| With the development of society and continually progress of science and technology,Stochastic ordinary differential equations(SODEs)and stochastic partial differential equations(SPDEs)are playing an increasingly important role in the fields of biology,finance,physics,chemistry,medicine,and engineering,etc.In general,the analytical solutions of SODEs and SPDEs are difficult to obtain.Therefore,it is of great theoret-ical significance and application value to construct appropriate numerical algorithms for SDEs and SPDEs.The main research content of this thesis consists of two parts.In the first part,the strong convergence and error analysis of numerical algorithms for two kinds of jump diffusion stochastic differential equations(JSDEs)are studied.Firstly,we study a class of JSDEs whose drift coefficients only satisfy the one-sided Lipschitz condition and the local Lipschitz condition.Based on the existence and uniqueness of solution to the underlying problem,a class of stochastic split-step ?-scheme is proposed and the convergence of the proposed scheme is strictly analyzed.Finally,the numerical results are conducted to verify the theoretical results;For a class of jump-diffusion CIR and CEV models,we first study the properties of the analytical solution.Then a novel time-stepping scheme,called transformed jump-adapted backward Euler method,is developed for the considered models.The proposed scheme is able to preserve the positivity of the underlying problems.Furthermore,its strong convergence rate of order one is recovered for the considered models with non-Lipschitz diffusion coefficients.Numerical examples are finally reported to confirm our theoretical findings.The second part deals with the construction and error analysis of numerical al-gorithms for SPDEs.More precisely,for a class of SPDEs driven by Gaussian and non-Gaussian noises,we first study the regularity for the mild solution on the basis of the existence and uniqueness of the solution.Next,we propose a semidiscrete finite el-ement scheme in space and a fully discrete linear implicit Euler scheme for the SPDEs,and rigorously obtain their error estimates;We also consider mean-square numerical approximations of stochastic partial differential equations within the variational formu-lation.Under suitably spatial regularity condition assumed for the exact solution and with the aid of Ito's formula,we first study the time regularity of the solution.Both semidiscrete and fully discrete schemes are proposed for the considered problem.The strong convergence of the proposed schemes with sharp rate is rigorously analyzed.We also consider the numerical approxiation for a class of backward SPDEs.We propose a spatial semidiscrete scheme.Strong convergence and error estimates in the sense of mean square are obtained.The main contribution and innovation:(1)For a class of stochastic split-step ?-scheme for JSDEs,we rigorously prove the the mean-square convergence of the proposed scheme under the condition that the drift coefficient only satisfies one-sided Lipschitz condition.The study result has been published in Adv.Appl.Math.Mech.[118].(2)Investigate numerical approximation of a class of jump-diffusion CIR and CEV models.The difficulties of identifying convergence rates essentially rely on the fact that the diffusion coefficients are non-globally Lipschitz.We first study the properties of the analytical solution(existence and uniqueness,positivity,bounded moments).Then a novel time-stepping scheme,called transformed jump-adapted backward Euler method,is developed.The proposed scheme is rigorously proved to be able to preserve the positivity of the underlying problems and is strong convergent with order one.The study result has been published in Numer.Algo-rithms[117].(3)Study the mean square error of numerical methods for SPDEs driven by Gaussian and non-Gaussian noises.We first study the regularity for the mild solution.Next,we propose a semidiscrete finite element scheme in space and a fully discrete linear implicit Euler scheme for the SPDEs,and rigorously obtain their error estimates.The study result has been published in Appl.Math.Comput.[120].(4)Consider mean-square numerical approximations of stochastic partial differential equations within the variational formulation.Both semidiscrete and fully dis-crete schemes are proposed for the considered problem.We give a strict proof of strong convergence with sharp rate for the proposed semidiscrete and fully discrete schemes.The study results has been finished[77]and[78].(5)Since the numerical research work for backward SPDEs is rather rare up to now,we study numerical approximation for a class of backward SPDEs.The semidiscrete discretization in space is done by piecewise linear finite element method.Strong convergence and error estimates in the sense of mean square are obtained.The study result has been finished[119].The framework:The thesis is divided into seven chapters.Chapter 1.IntroductionIn Chapter 1,we make a brief introduction of the background,research motivation,development of our topic in the following chapters,and re-lated concepts of stochastic computation.Chapter 2.PreliminariesIn Chapter 2,we give a brief introduce to infinite dimensional stochastic analysis,including Hilbert space valued random variables,Hilbert space valued stochastic processes,Gaussian measure and infinite dimensional Wiener process,Poisson random measure,Ito stochastic integral with respect to infinite dimensional Wiener process and Ito stochastic integral with respect to compensated Poisson random measure.We also collect some important formula including different version Ito formulas and some important inequalities,which are used in the subsequent chapters.Chapter 3.Error analysis for stochastic split-step ?-scheme for jump-diffusion SDEsIn Chapter 3,we rigorously analyze the mean-square convergence error of the stochastic split-step ?-scheme for nonlinear jump-diffusion stochastic differential equations.Under some standard assumptions,we rigorously prove that the strong rate of convergence of the split-step ?-scheme in strong sense is one half.Some numerical experiments are carried out to assert our theoretical result.This chapter is mainly based on the papers:XU YANG AND WEIDONG ZHAO,Strong convergence analysis of split-step ?-scheme for nonlinear stochastic differential equations with jumps,Adv.Appl.Math.Mech.,8(6),pp.1004-1022,2016.(SCI)Chapter 4.The positivity-preserving algorithm for two classes financial mod-els and its convergence analysis In Chapter 4.a new transformed jump-adapted backward Euler method(TJABEM)has been constructed and analyzed for a class of jump-extended CIR and CEV models.This kind of positivity preserving method is proved to possess first order strong convergence Under certain reason-able assumptions.Numerical examples are finally reported to confirm our theoretical findings.This chapter is mainly based on the paper:XU YANG AND XIAOJIE WANG,A transformed jump-adapted back-ward Euler method for jump-extended CIR and CEV models,Numer.Algorithms,74,pp.387-404,2017.(SCI)Chapter 5.Numerical methods for SPDEs and their error analysis within the semigroup frameworkIn Chapter 5,we investigate the mean square error of numerical methods for SPDEs driven by Gaussian and non-Gaussian noises.The Gaussian noise considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Levy process.As the models consider the influ-ences of Gaussian and non-Gaussian noises simultaneously,this makes the models more realistic when the models are also influenced by some randomly abrupt factors,but more complicated.As a consequence,the numerical analysis of the problems becomes more involved.We first study the regularity for the mild solution.Next,we propose a semidiscrete fi-nite element scheme in space and a fully discrete linear implicit Euler scheme for the SPDEs,and rigorously obtain their error estimates.This chapter is mainly based on the paper:XU YANG AND WEIDONG ZHAO,Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises,Appl.Math.Comput.,332,pp.58-75,2018.(SCI)Chapter 6.Numerical method for SPDEs within the variational formula-tionIn Chapter 6,we study strong approximation methods for a class of stochastic partial differential equations within the variational formula-tion.We first prove the existence and uniqueness of the variational so-lution,then under suitably spatial regularity condition assumed for the exact solution we further study the time regularity of the solution.Both semidiscrete and fully discrete schemes are proposed for the considered problem.With the obtained time regularity and some well-known er-ror estimates from deterministic PDEs,we give a strict proof of strong convergence with error estimates for the proposed schemes.This chapter is mainly based on the paper:STIG LARSSON,XU YANG,AND WEIDONG ZHAO,Convergence es-timates for spatially semidiscrete approximation of stochastic partial differential equations,Completed.STIG LARSSON,XU YANG,AND WEIDONG ZHAO,Strong con-vergence analysis for numerical approximation of stochastic partial differential equations,Completed.Chapter 7.Spatially semidiscrete approximation of backward SPDEs In Chapter 7,we study the semidiscrete approximation of nonlinear back-ward SPDEs The discretization in space is done by piecewise linear finite element method.We rigorously prove the strong convergence for the spa-tially semidiscrete approximation with precise convergence rates given.This chapter is mainly based on the paper:XU YANGAND WEIDONG ZHAO,Convergence estimates of semidis-crete finite element method for nonlinear backward stochastic partial differential equations,Submitted.The main results:Chapter 3:we consider a class of stochastic split-step ?-scheme for JS-DEs,and prove the strong convergence error of the proposed numerical scheme under weaker condition.We consider jump-diffusion Ito stochastic differential equations(JSDEs)of the form where X(t):=lims?t-X(s),f:Rm Rm,g:Rm ?Rm×d and h:Rm?Rm,m,d ?N+.Here W(t)is a standard d-dimensional Wiener process,and N(t)is a scalar Poisson process(independent of W(t)with intensity ?>0,both defined on a complete probability space(?,F,F,P)with a filtration F satisfying the usual conditions.For the time interval[0,T],we introduce the following time partition:0 = t0<t1<…<tM-1<tM = T,where M is a positive integer.Let ?tn = tn+1-tn.For simplicity,we consider the uniform partition of the interval[0,T],i.e.,?t = ?tn = T/M.Scheme 0.1(stochastic split-step ?-scheme).given the initial value Y0 = X0,compute{Yn}n=1M by Yn*=Yn+?t?f(Yn*),Yn+1=Yn+?tf(Yn*)+g(Yn*)?Wn+h(Yn*)?Nn,n = 0,1,…,M-1,for n = 0,1,…,M-1,where ??[0,1]is a fixed parameter,Yn is the approximation of X(tn)at time tn,?Wn:= W(tn+1)-W(tn)and ?Nn =:N(tn+1)-N(tn)are the increments of Wiener process Wt and the Poisson process Nt,respectively.In particular,in the case ? = 0,the scheme(0.1)reduces to the standard Euler scheme for JSDEs,while,if ? = 1,the proposed scheme(0.1)is equivalent to the split-step backward Euler scheme,which has been introduced and discussed in[50,52].For the convenience of discussion,we define the continuous extension Y(t)of Yn on[tn,tn+1)byY(t):=Yn+(t-tn)f(Yn*)+g(Yn*)?Wn(t)+h(Yn*)?Nn((t),t?[tn,tn+1),(0.22)where ?Wn(t):= W(t)-W(tn)and ?Nn(t):= N(t)-N(tn).Equivalently,we can rewrite(0.22)in the integral form where and IF is the characteristic function of a set F,namely,Note that Y(tn)= Y(tn)= Yn,meaning that Y(t)and Y(t)coincide with the discrete solutions at the grid-points,hence we can study the error in Y(t)in the supremum norm.This will of course give an immediate bound for the error in the discrete ap-proximation.We state our strong convergence result in the following theorem.Theorem 0.1.Let X(t)and Y(t)be the solution of(0.21)and(0.22),respectively.Then under assumptions(3.2),(3.3),(3.4),(3.5),(3.7)and(3.8),if 1/2???1 and?t<?t0<1/2L with constant L in(3.6),we have the estimate where C is a positive constant independent of At.Chapter 4:we propose transformed jump-adapted backward Euler method(TJABEM)for a class of jump-extended CIR and CEV models and analyze the scheme theoretically and numerically.Consider jump-diffusion Ito stochastic differential equations(JSDEs)described by where Xt-:=lims?t-Xs.Wt is a scalar Wiener process and Nt is a scalar Poisson process with intensity ?>0,defined on a complete probability space(?,F,P),with a normal filtration F,and they are independent with each other,when ? = 1/2,(0.23)is usually called jump-extended CIR model;when ? ?(1/2,1),(0.23)is usually called jump-extended CEV model.Part I:Properties of the exact processTheorem 0.2.Under conditions(4.3)-(4.4),and for X0>0,the problem(0.23)admits a unique solution,which remains positive with probability 1,if condition(?)or(?)in Lemma 4.1 holds.Theorem 0.3.Suppose ?,?,?>0,(4.3)holds and assume x + g(x)? ?x,(?)x>0 for a constant ?>0.Then we have the following properties.(1)For the jump-extended CIR model with ?=1/2 and 2????2,there exists a constant Cp,T depending on p,?,?,?,L,T such that the(inverse)moments of the solution to(0.23)satisfy if p ?[?2-2??/?2,?).In particular when g(x)?0,(?)x>0,(0.24)remains valid for all p ?(-2??/?2,?).(2)For the jump-extended,CEV model with ??(1/2,1),the solution to(0.23)satisfies(0.24)for all p ?(-?,?).Part ?:The proposed TJABEM and numerical analysisWe first introduce a new trasformed process Yt= Xt1-?,which,by Ito's formula for jump diffusion,obeys the following SDEs:with Y0 = X01-? and f?(y)given by For the transformed problem(0.25),we introduce a jump-adapted backward Euler method(JABEM).Transforming the JAB EM back we obtain a numerical scheme for the original SDEs(0.23).To this end,we construct a jump-adapted time partition T={0 =t0<t1<…<tnT= T},produced by a superposition of the jump times {T1,T2,…} to a deterministic equidistant grid with time step-size ?t =T Here nT:= max{n ? {0,1,…}:tn ? T}.This way the adapted time discretization including all jump times is path-dependent and the maximum step-size of the jump-adapted partition is ?t.On the grid{Yt}t?[0,T]can be expressed as for k = 0,1,…nT-1.Accordingly we propose a jump-adapted backward Euler method(JABEM)for(0.25),defined through Y0= Y0 and for k = 0,1,…,nT-1,where ?tk:= tk+1-tk,?Wk:=W(tk+1)-W(tk),and ?Nk:= N(tk+1)?N(tk)? {0,1}.If tn+1 is a jump time,then ANk = 1.Otherwise ?Nk = 0.Transforming the numerical approximations Ytk back yields numerical approximations of(0.23):Xtk=(Ytk)1/(1-?),for k=0,1…,nT-1.(0.27)The scheme(0.26)-(0.27)is called a transformed jump-adapted backward Euler method(TJABEM)for(0.23).We have the following positivity preserving and strong convergence results.Theorem 0.4.Under conditions stated in Theorem 4.2 and Yt0>0,the scheme(0.26)is well-defined and positivity preserving,i.e.,with probability one,Ytk>0 implies Ytk+1>0,from which we have Xtk>0,k = 0,1,…,nT.Theorem 0.5.Let conditions(4.3)-(4.4)and Assumptions 4.1,4.2 be fulfilled.Then for any r ?[1,p*),there exists a constant Cp*,r>0(independent of ?t)such thatTheorem 0.6.Let k,?,?>0 and let conditions(4.3),(4.9)and Assumption 4.2 hold.Then the proposed scheme(0.26)-(0.27)for the jump-extended CIR model(0.23)with?= 1/2 is strongly convergent with order one in p-th mean:for any p ?[1,4k?-2?2/3?2)with ??? 3/2?2.In the particular case when g(x)? 0,(?)x>0,(0.28)holds for any p ?[1,4??/3?2)with ??>?2.Theorem 0.7.Let ?,?,?>0 and let conditions(4.3),(4.9)and Assumption 4.2 hold.Then the proposed scheme(0.26)-(0.27)for the jump-extended CEV model(0.23)with? ?(1/2,1)is strongly convergent with order one in p-th mean,i.e.,for any p ?[1,?).Chapter 5:Study numerical methods for SPDEs driven by Gaussian and non-Gaussian noises.The strong convergence of the proposed schemes are strictly proved and the corresponding error estimates are obtained.Given a complete filtered probability space(?,F,F,P),consider the following stochastic partial differential equations(SPDEs)with values in a separable Hilbert space(H,(·,·),?·?)where ? is the mark set defined by ?:= H\{0}.Here A:D(A)(?)H ? H is assumed to a linear operatorl which is not necessarily bounded.F:H?H,G:H ?L20,and f:?×H?H are deterministic mappings.W(t)is a Q-Wiener process with values in a separable Hilbert space(K,(·,·)K,?·?K).N(dz,dt)is a compensated Poisson random measure.Definition 0.1(mild solution).A stochastic process {Xt),t?[0,T]} is called the mild solution of(0.29)if1.X(t)is Ft-adapted on the filtered probability space(?,T,F,P),2.{X(t),t?[0,T]} is joint measurable and E[?0T?X(t)?2dt]<?,3.For arbitrary t ?[0,T],holds a..Part I:Regularity of the mild solutionWe first discuss the regularity of the mild solution Xt of the SPDE(0.29).Theorem 0.8(space regularity).Suppose Assumptions 5.1,5.2,5.3 and 5.4 hold.Let X be the mild solution of(0.29).If X0?L2(?,F0;H?),??[0,1),then for all t ?[0,T],X(t)? L2(?,H?)satisfying E[?X(t)??2]<?.Theorem 0.9(time regularity).Suppose Assumptions 5.1,5.2,5.3 and 5.4 hold.Let X be the mild solution of(0.29),If X0 E L2(?,F0;H?),??[0,1),then we have for 0 ? t1 ?t2?T,E[?X(ti)-X(t2)?2]<C(t2-t1)?.Part ?:Spatially semidiscrete scheme and its error estimates Let {Th} be a family of triangulations of D,indexed by the maximal mesh size h.For each Th,we construct a finite element space Sh,consisting of continuous piecewise linear functions such that Sh(?)H01(D).Based on the projection operators Ph and Ah defined in(5.22)and(5.23),respectively,we propose the spatially semidiscrete scheme for the problem(0.29)as:solve the process Xh(t)= Xh(·,t)?Sh satisfying dXh(t)+ AhXh(t)dt = PhF(Xh(t))dt + PhG(Xh(t))dW(t)for t ?(0,T]with the initial condition Xh(0)= PhX0.It is not hard to check that(0.30)satisfies the conditions in Theorem 0.14,hence the semidiscrete equation(0.30)also has a unique mild solution Xh ? Sh given byWe are going to prove the following theorem,which provides an estimate in mean square sense for the error between the solutions of the SPDE(0.29)and the spatially semidiscrete approximation(0.30).Theorem 0.10.Let X and Xh be the mild solutions of(0.29)and(0.30),respectively.Suppose that the Assumptions 5.1 and 5.2 hold.If X0 ? L2(?,F0;H?),??[0,1),then there exists a constant C independent of h,such that E[?Xh(t)-X(t)?2]?Ch2?,for all t ?[0,T].Part ?:Fully discrete Euler scheme and its error estimatesBased on the finite element spatial approximation scheme(0.30),together with the linear implicit Euler scheme in time,we propose the fully discrete linear implicit Euler scheme for solving SPDEs(0.29)as:for m = 0,1,…,M-1,solve Xh,m+1 ? Sh by with the initial value Xh,0 = PhX0,where Eh,k is defined in(5.29)and Xh,m denotes the approximation of the solution Xh(t)of(0.30)at time tm.For the above fully discrete linear implicit Euler scheme,we state our second main result in the following theorem.Theorem 0.11.Let Xh,m and X be the mild solutions of(0.32)and(0.29),respectively.Suppose Assumptions 5.1-5.4 hold.If X0 ?L2(?,F0;H?),??[0,1),then there exists a constant C independent of h and k,such thatE[?Xh,m-X(tm)?2]?C(max(h2?,h4/k)+k?),m = 0,1,2,…,M.Chapter 6:study strong approximation methods for a class of stochastic partial differential equations within the variational formulation.Consider stochastic partial differential equations(SPDEs)of the following form:where D is a bounded convex domain in Rd with polygonal boundary(?)D,the operator L defined by Lu:= ?j,kd=1(?)/(?)xj(ajk(x)(?)u/(?)xk)+a0(x)u is a second order elliptic opera-tor,and W is a,standard Q-Wiener process with values in a separable Hilbert space(K,(·,·)K,?·?K)defined on a complete probability space(?,F,F,P)with normal filtration F.Define the Hilbert space H = L2(D),equipped with the usual inner product(·,·)and norm ?·?.Let A =-L with D(A)= H2(D)? H01(D),where Hm(D)denotes the standard Sobolev space of integer order m?1 and H01(D):= {??H1(D):?=0 on(?)D}.Set V = H01(D).We identify H with its dual H*,and denote by V*the dual of V.Then we have the Gelfand triplet V?H ? V*with compact and dense embeddings.The duality paring between V and V*is denoted by<·,·>.Obviously,the following holds:(u,v)=(u,v),u ? H,v ?V.Now,the problem(0.33)can be formulated in an abstract form in Ito's sense as follows:At this point,we introduce the definition of variational solution to(0.34).We refer the reader to[23]for more details.Definition 0.2(Variational solution).The process u is called a variational solution to(0.34)if u ?LF2((0,T);V)and,for any ? ? V,the following equation holds for each t ?[0,T]a.s.Part ?:The wellposedness of the problem and the regularity of variational solutionTheorem 0.12.Let Assumptions 6.1-6.3 hold.Then(0.34)admits a unique varia-tional solution satisfying i.e.,u ?LF2((0,T);V)?LF2(?;C([0,T];H)).Next we are to derive the time regularity of the solution u.Theoem 0.13.Under the same assumptions of Theorem 0.12,let u be the variational solution of SPDE(0.34).Additionally,we assume u ? CF([0,T];L2(?,H1)).Then,for any s,t ?[0,T]with s<t,we haveTheorem 0.14.Under the same assumptions of Theorem 0.12,let u be the vari-ational solution,of(0.34).Additionally,we suppose that Assumption 6.4 holds and u ? CF([0,T];L2(?,H2)).Then,for any s,t ?[0,T]with s<t,we havePart ?:Spatially semidiscrete scheme and its error estimatesFrom the definition of the variational solution and the bilinear form a(·,·),we have(u(t),?)Let {Th} be a quasi-uniform family of partitions of D,indexed by the maximal mesh size h.For each Th,we construct a finite element space Sh,consisting of continuous piecewise linear functions such,that Sh V = H01(D).The spatially semidiscrete problem is then to find a Sh-valued process uh(t)= uh(·,t)such that(uh(t),?h)With Ah and Ph defined in(6.11)and(5.22),respectively,the spatially semidiscrete finite element scheme(0.36)can be rewritten as:duh(t)+ Ahuh(t)dt = Phf(uh(t))dt + Ph(g(uh(t))dW(t)),for t E(0,T]with the initial condition uh(0)= Phu0.The following theorem provides an estimate in the mean square sense for the error between the solutions of the SPDEs(0.34)and the spatially semidiscrete approximation(0.36).Theorem 0.15.Let u and uh be the solutions of(0.34)and(0.36),respectively.Sup-pose that Assumptions 6.1-6.3 hold.Assume thatu ?LF2((0,T);H1+?)?LF2(?;C([0,T];H?))for ? ?(0,1].Then there exists a constant C independent of h,such that Part ?:Fully discrete Euler scheme and its error estimates Let tn = nk,(n = 0,1,…,L)be a uniform partition of[0,T]with k = T/L.It follows from(0.35)thatDenote by uhn ? Sh the approximation of u(tn)at time tn.The proposed fully discrete finite element method for(0.34)is defined by where ?Wn = W(tn+1)-W(tn).Now We give the following theorem,which provides an estimate in the mean square sense for the error between the solutions of the SPDEs(0.34)and the fully discrete approximation(0.38).Theorem 0.16.Let u(tn)and uhn,n = 0,1,2,…,L,be the solutions of(0.34)and(0.38),respectively.Suppose that Assumptions 5.1-5.4 hold.Then there exists a constant C independent of h and k,such that if u ?LF2(?;C([0,T];H2)),Chapter 7:A space semidiscrete scheme for solving a class of backward SPDEs is proposed and the strong convergence of the scheme is analyzed strictly.Consider backward SPDEs with the following form:Here T>0 is fixed and D is a bounded convex domain in D(?)Rd,d = 1,2,3,with a sufficiently smooth boundary(?)D.The operator A defined by is a second order elliptic operator,and ?u denotes the gradient of u with respect to x,i.e.W = {W(t):t ?[0,T]} is a standard Wiener process defined on a completed proba-bility space(?,F,F,P).uT is a random fields,FT-measurable at each x.Define the Hilbert space H = L2(D)equipped with the usual inner product(·,·)and norm ?·?.Next,we introduce the space related to the fractional powers of the linear operator A.For ??R,define H?:= D(A?/2)with the norm ?·??:=?A?/2·?.It follows that H?(?)H(?)for ? ?(?).From the definition of fractional powers of the of A,it is well-known that H0=H,H0= H01(D)and H2 = H2(D)? H01(D)= D(A).Now,we formulate the problem(0.39)in an abstract form in Ito's sense with values in Hilbert space(H,(·.·),?·?)Let {Th}h?(0,1]be a family of triangulations of D1 indexed by the maximal mesh size h.Furthermore,assume that the family Let {Th}h?(0,1]is quasi-uniform.For each Th,we construct a finite element space Sh,consisting of continuous piecewise linear functions such that Sh(?)H01(D).The spatially semidiscrete problem for(0.40)is then to find functions uh(t)=uh(·,t)? Sh and vh(t)= vh(·,t)?Sh such that for all t E[0,T]With Ph and Ah defined in(5.22)and(6.11),respectively,the spatially semidiscrete finite element scheme(0.41)can be rewritten as:-duh(t)+ Ahuh(t)dt = Phf(t,?uh(t),uh(t),vh(t))dt-vh(t)dW(t)for t E[0,T)with the terminal condition uh(T)= PhuT.The following theorem provides an estimate in mean,square sense for the error between the solutions of backward SPDEs(0.40)and the.spatially semidiscrete ap-proximation(0.41).Theorem 0.17.Let(u,v)and(uh,vh)be the,mild solutions of(0.40)and(0.41),respectively.Suppose that the Assumptions 7.1 and 7.2 hold.If uT ? L2(?,FT;H1),then there exists a constant C independent of h,such that for all t?[0,T],. |
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