| Forward backward stochastic differential equations(FBSDEs)are a kind of powerful tool for studying uncertainty,which are composed of the forward stochastic differential equation(SDE)flow with given initial conditions and the backward stochastic differential equation(BSDE)with given terminal conditions defined on a filtered complete probability space.The dynamic mechanism of FBSDEs can reveal the laws of things’ development and reflect the characteristics of thought in understanding the world to a great extent.Bismut[17]in 1973 first introduced the concept of linear types of BSDEs.The existence and uniqueness of the adapted solution of the general BSDEs under certain conditions are originally proved by Pardoux and Peng in[91].1991,Peng in[96]obtained the nonlinear Feynman-Kac formula,which establishes the connections between FBSDEs and a family of linear parabolic partial differential equations(PDEs).Since then,FBSDEs have been extensively studied and they are applied in diverse fields,such as stochastic optimal control[57,67,95,110,137],mathematical finance[12,34,42,87],nonlinear expectation[50,98],the theory of PDEs[8,11,93]and so on.Generally,analytic solutions of FBSDEs can seldom be expressed in explicit closed-form,and thus,of particular concern in the applications of FBSDEs is to propose numerical methods.It is worth pointing that most of the existing methods for FBSDEs are only temporal semi-discrete ones[5,22,28,135,141].The involved conditional expectations have to be approximated in numerical simulations and applications of temporal semidiscrete schemes.In the most of fully discrete schemes for FBSDEs,the corresponding the high-accurate spatial interpolations are indispensable[31,142],which brings many difficulties and challenges to the research of fully discrete schemes for FBSDEs.In this dissertation,we use Sinc methods to approximate the conditioned mathematical expectation,and propose the non-interpolation fully discrete Sinc schemes for FBSDEs.Then we apply the schemes to the generalized Allen-Cahn equations.For BSDEs,we combine the Sinc quadrature rule and θ-discretization to propose a family of fully discrete Sinc-θ schemes.By reasonably selecting the parameters in the Sinc quadrature rule,the spatial interpolations are avoided.We also rigourously prove the error estimates of the Sinc-θ schemes.For decoupled FBSDEs,by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we present new fully discrete multistep schemes called "Sinc-multistep schemes" for FBSDEs.The stability and the K-th order error estimates in time for the K-step Sinc multistep schemes are theoretically proved.For the generalized Allen Cahn equation,we utilize the fully discrete Sinc schemes for FBSDEs to propose the Sinc schemes,and carry out numerical and theoretical analysis of the schemes.The main contributions and innovation1.By using Sinc numerical methods,we combine with the numerical scheme of forward SDEs and the integral variable transformation to construct the fully discrete Sinc approximation operator to conditional mathematical expectations.The most important feature of the operator is that no spatial interpolations are needed.The operator is the basis of proposing fully discrete Sinc schemes for FBSDEs.It preserve many good properties of conditional mathematical expectations,increase the efficiency of approximations of conditional mathematical expectations,and is very useful to the rigorous analysis of fully discrete Sinc schemes.2.By using the fully discrete Sinc approximation operator to conditional mathematical expectations and θ-discretization,we propose a new family of fully discrete Sinc-θ schemes for BSDEs.we perform rigorous stability analysis and error estimates,and this seems to be the first attempt for analyzing fully discrete schemes for BSDEs with a second-order convergence rate in time and spectral accuracy in space.The study results have been published in SIAM J.Numer.Anal[125].3.By combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we propose Sincmultistep schemes for FBSDEs.The schemes avoid spatial interpolations and admit the K-th order of convergence rate in time and exponential convergence rate in space.The stability and the K-th order error estimates in time for the K-step Sinc multistep schemes are theoretically proved.This seems to be the first time for analyzing fully time-space discrete multistep schemes for BSDEs.Moreover,for the FBSDEs in general form,we provide the error estimates for the Sinc-onestep scheme.The study results have been published in Adv.Appl.Math.Mech[126].4.We apply the Sinc-θ schemes and Sinc-multistep schemes to solving the generalized Allen-Cahn equations.This seems to be the first time for proposing fully discrete probabilistic schemes for the generalized Allen-Cahn equations by using FBSDEs.The most important features of the Sinc scheme are that no spatial interpolations are needed,and that up-wind schemes for advection terms are avoided.Allen-Cahn equations,and perform their theoretical analysis.We perform rigorous stability analysis and error estimates for our Sinc schemes,and prove the discrete maximum bound principle for Sinc-θ scheme.The study resultshave been finished[127].The structure of the dissertation The dissertation is divided into six chapters.Chapter 1.Introduction We briefly introduce the relevant research background and development status of FBSDEs and their numerical methods as well as our research motivation.We also sort out the structure of the thesis.Chapter 2.Preliminaries We introduce some basic Preliminaries needed in the dissertation.We first introduce some definitions,properties and important conclusions of FBSDEs,such as ito formula,Feynman-Kac formula,and then review the two kinds of temporal semidiscrete schemes of BSDEs in the existing literature:the integration discretization method and the differential discretization method.Finally,we briefly introduce th numerical schemes for SDEs,which is e necessary in numerically solving FBSDEs.Chapter 3.Sinc numerical methodsIn this chapter,we study Sinc approximations to the conditional mathematical expectations.We introduce basic knowledge of Sinc numerical methods,including Sinc approximations of functions and integrals in various domain.Then based on the truncated Sinc integral approximations,we study the Sinc approximation of mathematical expectations and its error estimates.Numerical examples are used to verify our theoretical results.Combinin g the numerical schemes of SDEs and the Sinc approximation of mathematical expectation,we propose the fully discrete Sinc approximation operator to conditional mathematical expectations,and rigorously prove that the operator almost preserves good properties of conditional mathematical expectation.Chapter 4.Sinc-θ for BSDEsIn this chapter,we study the fully discrete Sinc-θ schemes for BSDEs.Based on the temporal semi-discrete schemes,we use the Sinc approximation operator to conditional mathematic expectations to propose the fully discrete Sinc-θ schemes for BSDEs.By appropriately selecting the parameters in the Sinc quadrature rule,our schemes are shown to be very efficient as no spacial interpolations are needed,which reduces the computation complexity and is very useful to the theoretical analysis.The difference quotient operator is introduced to construct the error reference equations,and the stability as well as optimal error estimates of the schemes are strictly analyzed theoretically.The theoretical results show that the schemes can admit the secondorder convergence in time and the exponential convergence in space.Several numerical experiments are presented to verify the theoretical results and to demonstrate the efficiency and accuracy of our schemes.This chapter is mainly based on the paper:·Xu Wang,Weidong Zhao,and Tao Zhou,Sinc-θ schemes for backward stochastic differential equations,SIAM J.Numer.Anal.,60(4)(2022),pp.1799--1823.Chapter 5.Sinc-multistep schemes for FBSDEsIn this chapter,we study the fully discrete Sinc-multistep schemes for decoupled FBSDEs.By combining the multistep discretization in time and the Sinc approximation operator to conditional mathematic expectations,we propose new fully discrete Sincmultistep schemes for decoupled FBSDEs.The schemes can avoid spatial interpolations by using integration variable transformation and by appropriately selecting the parameters in the Sinc quadrature rule.By using numerical algebra theory and the properties of Sinc approximation operator,we obtain the stability results and the K-th order error estimates in time for the K-step Sinc multistep schemes for BSDEs(where the forward stochastic process is Brownian motion).For the FBSDEs in general form,we provide the error estimates for the Sinc-onestep scheme(K=1).Numerical examples are also presented to demonstrate the effectiveness,stability,and high order of convergence rate of the schemes.This chapter is mainly based on the paper:·Xu Wang and Weidong Zhao,Sinc-multistep schemes for forward backward stochastic differential equations,to appear in Adv.Appl.Math.Mech.,2022,DOI:0.4208/aamm.OA2022-0073.Chapter 6.The applications of Sinc schemes in the generalized Allen-Cahn equationIn this chapter,we study the applications of Sinc schemes in the generalized AllenCahn equations.We apply the Sinc schemes for FBSDEs to numerically solving the Allen-Cahn equations,and present the highly accurate Sinc schemes for the generalized Allen-Cahn equations.We perform rigorous stability analysis and error estimates for our Sinc schemes.Under some reasonable assumptions,we prove the discrete expectations of solutions to Sinc-multistep schemes are uniformly bounded.Meanwhile,the discrete maximum bound principle of Sinc-θ scheme is proved.Several numerical results are provided to verify the theoretical ones,and show the effectiveness,robustness and accuracy of the proposed schemes.This chapter is mainly based on the paper:·Xu Wang and Weidong Zhao,Fully discrete Sinc schemes for the generalized Allen-Cahn equations,finished,2022.The main results of the dissertationIn Chapter 3,we study the Sinc approximations of mathematical expectations and their error estimates,and construct the non-interpolation Sinc approximation operator to conditional mathematical expectations and study its properties.Assume that X is a normally distributed random variable,then the integrand of expectation E[v(X)]has the following representation:f(x)=g(x)ql(x)exp(-ax2+bx),(?)x∈R,(0.0.9)where g is a bounded function,the function ql(x)is a polynomial function of degree l.e denote by Ql the set of all functions that are of the form in(0.0.11).By the Sinc quadrature rule,we have E[v(X)]=TM1,M2(f,h)+ηM1,M2(f,h),(?)x∈R.For functions f∈Ql,the following theorem holds.Theorem 0.1.Assume f∈Ql with fixed l.For sufficiently small h,if there exists a positive number γ0 such that γ0≤2aM1h2-bh-n/M1 and γ0≤2aM2h2-bh-n/M2,then we have the estimate|ηM1,M2(f,h)|≤Cγ0(l+1)h((M1h)lexp(-aM12h2+bM1h)+(M2h)lexp(-aM22h2+bM2h)),where Cγ0 is a positive constant depending only on γ0 and the upper bound of f.In particular,when M1=M2=M,we have|ηM(f,h)|≤Cγ0(l+1)h(Mh)le-aM2h2+bMh.See Section 3.2 for the proof and numerical validation of the Theorem 0.1.Using the Sinc quadrature rule,we construct non-interpolation fully discrete Sinc approximation operator to approximate the conditional mathematical expectations.Assume that the numerical scheme of forward SDE is Xin+1=xi+Φ(tn,xi,Δt,ΔWn,1),then by the Feynman-Kac formula,the conditional expectation Etnxi[g(Xin+j)(j=1,...,K)can be written in the following form:where ξnn+j=(ΔWn,1,...,ΔWn+j-1,1)is a jd-dimensional random vector;the functionχ is defined by To get rid of spatial interpolation,we introduce the transformation φin,j and denote its inverse transformation ψin,j(r)=s.By using these two transformations and Sinc quadrature rule,and by selecting the parameter matrix H as with L being is a positive integer diagonal matrix,we have where with Γg,Mn,j,i和Γg,Mn,j,i being the Sine truncation errors.Here Jin(r)represents the determinant of the Jacobin matrix (?)ψin(r)/(?)r.Now by removing the truncation error Γg,Mn,i in(0.0.11),we can define an approximation operator Etnxi[·]of Etnxi[·]in the following definition.Definition 0.1.Given an m-dimensional function ζ on Rhd or Rd at time level tn+j,we define the approximation operator Etnxi[·]as where Xin+j=xi+χ(tn,xi,Δt,ζnn+j);the deterministic function χ is defined in(0.0.10).The operator Etnxi[·]has the following properties.Lemma 0.1.For ζ and η that are two m-dimensional functions on Rd at time level tn+1,the following properties hold.(ⅰ)(Linearity)For any real numbers a and b,Etnxi[aζ+bη]=aEtnxi[ζ]+bEtnxi[η].(ⅱ)(Maximum bound)|Etnxi[ζ]|≤‖ζ‖∞,where‖ζ‖∞ represents the maximum normof the function ζ.(ⅲ)(Monotonicity)If ζ≥η,then Etnxi[ζ]≥Etnxi[η].(ⅳ)(Cauchy-Schwarz inequality)‖Etnxi[ζη┬]‖2≤Etnxi[‖ζ‖2]Etnxi[‖η‖2].Particularly,if the SDE Xt is a d-dimensional Brownian motion,we haveSee Section 3.3 for the proof of Lemma 0.1.In Chapter 4,we propose Sinc-θ schemes for BSDEs,and rigorously prove their optimal error estimate;We also numerically analyze the schemes.Consider the BSDE defined on a filtered complete probability space(Ω,F,F,P):Yt=φ(XT)+∫tTf(s,Xs,Ys,Zs)ds-∫tTZsdWs,(?)t∈[0,T],(0.0.12)where φ:Rd→Rq is a deterministic function.Xt∈Rd is the terminal value of the following stochastic process:Xt=X0+Wt,0≤t≤T.Using the Sinc approximation operator Etnxi[·]anu θ-discretization,we propose the following Sinc-θ schemes for BSDEs.Scheme 0.1(Sinc-θ schemes).Let(Yin,Zin)be the numerical approximations of(Yt,Zt)of the BSDEs(0.0.12)at the time-space point(tn,xi).Given random variables(YN,ZN)on Rhd,for n＝N-1,…,0,we solve(Yin,Zin)for each xi∈Rhd via Here θ1,θ2∈[0,1],θ3∈(0,1];θ4∈[-1,1]is constrained by |θ4|≤θ3,Let(Ytnxi,Ztnxi)and(Yin,Zin)be the exact solution and the numerical solution of Scheme 0.1 at(tn,xi),respectively,then we denote eyn,i=Ytnxi-Yin,ezn,i=Ztnxi-Zin,eyn=Ytn-Yn,ezn=Ztn-Zn.We first provide the following error estimates.Theorem 0.2.Denote If the function f is Lipschitz with respect to Y and Z,then we have where C is a positive constant only depending on T,f;the positive constant C1=(3θ32+θ42)(θ3+θ4)/12(θ3-θ4)(θ32+θ42).By taking difference quotient on the error equations,we deduce the optimal estimate of ‖ezn‖∞.Theorem 0.3.If f∈Cb1,1,2,2,φ∈Cb2,then we have where C is a positive constant only depending on T,f and φ.Based on Theorem 0.2 and 0.3,we present the optimal error estimates of Sinc-θschemes with the explicit convergence rate.Theorem 0.4.Suppose that the terminal errors satisfy‖eyN‖∞=O((Δt)2),‖▽xeyN‖∞=O((Δt)2),‖ezN‖∞=O((Δt)2),‖▽xezN‖∞=O((Δt)2).Assume that the value Mh is big enough,and the parameter h in Sine approximations satisfies h=(?).If f∈Cb2,4,4,4,φ∈Cb3,then for sufficiently small Δt,we have Moreover,if f∈Cb2,5,5,5,φ∈Cb5,then for sufficiently small Δt,we haveTheorem 0.4 shows that Sinc-θ schemes can admit the second-order convergence rate in time and the exponential convergence rate in space.The numerical results are shown in Section 4.3.In Chapter 5,the Sinc-multistep schemes for FBSDEs are proposed,and are analyzed theoretically and numerically.Consider the following decoupled FBSDEs defined on a filtered complete probability space(Ω,F,F,P):where φ:Rd→Rq is a deterministic function.Based the temporal semi-discrete differential multistep schemes,we use the Sine approximation operator Etnxi[·]to propose the following Sinc-multistep schemes.Scheme 0.2(Sinc-multistep schemes).Let(Yin,Zin)be the numerical approximations of(Yt,Zt)of the BSDE in(0.0.13)at the time-space point(tn,xi).Given YN-j and ZN-j(j=0,1,...,K-1)on Rhd,for n=N-K,...,1,0 and each xi∈Rhd,solve Xin,j,Yin(j=1,...,K),Yin=Yn(xi)and Zin=Zn(xi)by where Yn+j=Yn+j(Xin,j),the coefficients {αj}j=0K.satisfy To analyze the Sinc-multistep schemes,we introduce the following lemma.Lemma 0.2.Let the matrix A=(?).If its characteristic polynomial P(λ)=∑i=0KαiλK-i satisfies root conditions,then there exists a nonsingular matrix B∈CK×K such that where ‖·‖*＝‖B·‖.If the forward process in the FBSDEs is in the form of Xt=X0+σWt(σ is a nonsingular constant matrix),then we provide the general error convergence results of Sinc-multistep schemes in the following theorem.Theorem 0.5.Let(Yt,Zt)and(Yn,Zn),n=0,1,...,N be the solution of the BSDE in(0.0.13)and Sinc-multistep Scheme 0.2,respectively.Assume the generator function f of the FBSDEs(1.0.1)is Lipschitz continuous with respect to Y and Z.Then for sufficiently small Δt,we have the error estimate where Ry,nK and Rz,nK are truncation errors in time direction;Γy,Mn,j,σ and Γy,w,Mn,j,σ are the Sinc integral truncation errors.Here C is a positive constant only depending on the terminal time T,the Lipschitz constant of f,the step number K,and the coefficients{αj}j=0K in the Sinc-multistep scheme.Based on Theorem 0.5,we provide the error estimates with explicit convergence rate of Sinc-multistep schemes.Theorem 0.6.Let(Yt,Zt)and(Yn,Zn),n=0,1,...,N be the solution of the BSDE in(0.0.13)and Sinc-multistep Scheme 0.2,respectively.Suppose that the parameter Mh in the Sinc quadrature rule is sufficiently big and the terminal errors satisfy(?)Et0[|YtN-j-YN-j|2]=O((Δt)2K).If f∈Cb1+K,1,2+2K,2+2K,and φ∈Cb3+2K,then for sufficiently small Δt,we have where C is a positive constant only depending on the terminal time T,the Lipschitz constant of f,the step number K,the coefficients {αj}j=0K in the Sinc-multistep schemes and the upper bounds of f and φ.Theorem 0.6 shows that the Sinc-multistep schemes for BSDEs can admit the K-th order convergence rate in time and the exponential convergence rate in space.For the Sinc-onestep scheme(K=1)for FBSDEs in the general form,we have the following error estimates.Theorem 0.7.Denote by the positive constant a the upper bound of the diffusion σ in FBSDEs(0.0.13).Suppose the terminal error ‖YtN-YN‖∞=O(Δt).If b,σ∈Cb2,4,f∈Cb2,3,3,3,φ∈Cb4,then for sufficiently small Δt and sufficiently big Mh,we have where C is a positive constant only depending on the terminal time T,the upper bound of the functions b,σ,f,φ and their derivatives.Theorem 0.7 shows that the Sinc-multistep scheme(K=1)for FBSDEs is convergent.The numerical results of Sinc-multistep schemes for FBSDEs are shown in Section 5.4.In Chapter 6,we study the applications of Sinc scheme for FBSDEs in the generalized Allen-Cahn equations,and theoretically and numerically analyze the proposed Sinc schemes.Consider the Allen-Cahn equation in the following general form:where Ω is a bounded/infinite domain in Rd;u represents the concentration of one of the two metallic components of the alloy;the nonlinear bulk force f(u)=F’(u)=u3-u with F(u)being a given double-well free energy density;v(t,x)is a given incompressible velocity field;the small constant 0<ε<<1 represents the inter-facial width.Let τ=T-t,by Feynman-Kac formula,the solution u(τ,·)of the PDE(0.0.15)can be represented as an L2-adapted solution Yτ(x)of the following FBSDEs defined on a filtered complete probability space(Ω,F,F,P):Based on Schemes 0.2,we propose the Sinc-multistep schemes for the generalized Allen-Cahn equations.Scheme 0.3.Let Yin be the numerical approximations for the solution Y of the BSDE in(0.0.16)at the time-space point(τn,xi).Assume that YN-l,l=1,...,K-1 on Rhd are given.For xi∈Rhd and for n=N-K,...,0,solve Xin,j and Yin=Yn(xi)by where Yn+j is the values of Yn+j at space points Xin,j;the coefficients {αj}j=0K are defined in(0.0.14).In our analysis of the Sinc-multistep schemes,we shall restrict our attention to the zero-velocity Allen-Cahn equation(v=0),and impose the Lipschitz continuous assumption on the bulk force function f.In Theorem 0.8,we obtain the expectation uniform bound principle of Sincmultistep schemes.Theorem 0.8(Expectation uniform bound principle).Assume that the bulk force f is uniformly Lipschitz continuous.If the given terminal values are bounded,for sufficiently small time step Δτ,then the discrete mathematical expectations of numerical solutions to Sinc-multistep schemes preserve a uniform bound,i.e.,where C is a positive constant only depending on the terminal time T,the Lipschitz constant of f,the step number K,and the coefficients {αj}j=0K in Sinc-multistep schemes.Define the perturbation error asδy,ζn=Yζn-Yn,δf,ζn=f(Yζn)-f(Yn).In Theorem 0.9,we obtain the stability results of Sinc-multistep schemes.Theorem 0.9.Let the assumptions in Theorem 0.8 hold,then for 0≤n≤N-K,we have where C is a positive constant only depending on the terminal time T,the Lipschitz constant of f,the step number K,and the coefficients {αj}j=0K.Using the stability results in Theorem 0.9,we provide error estimates of Sincmultistep schemes in the following theorem.Theorem 0.10.Let Yτnx and Yn(x)be the exact solution of the BSDE in(0.0.16)and the numerical solution of the Sinc-multistep Schemes 0.3 at time-space points(τn,x)for all x∈Rhd,respectively.Suppose that the parameter Mh in the Sine quadrature rule is sufficiently big and the terminal errors satisfy(?)Eτ0[|YτN-j-YN-j|]=O((Δτ)K).If f∈Cb2+2K,and φ∈Cb3+2K,then for sufficiently small Δτ we have where C is a positive constant only depending on the terminal time T,the Lipschitz constant of f,the srtep number K,the coefficients {αj}j=0K as well as the upper bound of the functions cp and f and their derivatives.By combining the predictor-corrector scheme for SDE and the Sinc-θ schemes 0.1 for BSDEs,we propose the following Sinc-θ schemes for the generalized Allen-Cahn equations.Scheme 0.4.Let Yin be the numerical approximations for the solution Y of the BSDE in(0.0.16)at the time-space point(τn,xi).Assume that YN on Rhd is given.For xi∈Rhd and for n=N-1,...,0,solve Xin+1 and Yin=Yn(xi)by where Xin+1=xi-εΔτv(T-τn,xi)+(?)εΔWn,1,θ1,θ2∈[0,1].In the theoretical analysis of Sinc-θ schemes,we get rid of the restrictions in the analysis of Sinc-multistep scheme.We prove the Sinc-θ schemes satisfy the discrete maximum bound principle strictly,and provide their stability analysis and error estimates.In theorem 0.11,we prove the discrete maximum bound principle of Sinc-θ schemes.Theorem 0.11(Discrete maximum bound principle).Assume that the time stepΔτ≤1/2-2θ1,then the Sinc-θ scheme preserves the maximum bound principle at the discrete level,i,e.,Define the perturbed errors asUsing the discrete maximum bound principle in Theorem 0.11,we obtain the stability results of Sinc-θ schemes.Theorem 0.12.Suppose the assumptions in Theorem 0.11 hold,then for 0≤n≤N-1,we have where C is a constant only depending on T and the parameter θ1.Using the stability results in Theorem 0.12,we provide error estimates of Sinc-θschemes in the following theorem.Theorem 0.13.Suppose the assumptions in Theorem 0.11 hold,then the numerical solution of the Sinc-θ scheme is convergent in the maximum norm.Moreover,If φ∈Cb2,we have where θ1,θ2∈[0,1],and C is a positive constant depending only on T and the upper bound of the derivatives of the terminal condition φ.Theorem 0.13 shows that Sinc-θ schemes for the generalized Allen-Cahn equations can admit the second-order convergence rate in time and the exponential convergence rate in space.We remark that several numerical results of schemes 0.3 and 0.4 are demonstrated in Section 6.4. |
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