)holds,provided all of the derivatives of f3,b3 and ?3 and all of the multiple lto integ rals appearing in(0.52)exist.Based on the Ito-Taylor expansion,we propose the strong Ito-Taylor scheme and weak Ito-Taylor scheme.Scheme 0.1(Strong order ? Ito-Taylor scheme).Xk+1=???I?[f?Xk(tk,Xk)]tk,tk+1,k=0.1,…,NT.(0.53)Scheme 0.2(Weak order ? Ito-Taylor scheme).Xk+1=?????I?[f?Xk(tk,Xk)]tk,tk+1,k=0,1,…,NT.(0.54)To rigorously obtain the global convergence rate of Scheme 0.1,we first give the relationship between the local and global convergence rates.Let {Xt,x(s)?t?s?T be the solution of MSDE(0.45)starting from the point,(t,X).By the(0.45)and(0.47),it also holds that Xt,x(t+?)=X+?t t+? aXt,x(s,Xt,x(s))ds+?t t+? ?Xt,x(s,Xt,X(s))dWs(0.55)for 0???T-t.Let Xt,X(t+h)be the one-step approximation of Xt,x(t+h.).We have the following theorem.Theorem 0.3.Let Xt.x(t+h)be defined as(0.55).If Xt,x(t+h)satisfies|E[Xt,X(t+h)-Xt.x(t+h)|Ft]|?C*(1+E[|X|2]+|X|2)1/2hp1,(0.56a)(E[|Xt,x(t+h)-Xt,x(t+h)|2|Ft])1/2?C*(1+E[|X|2]+|X|2)1/2(0.56b)for t ?[t0,T-h]and x ?Rd,where p1 and p2 are parameters satisfying p2?1/2,P1?P2+1/2,and C*a positive constant independent of h,Xt,x(t+h)and Xt,x(t+h).Then for k=1,…,NT,it holds that(E[|Xt0,x0(tk)-Xt0,X0(tk)|2])1/2?C(1+E[|X0|2])1/2hp2-1/2(0.57)where Xt0,x0(t0)=X0 and Xt0,X0(tk)is the solution of the one-step scheme Xt0,0(tk)=Xtk-1,Xt0,X0(tk-1)(tk),(0.58)and C a constant independent of h,Xt,x(t-h)and Xt,x[t+h).We have the following strong convergence results about Scheme 0.3.Theorem 0.4.Let X(t)and Xk be the solutions of the MSDE(0.45)and the strong or-der ? Ito-Taylor scheme(0.53)with X(t0)= X0,respectively.Suppose fxt,x(s,Xt,x(s))=Xt,x(s)has the Ito-Taylor expansion(0.52)with A = A?=and|f??(t,x)|?C(1+E[|?t|2]+|x|2)1/2,for ? ?A? ?B(A?),(t,x)?[0,T]×Rd and 3 defined by(0.48).Then.it holds that max k??1,2,…,NT?E[|Xtk-Xk|2]?C(1+E[|X0|2](?t)2?.The numerical experiments are shown in § 3.4.Chapter 4:we propose a class of explicit ?-schemes for solving MBSDEs,and rigorously analyze the stability and error estimates of the schemes.We consider the following MBSDE:(?)wehere ?(x):Rd?RP is the deterministic functions,and XT0,X0 is the process start from the point X0:Xt0,X0=X0+Wt,0?t?T.(0.60)We propose the following explicit ?-scheme for solving the MBSDE(0.59).Scheme 0.3.Gi.ven the termina.l con.ditions YN,X0 and ZN,X0,we solve random vari-ables Yn,Y0 and Zn,Y0 with n =N-1,…,0 by(?)where fnx0,X0=E[f(tn,Yn,x0,Zn,x0y,z)]|y=Yn,X0,2=Zn,X0,and |?4|??3.Let-?f,(?yN,X0,?zN,X0)be perturbations on the generator f and the terminal condi-tions(YN,X0,ZN,X0).Denote by Y?n,X0,Y?n,X0and Z?n,X0 the solutions of Scheme 0.3 with perturbations,and define the perturbation errors of Scheme 0.3 as:For the stability of Scheme 0.3 in the case where X0=x0,we have the following theorem.Theorem 0.5.Assume that f(t,y',z',y,z)is uniformly Lipschitz continuous with respect to(y',z',y,z)with Lipschitz constant L.Let c0 be a time partition regularity parameter.Then for sufficiently small time step ?t,it holds that(for 0?n?N-1)(?)where C is a positive constant depending on c0,L,T,?3 and ?4.Based on Theorem 0.5,we obtain the stability result in the general cases.Theorem 0.6.Under similar assumptions as in Theorem.0.5,for 0?n?N-1,it holds that(?)where C is a positive constant depending on c0,L.T,?3 an.d ?4.Based on its stability results,we prove the error estimates of Scheme 0.3 in the following theorem.Theorem 0.7.Let(Yt0,X0,Zt0,X0)and(Yn,X0,Zn,X0)be the solution of the MBSDE(0.59)and the numerical solution by Scheme 0.3,respectively.Then for parameters?i?[0,1](i-2),?3?(0,1],and ?4 ?(-1,1]with constraint |?4|<?3,and sufficiently small time step ?t,we have the following conclusions.1.If f ? Cb1,3,3,3.3,??Cb3+a,and a?(0,1),we have(?)2.If f? Cb,2,5,5,5,5,??Cb5+a,and ??(0,1),then we have(?)Here,C is a positive constant depending on c0,x0,X0,L,T,the upper bounds of the derivatives of the functions ?and f.and the parameters ?i(i=1...4).The numerical experiments are shown in § 4.4.Chapter 5:we propose an explicit prediction-correction scheme for solving the decoupled MFBSDEs,and rigorously analyze the stability and error estimates of the scheme.We consider the following decoupled MFBSDEs:We first give following t.hree Ito-Taylor type schemes for solving MSDEs.· The Euler scheme:Xn+1X0=XnX0+bXnx0(tn,XnX0)?tn+?Xnx0(tn,XnX0)?Wn.· The Milstein scheme:· The weak order 2.0 Ito-Taylor scheme Where,?zn=?Wn?tn-?tn tn+1?tnsdzdWs.To propose the prediction-correction scheme,we define the following stochastic processes:?Wtn,s = 2?Wtn,s-3/?tn ?tn s(r-tn)dWr.(0.68)we propose the following prediction-correction scheme for MFBSDEs.Scheme 0.4.Step 1.Let X0 = x0,and solve MSDEs in(0.64)by Ito-Taylor to obtain?Xnx0,n=0.1,…,N?;Step 2.Give-n.initial value Xor,te?rminal co-nditions YNX0 and ZNX0,form=N-1,…,0,solve YnXO =Yn(XnX0)and ZnX0=Zn(XnX0)by(?)where,?nx=(Xnx,Ynx,Znx),and fnx0,X0-E[f(tn,?nx0,x)]|x-?nX0Let ?f,(?yN,X0,?zN,X0)be perturbations on the generator f and the terminal condi-tions(YN,X0,ZN,X0).Denote by Y?n,X0,Y?n,X0and Z?n,X0 the solutions of Scheme 0.4 with perturbations,and define the perturbation errors of Scheme 0.4 as?yn,X0=Yn,?X0-YnX0,?zn,X0=Zn,?X0-ZnX0,?y,X0=Yn,?X0-YnX0We state our main result for the stability of Scheme 0.4 in the following two theorems.Theorem 0.8.Assume that the furnction f(t,x',y',z',x,y,z)is uniformly Lipschitz continuous w.r.t.(x',y'.z',x,y,z)with Lipschitz constant L.Let c0 be the time parti-tion regularity parameter.Then,for sufficiently small time step At,we have that for any 0?n?N-1(?)where C is a positive constant depending on c0 L and T.Theorem 0.9.Under the conditions in.Theorem 0.8,for sufficiently small time step At,we have that for any 0?n?N-1(?)where C is a positive constant depending on c0,L and T.We give the following error estimates result.Theorem 0.10.Under reasonable assumptions,for the sufficiently small time step?t,it holds that E[|eyn,X0|2]+?t?i=nN-1E[|ezi,X0|2]?C(?t2?+?t2??t4),(0.74)where ? and ? are defined sin Assumption 5.1,where C is a positive constant depending on c0,L and T.The numerical experiments are shown in § 5.4.Chapter 6:we propose an explicit multistep numerical scheme for solv-ing the decoupled MFBSDEs and rigorously analyze the error estimates of the proposed scheme.We consider the following decoupled MFBSDEs:(?)where?=E[?(XT0,x0,?)]|?=XT0,X0.To propose the multistep scheme for MFBSDEs.we first introduce the backward orthogonal polynomials[113].??0.15(Backward orthogonal polynomials).We call a set of poly'nomials ?Qi(s)?i=0L defined on the interval[0,1]the backward orthogonal polynomials,if for each i =0,1,…,L,it holds that?01Qi(s)ds=1,?01Qi(s)sjds=0,1?j?i.Define the new backward orthogonal polynomial set ?Pi(s)?i=0L on[a,b]by Pi(s)=Qi(s-a/b-a)We propose the multistep scheme for solving the decoupled MFBSDEs(0.75)as follows.Scheme 0.5.Step 1.Let X0 =x0,and solve MSDEs in(0,75)by Ito-Taylor scheme to obtain {Xnx0,n-0,1,…,N?;Step 2.Let K = max{Ky,Kf,Kz}.Given.initial value X0 and terminal conditions YN-iX0 and ZN-iX0,i=0,1,…,K-1.Forn=N-K,…,1,0 solve YnX0?Yn(XnX0)and ZnX0=Zn(XnX0)by(?)where bKy,jn and BKy,jn are the coefficients of Lagrange interpolation polynomial:(?)We state our main result for the stability and error estimates of Scheme 0.5 in the following theorems.Theorem 0.11.Let(Xt.Yt,Zt),t?[0,T]and(Xn,Yn,Zn),n=0,1,…,N,be the exact solution.of(0.75)and numerical solutions of Scheme 0.5,respectively.Suppose that f is uniformly Lipschitz continuous with constant L.Let B=max 1?i?ky,1?j?Kf?bKy,jn,BKf.km? and PKz=max 0?s?Kc0 Qkz(s)with,c0 being the time partition regularity parameter.then for sufficiently small ?t and all n = N-K,…,0,we have(?)where C>0 is a constant depending on c0,T,L,B,K,PKz and Qkz(0).Theorem 0.12.Urnder the conditions of Theorem 0.11,for sufficiently small ?t and,all n= N-K,…,0,we have(?)where C>0 is a constant depending on c0,T,L,B,K,Pkz and Qkz(0).Theorem 0.13.Under reasonable ossumptions,for sufficiently small time step ?t it holds that(?)where ? and ? are defined in Assumption 5.1,and C>0 depends on c0,T,L,B,K,PKz,Qkz(0),X0,X0,x0,and the upper bounds of the derivatives of b,?,f and ?.The numerical experiments are shown in § 6.4.Chapter 7:we propose a second-order scheme for solving the decou?pled MFBSDEJs,and numerically analyze the stability and second-order convergence rate of the scheme.We consider the following decoupled MFBSDEJs:(?)where?= E?(XT0,x0,x)]|x-XT0,X0,?s0,X0=(Xs0,X0,Ys0,X0,Zs0,X0,?s0,x0)are the unknown terms.To propose the second-order scheme,we define the followiag two stochastic pro-cesses:(?)where p(r)=2-3/?tn(r-?tn).We propose the following Euler scheme for MSDEJs and second order scheme for MFBSDEJs(0.82),respectively.Scheme 0.6.Given initial values x0 and X0,for n=1,2,…,N,solve XnX0 by(?)Scheme 0.7.Step 1:Let X0 = x0,and use Euler scheme 0.6 to solve MSDEJ in(0.82)to obtain Xnx0,n0.1,…,N;Step 2:Given initial value X0,and terminal conditions YNX0,ZNX0 and ?NX0,for n =The numerical experiments are shown in § 7.2.