arXiv:1408.6471v3 [math.PR] 7 Apr 2016 (1.1) 06 o.2,N.2 1147–1207 2, No. DOI: 26, Probability Vol. Applied 2016, of Annals The SE on (SDE) ue cee rcinlcluu,Mlivncluu,fo calculus, Malliavin calculus, fractional scheme, Euler
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HU, Y. LIU AND D. NUALART where x Rd, B = (B1,...,Bm) is an m-dimensional fractional Brown- ian motion∈ (fBm) with Hurst parameter H ( 1 , 1) and b, σ1,...,σm : Rd ∈ 2 → Rd are continuous functions. The above stochastic integrals are pathwise Riemann–Stieltjes integrals. If σ1,...,σm are continuously differentiable and their partial derivatives are bounded and locally H¨older continuous of or- der δ> 1 1 and b is Lipschitz, then equation (1.1) has a unique solution H − which is H¨older continuous of order γ for any 0 <γ t m t n n j n j (1.2) Xt = x + b(Xη(s)) ds + σ (Xη(s)) dBs, t [0, T ]. 0 0 ∈ Z Xj=1 Z This scheme can also be written as m n n n j n j j X = X + b(X )(t tk)+ σ (X )(B B ), t tk tk − tk t − tk Xj=1 for t t t , k = 0, 1,...,n 1 and Xn = x. It was proved by Mishura k ≤ ≤ k+1 − 0 [17] that for any real number ε> 0 there exists a random variable Cε such that almost surely, n 1 2H+ε sup Xt Xt Cεn − . 0 t T | − |≤ ≤ ≤ 1 2H Moreover, the convergence rate n − is sharp for this scheme, in the sense 2H 1 n that n − [Xt Xt] converges almost surely to a finite and nonzero limit. This has been proved− in the one-dimensional case by Nourdin and Neuenkirch in [18] using the Doss representation of the solution; see also Theorem 10.1 below. Notice that while H tends to 1 , the convergence rate 2H 1 of the 2 − numerical scheme (1.2) deteriorates, and so it is not a proper extension of 1 the Euler–Maruyama scheme for the case H = 2 ; see, for example, [7, 12]. This is not surprising because the limit H 1 of the SDE (1.1) corresponds → 2 to a Stratonovich SDE driven by standard Brownian motion, while the Euler scheme (1.2) is the extension of the classical Euler scheme for the ItˆoSDEs. It is then natural to ask the following question: Can we find a numerical scheme that generalizes the Euler–Maruyama scheme to the fBm case? EULER SCHEMES OF SDE DRIVEN BY FBM 3 In this paper we introduce the following new approximation scheme that we call a modified Euler scheme: t m t n j n j Xt = x + b(Xη(s)) ds + σ (Xη(s)) dBs Z0 j=1 Z0 (1.3) X m t j j n 2H 1 + H ( σ σ )(Xη(s))(s η(s)) − ds, 0 ∇ − Xj=1 Z or m n n n j n j j X = X + b(X )(t tk)+ σ (X )(B B ) t tk tk − tk t − tk Xj=1 m 1 + ( σjσj)(Xn )(t t )2H , 2 ∇ tk − k Xj=1 n j for any t [tk,tk+1] and X0 = x. Here σ denotes the d d matrix ∂σj,i ∈ j j i d ∂σj,i j,k∇ × ( )1 i,k d, and ( σ σ ) = k=1 σ . ∂xk ≤ ≤ ∇ ∂xk Notice that if we formally set H = 1 and replace B by a standard Brow- P 2 nian motion W , this is the classical Euler scheme for the Stratonovich SDE, t m t j j Xt = x + b(Xs) ds + σ (Xs) dWs 0 0 Z Xj=1 Z t m t t m j j 1 j j = x + b(Xs) ds + σ (Xs)δWs + ( σ σ )(Xs) ds. 0 0 2 0 ∇ Z Xj=1 Z Z Xj=1 In the above and throughout this paper, d denotes the Stratonovich integral, and δ denotes the Itˆo(or Skorohod) integral. For our new modified Euler scheme (1.3) we shall prove the following estimate: E n p 1/p 1 (1.4) sup ( Xt Xt ) Cγn− , 0 t T | − | ≤ ≤ ≤ for any p 1, where ≥ 2H 1/2 1 3 n − , if 2 m nt/T ⌊ ⌋ tk+1 s n i,j i j n (1.6) Xt Xt = f (tk) δBuδBs + Rt , − tk tk i,jX=1 Xk=0 Z Z where x denotes the integer part of a real number x. The weighted quadratic variation⌊ ⌋ term provides the desired rate of convergence in Lp. To further study this new scheme and compare it to the classical Brownian motion case, it is natural to ask the following questions: Is the above rate n of convergence (1.4) exact or not? Namely, does the quantity γn(Xt Xt ) have a nonzero limit? If yes, how does one identify the limit, and is− there a similarity to the classical Brownian motion case (see [10, 13])? In the second part of the paper, we give a complete answer to these questions. The weighted variation term in (1.6) is still a key ingredient in our study of the scheme. As in the Breuer–Major theorem, there is a different behavior in 1 3 3 1 3 n the cases H ( 2 , 4 ] and H ( 4 , 1). If H ( 2 , 4 ], we show that γn(Xt Xt ) converges stably∈ to the solution∈ of a linear∈ stochastic differential equation− driven by a matrix-valued Brownian motion W independent of B. The main tools in this case are Malliavin calculus and the fourth moment theorem. We will also make use of a recent limit theorem in law for weighted sums proved 3 n in [3]. In the case H ( 4 , 1), we show the convergence of γn(Xt Xt ) in Lp to the solution of∈ a linear stochastic differential equation driven− by a matrix-valued Rosenblatt process. Again we use the technique of Malliavin calculus and the convergence in Lp of weighted sums, which is obtained applying the approach introduced in [3]. We refer to [20] for a discussion on the asymptotic behavior of some weighted Hermite variations of one- dimensional fBm, which are related with the results proved here. We also consider a weak approximation result for our new numerical 1 scheme. In this case, the rate is n− for all values of H. More precisely, we are able to show that n[E(f(X )) E(f(Xn))] converges to a finite nonzero t − t EULER SCHEMES OF SDE DRIVEN BY FBM 5 limit which can be explicitly computed. This extends the result of [31] to 1 H> 2 . Let us mention that the techniques of Malliavin calculus also allow us to provide an alternative and simpler proof of the fact that the rate of con- 1 2H vergence of the numerical scheme (1.2) is of the order n − , and this rate is optimal, extending to the multidimensional case the results by Neuenkirch and Nourdin [18]. If the driven process is a standard Brownian motion, similar problems have been studied in [10, 13] and the references therein. See also [2] for the precise L2-limit and also for a discussion on the “best” partition. In the case 1 1 4 2. Preliminaries and notation. Throughout the paper we consider a fixed time interval [0, T ]. To simplify the presentation we only deal with the uni- form partition of this interval; that is, for each n 1 and i = 0, 1,...,n, we iT ≥ set ti = n . We use C and K to represent constants that are independent of n and whose values may change from line to line. 2.1. Elements of fractional calculus. In this subsection we introduce the definitions of the fractional integral and derivative operators, and we review some properties of these operators. 6 Y. HU, Y. LIU AND D. NUALART Let a, b [0, T ] with a < b, and let β (0, 1). We denote by Cβ(a, b) the space of β-H¨older∈ continuous functions on∈ the interval [a, b]. For a function x : [0, T ] R, x denotes the β-H¨older seminorm of x on [a, b], that is, → k ka,b,β x x x = sup | u − v|; a u < v b . k ka,b,β (v u)β ≤ ≤ − We will also make use of the following seminorm: x x (2.1) x = sup | u − v|; a u < v b, η(u)= u . k ka,b,β,n (v u)β ≤ ≤ − Recall that for each n 1 and i = 0, 1,...,n, t = iT and η(t)= t when ≥ i n i t t Proposition 2.1. Suppose that f Cλ(a, b) and g Cµ(a, b) with λ + ∈ ∈ b µ> 1. Let λ > α and µ> 1 α. Then the Riemann–Stieltjes integral a f dg exists, and it can be expressed− as R b b α α 1 α (2.4) f dg = ( 1) Da+f(t)D − gb (t) dt, − b − Za Za − where gb (t)= 1(a,b)(t)(g(t) g(b )). − − − The notion of H¨older continuity and the above result on the existence of Riemann–Stieltjes integrals can be generalized to functions taking values in some normed spaces. We fix a probability space (Ω, F , P ) and denote by the norm in the space Lp := Lp(Ω), where p 1. k · kp ≥ Definition 2.1. Let f = f(t),t [0, T ] be a stochastic process such that f(t) Lp for all t [0, T ].{ We say∈ that f} is H¨older continuous of order β> 0 in L∈p if ∈ (2.5) f(t) f(s) C t s β, k − kp ≤ | − | for all s,t [0, T ]. ∈ The following result shows that with proper H¨older continuity assump- T tions on f and g, the Riemann–Stieltjes integral 0 f dg exists, and equation (2.4) holds. R Proposition 2.2. Let the positive numbers p , λ, µ, p, q satisfy p 1, 0 0 ≥ λ + µ> 1, 1 + 1 = 1 and p p> 1 , p q> 1 . Assume that f = f(t),t [0, T p q 0 µ 0 λ { ∈ } and g = g(t),t [0, T ] are H¨older continuous stochastic processes of order { ∈ } µ and λ in Lp0p and Lp0q, respectively, and f(0) Lp0p. Let π :0= t < ∈ 0 t1 < 2.2. Elements of Malliavin calculus. We briefly recall some basic facts about the stochastic calculus of variations with respect to an fBm. We refer 1 m the reader to [22] for further details. Let B = (Bt ,...,Bt ),t [0, T ] be an m-dimensional fBm with Hurst parameter H{ ( 1 , 1), defined∈ on some} ∈ 2 8 Y. HU, Y. LIU AND D. NUALART complete probability space (Ω, F , P ). Namely, B is a mean zero Gaussian process with covariance E(BiBj)= 1 (t2H + s2H t s 2H )δ , i, j = 1,...,m, t s 2 − | − | ij for all s,t [0, T ], where δij is the Kronecker symbol. Let be∈ the Hilbert space defined as the closure of the set of step func- tions onH [0, T ] with respect to the scalar product 1 2H 2H 2H 1[0,t], 1[0,s] = (t + s t s ). h iH 2 − | − | It is easy to see that the covariance of fBm can be written as t s 2H 2 α u v − du dv, H | − | Z0 Z0 where α = H(2H 1). This implies that H − T T 2H 2 ψ, φ = αH ψuφv u v − du dv h iH | − | Z0 Z0 for any pair of step functions φ and ψ on [0, T ]. l The elements of the Hilbert space , or more generally, of the space ⊗ may not be functions, but distributions;H see [28] and [29]. We can findH a l l linear space of functions contained in ⊗ in the following way: Let ⊗ be the linear space of measurable functionsH φ on [0, T ]l Rl such that|H| ⊂ 2 l 2H 2 2H 2 φ ⊗l := αH φu φv u1 v1 − ul vl − du dv < , k k 2l | || || − | ···| − | ∞ |H| Z[0,T ] l 1/H l where u = (u1, . . ., ul), v = (v1, . . ., vl) [0, T ] . Suppose φ L ([0, T ] ). The following estimate holds: ∈ ∈ (2.6) φ ⊗l bH,l φ L1/H ([0,T ]l) k k|H| ≤ k k for some constant bH,l > 0; the case l = 1 was proved in [16], and the exten- sion to general case is easy; see [9], equation (2.5). 1 m The mapping 1[0,t1] 1[0,tm] (Bt1 ,...,Btm ) can be extended to a linear isometry between×···×m and the7→ Gaussian space spanned by B. We de- note this isometry by h HB(h). In this way, B(h), h m is an isonormal Gaussian process indexed7→ by the Hilbert space{ m. ∈H } Let be the set of smooth and cylindrical randomH variables of the form S F = f(Bs1 ,...,BsN ), Rm N where N 1 and f Cb∞( × ). For each j = 1,...,m and t [0, T ], the derivative≥ operator D∈jF on F is defined as the -valued random∈ vari- able ∈ S H N ∂f DjF = (B ,...,B )1 (t), t [0, T ]. t j s1 sN [0,si] ∈ ∂xi Xi=1 EULER SCHEMES OF SDE DRIVEN BY FBM 9 We can iterate this procedure to define higher order derivatives Dj1,...,jl F which take values on l. For any p 1 and any integer k 1, we define H⊗ ≥ ≥ the Sobolev space Dk,p as the closure of with respect to the norm S k m p/2 p E p E j1,...,jl 2 F k,p = [ F ]+ D F ⊗l . k k | | " k kH ! # Xl=1 j1,...,jXl=1 If V is a Hilbert space, Dk,p(V ) denotes the corresponding Sobolev space of V -valued random variables. For any j = 1,...,m, we denote by δj the adjoint of the derivative operator Dj. We say u Dom δj if there is a δj (u) L2(Ω) such that for any F D1,2 ∈ ∈ ∈ the following duality relationship holds: (2.7) E( u, DjF )= E(δj (u)F ). h iH The random variable δj(u) is also called the Skorohod integral of u with j j T j respect to the fBm B , and we use the notation δ (u)= 0 utδBt . Let F D1,2 and u be in the domain of δj such that F u L2(Ω; ). Then ∈ R∈ H (see [23]) F u belongs to the domain of δj , and the following equality holds: (2.8) δj (F u)= F δj(u) DjF, u , − h iH provided the right-hand side of (2.8) is square integrable. Suppose that u = u ,t [0, T ] is a stochastic process whose trajecto- { t ∈ } ries are H¨older continuous of order γ> 1 H. Then, for any j = 1,...,m, T j − the Riemann–Stieltjes integral 0 ut dBt exists. On the other hand, if u 1,2 j ∈ D ( ) and the derivative Dsu exists and satisfies almost surely H Rt T T j 2H 2 D u t s − ds dt < , | s t|| − | ∞ Z0 Z0 and E( Dju 2 ) < , then (see Proposition 5.2.3 in [23]) T u δBj k kL1/H ([0,T ]2) ∞ 0 t t exists, and we have the following relationship between these two stochastic R integrals: T T T T j j j 2H 2 (2.9) u dB = u δB + α D u t s − ds dt. t t t t H s t| − | Z0 Z0 Z0 Z0 The following result is Meyer’s inequality for the Skorohod integral; see, for example, Proposition 1.5.7 of [23]. Given p> 1 and an integer k 1, ≥ there is a constant ck,p such that k Dk,p k (2.10) δ (u) p ck,p u Dk,p( ⊗k) for all u ( ⊗ ). k k ≤ k k H ∈ H 10 Y. HU, Y. LIU AND D. NUALART Applying (2.6) and then the Minkowski inequality to the right-hand side of (2.10) yields k δ (u) p C u p L1/H ([0,T ]p) (2.11) k k ≤ kk k k k m j1,...,jl + C D u 1/H p+l kk kpkL ([0,T ] ) Xl=1 j1,...,jXl=1 for all u Dk,p( k), provided pH 1. ∈ H⊗ ≥ 2.3. Stable convergence. Let Yn, n N be a sequence of random variables defined on a probability space (Ω, F , P∈) with values in a Polish space (E, E ). We say that Yn converges stably to the limit Y , where Y is defined on an extension of the original probability space (Ω′, F ′, P ′), if and only if for any bounded F -measurable random variable Z, it holds that (Y ,Z) (Y,Z) n ⇒ as n , where denotes the convergence in law. Note→∞ that stable⇒ convergence is stronger than weak convergence but weaker than convergence in probability. We refer to [11] and [1] for more details on this concept. 2.4. A matrix-valued Brownian motion. The aim of this subsection is to define a matrix-valued Brownian motion that will play a fundamental role in our central limit theorem. First, we introduce two constants Q and R which depend on H. 2 2H 2 Denote by µ the measure on R with density s t − . Define, for each p Z, | − | ∈ 1 p+1 t s Q(p)= T 4H µ(dv du)µ(ds dt) Z0 Zp Z0 Zp and 1 p+1 1 s R(p)= T 4H µ(dv du)µ(ds dt). Z0 Zp Zt Zp 1 3 It is not difficult to check that for 2 (2.12) Q = Q(p),R = R(p), p Z p Z X∈ X∈ EULER SCHEMES OF SDE DRIVEN BY FBM 11 for the case H ( 1 , 3 ), and ∈ 2 4 4H 4H p n Q(p) T p n R(p) T Q = lim | |≤ = ,R = lim | |≤ = , n log n 2 n log n 2 →∞ P →∞ P 3 for the case H = 4 . Lemma 2.1. The constants Q and R satisfy R Q. ≤ 3 Proof. If H = 4 , we see from (2.12) that these two constants are both 4H equal to T . Suppose H ( 1 , 3 ). Consider the functions on R2 defined by 2 ∈ 2 4 ϕp(v,s)= 1 p v s p+1 , ψp(v,s)= 1 p s v p+1 , p Z. Then { ≤ ≤ ≤ } { ≤ ≤ ≤ } ∈ n 1 2 1 − (ϕp ψp) n − p=0 L2(R2,µ) X n 1 2 − = ( 1 p v s p+1 , 1 q v s q+1 L2(R2,µ) n h { ≤ ≤ ≤ } { ≤ ≤ ≤ }i p,qX=0 1 p v s p+1 , 1 q s v q+1 L2(R2,µ)) − h { ≤ ≤ ≤ } { ≤ ≤ ≤ }i n 1 2 − = (Q(p q) R(p q)). n − − − p,q=0 X It is easy to see that the above is equal to n 1 j 2 − (Q(k) R(k)). n − j=0 k= j X X− It then follows from a Ces`aro limit argument that the quantity in the right- hand side of the above converges to 2(Q R) as n tends to infinity. Therefore, Q R. − ≥ 0,ij 0,ij 1,ij 1,ij Let W = Wt ,t [0, T ] , i j, i, j = 1,...,m and W = Wt ,t [0, T ] , i, j = 1,...,m{ be∈ independent} ≤ standard Brownian motions.{ When i>∈ } 0,ij 0,ji ij j, we definef Wt f= Wt . The matrix-valued Brownian motionf (Wf)1 i,j m, i, j = 1,...,m is defined as follows: ≤ ≤ f f α W ii = H ( Q + RW 1,ii) √T p and f α W ij = H ( Q RW 1,ij + √RW 0,ij) when i = j. √T − 6 p f f 12 Y. HU, Y. LIU AND D. NUALART Fig. 1. Simulation of q and r. Notice that this definition makes sense because R Q. The random matrix 3 ≤ Wt is not symmetric when H< 4 ; see the plot and table below. For i, j, i′, j′ = E ij i′j′ 1,...,m, the covariance (Wt Ws ) is equal to 2 αH (t s) ∧ (Rδ ′ δ ′ + Qδ ′ δ ′ ), T ji ij jj ii where δ is the Kronecker function. In the Figure 1 and Table 1, we consider two quantities for H ( 1 , 3 ), ∈ 2 4 α2 α2 q = H Q and r = H R. T 4H T 4H 1 We see that the values of q and r approach 0.5 and 0 as H tends to 2 , 3 respectively, and both of them tend to infinity when H gets closer to 4 . 2.5. A matrix-valued generalized Rosenblatt process. In this subsection we introduce a generalized Rosenblatt process which will appear in the lim- 3 iting result proved in Section 8 when H > 4 . Consider an m-dimensional 1 m 3 fBm Bt = (Bt ,...,Bt ) with Hurst parameter H ( 4 , 1). Define for i1, i2 1,...,m, ∈ ∈ nt/T ⌊ ⌋ tj+1 i1,i2 i1 i1 i2 Zn (t) := n (Bs Btj )δBs . tj − Xj=1 Z EULER SCHEMES OF SDE DRIVEN BY FBM 13 Table 1 Simulation of q and r H 0.5010 0.5260 0.5510 0.6010 0.6260 0.6510 0.7010 0.7260 q 0.4990 0.4763 0.4580 0.4369 0.4375 0.4522 0.5669 0.7290 r 9.9868 10−4 0.0256 0.0503 0.1053 0.1400 0.1845 0.3689 0.6149 × When i1 = i2 = i, we can write nt/T 2H ⌊ ⌋ i,i T n,i Zn (t)= 2H 1 H2(ξj ), 2n − Xj=1 where H (x)= x2 1 is the second degree Hermite polynomial and ξn,i = 2 − j T H nH (Bi Bi ). It is well known (see [20]) that for each i = 1,...,m, − tj+1 − tj i,i 2 the process Zn (t) converges in L to the Rosenblatt process R(t). We refer the reader to [30] and [32] for further details on the Rosenblatt process. tj+1 i1 i1 i2 When i1 = i2, the stochastic integral (B B )δB cannot be writ- 6 tj s − tj s ten as the second Hermite polynomial of a Gaussian random variable. Never- i1,i2 R 2 theless, the process Zn (t) is still convergent in L . Indeed, for any positive integers n and n′, we have E i1i2 i1i2 (Zn (t)Zn′ (t)) nt/T n′t/T ⌊ ⌋ ⌊ ⌋ ((k+1)/n)T E i1 i1 i2 = nn′ (Bs B(k/n)T )δBs ′ (k/n)T − Xk=0 kX=0 Z ((k+1)/n′)T i1 i1 i2 (Bs B(k/n′)T )δBs × ′ − Z(k/n )T ′ nt/T n t/T ′ ′ ⌊ ⌋ ⌊ ⌋ ((k+1)/n)T ((k +1)/n )T t s 2 = nn′αH µ(dv du) ′ ′ ′ ′ ′ (k/n)T (k /n )T (k/n)T (k /n )T Xk=0 kX=0 Z Z Z Z µ(ds dt) × 2 2 t t T αH 4H 4 u v − du dv → 4 | − | Z0 Z0 4H 2 = cH t − , T 2H2(2H 1) as n′,n + , where cH = 4(4H 3)− . This allows us to conclude that → ∞ − i1i2 2 i1i2 2 Zn (t) is a Cauchy sequence in L . We denote by Zt the L -limit of i1i2 i1i2 Zn (t). Then Zt can be considered a generalized Rosenblatt process. 14 Y. HU, Y. LIU AND D. NUALART It is easy to show that i1i2 i1i2 2 4H 2 E[ Z Z ] C t s − , | t − s | ≤ | − | and by the hypercontractivity property, we deduce i1i2 i1i2 p p(2H 1) (2.13) E[ Z Z ] C t s − | t − s | ≤ p| − | for any p 2 and s,t [0, T ]. By the Kolmogorov continuity criterion this implies that≥ Zi1i2 has∈ a H¨older continuous version of exponent λ for any λ< 2H 1. − 3. Estimates for solutions of some SDEs. The purpose of this section is to provide upper bounds for the H¨older seminorms of solutions of two types of SDEs. The first type [see (3.1)] covers equation (1.1) and its Malliavin derivatives, as well as all the other SDEs involving only continuous inte- grands which we will encounter in this paper. The second type [see (3.13)] deals with the case where the integrands are step processes. These SDEs arise from approximation schemes such as (1.2) and (1.3). k RM RN For any integers k,N,M 1, we denote by Cb ( ; ) the space of k times continuously differentiable≥ functions f : RM RN which are bounded together with their first k partial derivatives. If→N = 1, we simply write k RM Cb ( ). In order to simplify the notation we only consider the case when the fBm is one-dimensional, that is, m = 1. All results of this section can be generalized to the case m> 1. Throughout the remainder of the paper we let β be any 1 number satisfying 2 <β M Lemma 3.1. Fix τ [0, T ). Let V = Vt,t [τ, T ] be an R -valued processes satisfying ∈ { ∈ } t t 1 2 (3.1) Vt = St + [g1(Vu)+ Uu Vu] du + [g2(Vu)+ UuVu] dBu, Zτ Zτ RM RM 1 RM RM i i where g1 Cb( ; ), g2 Cb ( ; ) and U = Ut ,t [τ, T ] , i = 1, 2, ∈ ∈M M M { ∈ } and S = St, [τ, T ] are R × -valued and R -valued processes, respec- tively. We{ assume∈ that} S has β-H¨older continuous trajectories, and the pro- cesses U i, i = 1, 2, are uniformly bounded by a constant C. 1 2 (i) If U = U = 0, then we can find constants K and K′ such that (t s)β B K, τ s β (ii) Suppose that there exist constants K0 and K0′ such that (t s) B β K , τ s K B 1/β (3.3) max V , V β Ke k kβ ( Sτ + S β + 1). {k k∞ k k }≤ | | k k Proof. The proof follows the approach used, for instance, by Hu and Nualart [8]. Let τ s (3.7) c1 = max C, g1 , g2 , g2 . { k k∞ k k∞ k∇ k∞} 1 1 β Take K = 2 (K2c1)− . Then for any τ s β 1 1 If (t s) B β 4 (K2c1)− , then the coefficient of V s,t,β on the right- − k k ≤ 1 k k hand side of (3.8) is less or equal than 2 . Thus we obtain 1 β V s,t,β 2 S β + 2c1(1+ V s,t, )(t s) − k k ≤ k k k k ∞ − + 2K1c1(1 + V s,t, ) B β k k ∞ k k 2 β + 2K2 V s,t, U s,t,β B β(t s) . k k ∞k k k k − β On the other hand, assuming (t s) B β K0 and applying (3.2), we obtain − k k ≤ (3.9) V s,t,β 2 S β + C1(1 + B β)(1 + V s,t, ), k k ≤ k k k k k k ∞ for some constant C1. This implies β β V s,t, Vs + 2(t s) S β + C1(t s) (1+ B β)(1 + V s,t, ). k k ∞ ≤ | | − k k − k k k k ∞ β 1 β 1 1 Assuming (t s) B β and (t s) , we obtain − k k ≤ 4C1 − ≤ 4C1 ∧ 2 (3.10) V s,t, 2 Vs + 2 S β + 1. k k ∞ ≤ | | k k 1 1 1 1/β 1 1 1/β Take ∆ = [ B − min( , K , )] ( ) . We divide the in- β 4K2c1 0 4C1 4C1 2 k k T τ ∧ ∧ terval [τ, T ] into N = − + 1 subintervals and denote by s ,s ,...,s the ⌊ ∆ ⌋ 1 2 N left endpoints of these intervals and sN+1 = T . Applying inequality (3.10) to each interval [si,si+1] for i = 1,...,N yields N+1 (3.11) V 2 ( Sτ + 2 S β + 1). k k∞ ≤ | | k k From the definition of ∆ we get T (3.12) N 1+ 1+ T max(C ,C B 1/β), ≤ ∆ ≤ 2 3k kβ for some constants C2 and C3. From inequalities (3.11) and (3.12) we obtain the desired estimate for V . If t,s [τ, T ] satisfy 0k kt∞ s ∆, then from (3.9) and from the upper ∈ ≤ − ≤Vt Vs bound of V we can estimate − β by the right-hand side of (3.3) for k k∞ (t s) some constant K. On the other hand,− if t s> ∆, then − Vt Vs 1 | − | 2 V ∆− . (t s)β ≤ k k∞ − We can obtain a similar estimate from the upper bound of V and from k k∞ the definition of ∆. This gives then the desired estimate for V β, and hence we complete the proof of (ii). k k For the second lemma we fix n and consider the partition of [0, T ] given by t = i T , i = 0, 1,...,n. Define η(t)= t if t t i Lemma 3.2. Suppose that S, gi, U , i = 1, 2 are the same as in Lemma 3.1. M Let g C([0, T ]). Let V = Vt,t [τ, T ] be an R -valued processes satisfy- ing the∈ equation { ∈ } t ε(τ) ∨ 1 Vt = St + [g1(Vη(u))+ Uη(u)Vη(u)]g(u η(u)) du ε(τ) − (3.13) Z t ε(τ) ∨ 2 + [g2(Vη(u))+ Uη(u)Vη(u)] dBu. Zε(τ) 1 2 (i) If U = U = 0, then we can find constants K and K′ such that (t s)β B K, τ s Step 1. In the case U 1 = U 2 =0, (3.15) becomes V k ks,t,β,n 1 β β S β + c1 g (t s) − + K1c1 B β + K3c1 V s,t,β,n B β(t s) , ≤ k k k k∞ − k k k k k k − where c is defined in (3.7). Taking K = 1 (K c ) 1, for any τ s (3.17) V s,t, 2 Vs + 2 S β + 1, k k ∞ ≤ | | k k provided s = η(s), and both t s and (t s)β B are bounded by some − − k kβ constant C4. 1/β 1/β Take ∆ = (C4 B β− ) C4. We are going to consider two cases de- k k ∧ 2T pending on the relation between ∆ and n . 2T 2(T ε(τ)) If ∆ > n , we take N = −∆ and divide the interval [ε(τ), ε(τ)+ ∆ ⌊ ∆ ⌋ N 2 ] into N subintervals of length 2 . Since the length of each of these subin- T tervals is larger than n , we are able to choose N points s1,s2,...,sN from each of these intervals such that s1 = ε(τ) and η(si)= si, i = 1, 2,...,N. On the other hand, we have s s ∆ for all i = 1,...,N 1. Applying in- i+1 − i ≤ − equality (3.17) to each of the intervals [s1,s2], [s2,s3],..., [sN 1,sN ], [sN , T ] yields − N+1 (3.18) V ε(τ),T, 2 ( Sε(τ) + 2 S β + 1). k k ∞ ≤ | | k k From the definition of ∆ we have 2T (3.19) N K + K B 1/β, ≤ ∆ ≤ k kβ for some constant K depending on T and C4. From (3.18) and (3.19) and taking into account that β (3.20) V τ,ε(τ), = S τ,ε(τ), Sτ + T S β , k k ∞ k k ∞ ≤ | | k k we obtain the desired estimate for V . k k∞ EULER SCHEMES OF SDE DRIVEN BY FBM 19 2T 2T 1/β If ∆ n , that is, when n ∆ K + K B β , then by equation (3.13) we have≤ ≤ ≤ k k Vt Vη(t) + St Sη(t) + (c1 + C Vη(t) ) g (T/n) | | ≤ | | | − | | | k k∞ + (c + C V ) B (T/n)β 1 | η(t)| k kβ A + B V , ≤ n n| η(t)| for any t [τ, T ], where ∈ β β An = S β(T/n) + c1 g (T/n)+ c1 B β(T/n) k k k k∞ k k and β Bn =1+ C g (T/n)+ C B β(T/n) . k k∞ k k Iterating this estimate, we obtain n n 1 V ε(τ),T, Sε(τ) Bn + nAnBn− (3.21) k k ∞ ≤ | | K B 1/β K( S + S + 1)e k kβ , ≤ | ε(τ)| k kβ for some constant K independent of n, where we have used the inequality n K(1+ B )n1−β B e k kβ , n ≤ and the fact that n K + K B 1/β for some constant K. Taking (3.20) into ≤ k kβ account, we obtain the desired upper bound for V . In order to show the upper bound for V k k,∞ we notice that if 0 k kτ,T,β ≤ t s ∆, then from (3.16) and from the upper bound of V τ,T, , we have − ≤ k k ∞ K B 1/β V K( S + S + 1)e k kβ , k kε(s),t,β,n ≤ | τ | k kβ for some constant K. Thus V V Vε(s) Vs | t − s| V + | − | (t s)β ≤ k kε(s),t,β,n (ε(s) s)β − − K B 1/β K( S + S + 1)e k kβ . ≤ | τ | k kβ If t s ∆, we can obtain the upper bound of V β by an argument similar to that− ≥ in the proof of Lemma 3.1. The proof ofk (ii)k is now complete. The following result gives upper bounds for the norm of Malliavin deriva- tives of the solutions of the two types of SDEs, (3.1) and (3.13). Given N,2 a process P = Pt,t [τ, T ] such that Pt D , for each t and some { ∈D } ∈ N 1, we denote by N∗ P the maximum of the supnorms of the functions ≥ N D Pr0 , Dr1 Pr0 ,...,Dr ,...,r Pr0 over r0, . . . , rN [τ, T ], and denote by N P 1 N D ∈ the maximum of the random variable N∗ P and the supnorms of P β , N k k Dr1 P r1,T,β,..., Dr ,...,r P r1 rN ,T,β over r0, . . . , rN [τ, T ]. If N = 0, k k Dk 1 N k ∨···∨D ∈ we simply write 0∗P = P and 0P = max( P , P β). k k∞ k k∞ k k 20 Y. HU, Y. LIU AND D. NUALART Lemma 3.3. (i) Let V be the solution of equation (3.1). Assume that 1 2 g1 = g2 = 0. Suppose that U are U are uniformly bounded by a constant C, β and assume that there exist constants K0 and K0′ such that (t s) B β K , τ s Proof. We first show point (i). The upper bounds of V and V β k k∞ k k follow from Lemma 3.1(ii). The Malliavin derivative DrVt satisfies the equa- tion (see Proposition 7 in [26]) t t (1) 1 2 DrVt = St + UuDrVu du + UuDrVu dBu Zr Zr while t [r τ, T ] and D V = 0 otherwise, where ∈ ∨ r t t t (1) 2 1 2 (3.23) St := DrSt + Ur Vr + [DrUu ]Vu du + [DrUu]Vu dBu Zr Zr for t [r τ, T ]. Lemma 3.1(ii) applied to the time interval [r, T ], where r τ,∈ implies∨ that ≥ 1/β K B β (1) (1) max DrV r,T, , DrV r,T,β Ke k k ( Sr + S r,T,β + 1). {k k ∞ k k }≤ | | k k Therefore, to obtain the desired upper bound it suffices to show that there (1) (1) exists a constant K independent of r such that both S r,T, and S r,T,β 1/β k k ∞ k k K B are less than or equal to Ke k kβ . Applying Lemma A.1(ii) to the second 2 2 integral in (3.23) and noticing that DrU , DrU r,T,β, V , V r,T,β 1/β k k∞ k k k k∞ k k K B (1) are bounded by Ke k kβ , we see that the upper bound of S is ∞ K B 1/β k k bounded by Ke k kβ . On the other hand, in order to show the upper S(1) S(1) (1) t − s bound for S r,T,β, we calculate (t s)β using (3.23) to obtain k k − (1) (1) t St Ss β 1 − D S + (t s)− [D U ]V du (t s)β ≤ k r kr,T,β − r u u − Zs t β 2 + (t s)− [D U ]V dB . − r u u u Zs EULER SCHEMES OF SDE DRIVEN BY FBM 21 Now we can estimate each term of the above right-hand side as before. (1) Taking the supremum over s,t [r, T ] yields the upper bound of S r,T,β. We turn to the second derivative.∈ As before, we are able to findk thek equa- 2 2 tion of Dr1,r2 Vt; see Proposition 7 in [26]. The estimates of Dr1,r2 Vt can then be obtained in the same way as above by applying Lemma 3.1(ii) and the estimates that we just obtained for Vt and DsVt, as well as the assumptions on S and U i. The estimates of the higher order derivatives of V can be obtained analogously. The proof of (ii) follows along the same lines, except that we use Lem- ma 3.2(ii) and Lemma A.1(i) instead of Lemma 3.1(ii) and Lemma A.1(ii). 1 Remark 3.2. Since β > 2 , from Fernique’s theorem we know that K B 1/β Ke k kβ has finite moments of any order. So Lemma 3.3 implies that the uniform norms and H¨older seminorms of the solutions of (3.1) and (3.13) and their Malliavin derivatives have finite moments of any order. We will need this fact in many of our arguments. The next proposition is an immediate consequence of Lemma 3.3. Recall D D that the random variables N∗ P and N P are defined in Section 3. Proposition 3.1. Let X be the solution of equation (1.1), and let Xn be the solution of the Euler scheme (1.2). Fix N 0, and suppose that b CN (Rd, Rd),σ CN+1(Rd, Rd) (recall that we assume≥ m = 1). Then ∈ b ∈ b there exists a positive constant K such that the random variables DN X 1/β n K B and DN X are bounded by Ke k kβ for all n N. If we further assume N+2 Rd Rd ∈ σ Cb ( , ), then the same upper bound holds for the modified Euler scheme∈ (1.3). Proof. We first consider the process X, the solution to equation (1.1). The upper bounds for X and X β follow from Lemma 3.1(ii). The k k∞ k k Malliavin derivative DrXt satisfies the following linear stochastic differential equation: t t (3.24) D X = σ(X )+ b(X )D X du + σ(X )D X dB , r t r ∇ u r u ∇ u r u u Zr Zr while 0 < r t T , and D X = 0 otherwise. Then it suffices to show that ≤ ≤ r t K B 1/β (3.25) sup DM (DrX) Ke k kβ , r [0,T ] ≤ ∈ for M = N 1. We can prove estimate (3.25) by induction on N 1. Set S = σ(X ),−U 1 = b(X ) and U 2 = σ(X ). Applying Lemma 3.1(i)≥ to X t r t ∇ t t ∇ t 22 Y. HU, Y. LIU AND D. NUALART we obtain that U 2 satisfies (3.22). Therefore, Lemma 3.3 implies that (3.25) holds for M = 0. Now we assume that K B 1/β sup DM (DrX) Ke k kβ r [0,T ] ≤ ∈ for some 0 M N 2. It is then easy to see that ≤ ≤ − 1/β 1 2 K B D ∗ (U ) D (U ) D (S) Ke k kβ , M+1 ∨ M+1 ∨ M+1 ≤ N Rd Rd N+1 Rd Rd taking into account that b Cb ( ; ), σ Cb ( ; ), which enables us to apply Lemma 3.3 to (∈3.24) to obtain the∈ upper bound of the quantity D supr [0,T ] M+1(DrX). The∈ estimates of the Euler scheme and the modified Euler scheme and their derivatives can be obtained in the same way. We omit the proof, and we only point out that one more derivative of σ is needed for the modified Euler scheme because the function σ is involved in its equation. ∇ 4. Rate of convergence for the modified Euler scheme and related pro- cesses. The main result of this section is the convergence rate of the scheme defined by (1.3) to the solution of the SDE (1.1). Recall that γn is the func- tion of n defined in (1.5). Theorem 4.1. Let X and Xn be solutions to equations (1.1) and (1.3), 3 Rd Rd 4 Rd Rd m respectively. We assume b Cb ( ; ), σ Cb ( ; × ). Then for any p 1 there exists a constant∈ C independent∈ of n (but dependent on p) such that≥ E n p 1/p 1 sup [ Xt Xt ] Cγn− . 0 t T | − | ≤ ≤ ≤ Proof. Denote Y := X Xn. Notice that Y depends on n, but for notational simplicity we shall− omit the explicit dependence on n for Y and some other processes when there is no ambiguity. The idea of the proof is to decompose Y into seven terms [see (4.7) below] and then study their convergence rate individually. Step 1. By the definitions of the processes X and Xn, we have t Y = [b(X ) b(Xn)+ b(Xn) b(Xn )] ds t s − s s − η(s) Z0 m t j j n j n j n j + [σ (Xs) σ (Xs )+ σ (Xs ) σ (Xη(s))] dBs 0 − − Xj=1 Z m t j j n 2H 1 H ( σ σ )(Xη(s))(s η(s)) − ds. − 0 ∇ − Xj=1 Z EULER SCHEMES OF SDE DRIVEN BY FBM 23 By denoting 1 σj(s) = ( σjσj)(Xn ), b (s)= b(θX + (1 θ)Xn) dθ, 0 ∇ η(s) 1 ∇ s − s Z0 1 σj(s)= σj(θX + (1 θ)Xn) dθ, 1 ∇ s − s Z0 we can write t m t t j j n n Yt = b1(s)Ys ds + σ1(s)Ys dBs + [b(Xs ) b(Xη(s))] ds 0 0 0 − Z Xj=1 Z Z m t m t j n j n j j 2H 1 + [σ (Xs ) σ (Xη(s))] dBs H σ0(s)(s η(s)) − ds. 0 − − 0 − Xj=1 Z Xj=1 Z n n Let Λ = Λt ,t [0, T ] be the d d matrix-valued solution of the fol- lowing linear{ SDE:∈ } × t m t n n j n j (4.1) Λt = I + b1(s)Λs ds + σ1(s)Λs dBs , 0 0 Z Xj=1 Z where I is the d d identity matrix. Applying the chain rule for the Young integral to ΓnΛn×, where Γn, t [0, T ] is the unique solution of the equation t t t ∈ t m t n n n j j (4.2) Γt = I Γs b1(s) ds Γs σ1(s) dBs, − 0 − 0 Z Xj=1 Z n n n n for t [0, T ], we see that Γt Λt = Λt Γt = I for all t [0, T ]. Therefore, n ∈1 n ∈ (Λt )− exists and coincides with Γt . n We can express the process Yt in terms of Λt as follows: t Y = ΛnΓn[b(Xn) b(Xn )] ds t t s s − η(s) Z0 m t n n j n j n j (4.3) + Λt Γs [σ (Xs ) σ (Xη(s))] dBs 0 − Xj=1 Z m t n n j 2H 1 H Λt Γs σ0(s)(s η(s)) − ds. − 0 − Xj=1 Z The first two terms in the right-hand side of equation (4.3) can be further decomposed as follows: t ΛnΓn[σj(Xn) σj(Xn )] dBj t s s − η(s) s Z0 24 Y. HU, Y. LIU AND D. NUALART t = ΛnΓnbj (s)(s η(s)) dBj t s 2 − s Z0 m t n n j,i i i j (4.4) + Λt Γs σ2 (s)(Bs Bη(s)) dBs 0 − Xi=1 Z t + ΛnΓnσj(s)(s η(s))2H dBj t s 3 − s Z0 m := I2,j(t)+ I3,j,i(t)+ I4,j(t), Xi=1 where 1 bj (s)= σj(θXn + (1 θ)Xn )b(Xn ) dθ, 2 ∇ s − η(s) η(s) Z0 1 σj,i(s)= σj(θXn + (1 θ)Xn )σi(Xn ) dθ, 2 ∇ s − η(s) η(s) Z0 1 m j 1 j n n l σ3(s)= σ (θXs + (1 θ)Xη(s)) σ0(s) dθ 2 0 ∇ − Z Xl=1 and t Λn Γn[b(Xn) b(Xn )] ds t s s − η(s) Z0 t m n n n j n j j (4.5) = Λt Γs b3(s) b(Xη(s))(s η(s)) + σ (Xη(s))(Bs Bη(s)) 0 " − − Z Xj=1 m 1 j 2H + σ0(s)(s η(s)) ds 2 − # Xj=1 m := I11(t)+ I12,j(t)+ I13(t), Xj=1 where b (s)= 1 b(θXn + (1 θ)Xn ) dθ. We also denote 3 0 ∇ s − η(s) t R n n j 2H 1 (4.6) I (t)= HΛ Γ σ (s)(s η(s)) − ds. 5,j − t s 0 − Z0 Substituting equations (4.4), (4.5) and (4.6) into (4.3) yields m m m m m (4.7) Y = I11 + I12,j + I13 + I2,j + I3,j,i + I4,j + I5,j. Xj=1 Xj=1 j,iX=1 Xj=1 Xj=1 EULER SCHEMES OF SDE DRIVEN BY FBM 25 n n Step 2. Denote by (Λ )i, i = 1,...,d, the ith columns of Λ . We claim that n 1 (Λ )i satisfy the conditions in Lemma 3.3 with M = d, τ = 0, Ut = b1(t), 2 j 2 Ut = σ1(t) and N =2. We first show that U satisfies (3.22). Taking into 3 Rd Rd 4 Rd Rd m account that b Cb ( ; ), σ Cb ( ; × ), it suffices to show that both X and Xn satisfy∈ (3.22). This∈ is clear for X because of Lemma 3.1(i). It follows from Lemma 3.2(i) that there exist constants K and K′ such that (t s)β B K, 0 s m nt/T ⌊ ⌋ tk+1 t s (4.12) E (t) = Λn F n,i,j ∧ δBi δBj, 2,j t tk u s tk tk Xi=1 Xk=0 Z Z n,i,j n j i n T for t [0, T ], where Ft =Γt ( σ σ )(Xt ), and we define tn+1 = (n + 1) n . From∈ (4.8) and Proposition 3.1,∇ we have 1/β n,i,j n,i,j n,i,j K B (4.13) max F , D F , D D F Ke k kβ . {| t | | r1 t | | r2 r1 |} ≤ Hence, applying estimate (A.8) from Lemma A.4 to E2,j(t), we obtain E( E (t) p)1/p Cγ 1. The proof is now complete. | 2,j | ≤ n− The following result provides a rate of convergence for the Malliavin derivatives of the modified scheme and some related processes. Recall that 1 β satisfies 2 <β Lemma 4.1. Let X and Xn be the processes defined by (1.1) and (1.3), respectively. Suppose that σ C5(Rd; Rd m), b C4(Rd; Rd). Let p 1. Then: ∈ b × ∈ b ≥ n (i) There exists a constant C such that the quantities DsXt DsXt p, n n k − 1 k2β DrDsXt DrDsXt p, DuDrDsXt DuDrDsXt p are less than Cn − fork all u, r,− s, t [0, T ]kandk n N. − k ∈ ∈ EULER SCHEMES OF SDE DRIVEN BY FBM 27 (ii) Let V and V n be d-dimensional processes satisfying the equations t m t j j Vt = V0 + f1(Xu, Xu)Vu du + f2 (Xu, Xu)Vu dBu, 0 0 Z Xj=1 Z t m t n n n j n n j Vt = V0 + f1(Xu, Xu )Vu du + f2 (Xu, Xu )Vu dBu, 0 0 Z Xj=1 Z 3 Rd Rd Rd d j 4 Rd Rd Rd d where f1 Cb ( ; × ) and f2 Cb ( ; × ). Then there ex- ∈ × ∈ × n n ists a constant C such that the quantities Vt Vt p, DsVt DsVt p, D D V D D V n are less than Cn1 2β kfor− all r,s,tk k[0, T ] −and n kN. k r s t − r s t kp − ∈ ∈ Remark 4.1. The above results still hold when the approximation pro- cess Xn is replaced by the one defined by the recursive scheme (1.2). The proof follows exactly along the same lines. Proof of Lemma 4.1. (i) Taking the Malliavin derivative in both sides of (4.3), we obtain t D (X Xn)= D [ΛnΓn(b(Xn) b(Xn ))] ds r t − t r t s s − η(s) Z0 m t n n j n j n j + Dr[Λt Γs (σ (Xs ) σ (Xη(s)))] dBs 0 − Xj=1 Z m + ΛnΓn(σj(Xn) σj(Xn )) t r r − η(r) Xj=1 m t n n j 2H 1 H Dr[Λt Γs σ0(s)](s η(s)) − ds. − 0 − Xj=1 Z Proposition 3.1 and equation (4.8) imply that the first, third and last terms p 1 2H of the above right-hand side have L -norms bounded by Cn − . Applying estimate (A.16) from Lemma A.5 to the second term and noticing that X β and supr [0,T ] DrX β have finite moments of any order, we see that k k p ∈ k k 1 2β its L -norm is also bounded by Cn − . Similarly, we can take the second derivative in (4.3) and then estimate each term individually as before to obtain that the upper bound of DrDsXt n 1 2β k − DrDsXt p is bounded by Cn − . (ii) Usingk the chain rule for Young’s integral we derive the following ex- plicit expression for V V n: t − t t n 1 n n V V = Υ Υ− (f (X , X ) f (X , X ))V ds t − t t s 1 s s − 1 s s s Z0 28 Y. HU, Y. LIU AND D. NUALART (4.14) m t 1 j j n n j + ΥtΥs− (f2 (Xs, Xs) f2 (Xs, Xs ))Vs dBs , 0 − Xj=1 Z where Υ= Υ , t [0, T ] is the Rd d-valued process that satisfies { t ∈ } × t m t j j Υt = I + f1(Xs, Xs)Υt ds + f2 (Xs, Xs)Υt dBs. 0 0 Z Xj=1 Z Lemma 3.3 implies that there exists a constant K such that for all n N, u, r, s, t [0, T ], we have ∈ ∈ K B 1/β (4.15) max Υ , D Υ , D D Υ , D D D Υ Ke k kβ . { t s t r s t u r s t}≤ Therefore, applying estimate (A.4) to the second integral in (4.14) with ν = 0 and taking into account the estimate of Lemma 4.1(i), we obtain n 1 2β V V Cn − . k − kp ≤ Taking the Malliavin derivative on both sides of (4.14), and then applying estimates (A.4) from Lemmas A.3 and 4.1(i) as before, we can obtain the de- n n sired estimate for DsVt DsVt p. The estimate for DrDsVt DrDsVt p can be obtained ink a similar− way.k k − k We define Λt,t [0, T ] as the solution of the limiting equation of (4.1), that is, { ∈ } t m t j j (4.16) Λt = I + b(Xs)Λs ds + σ (Xs)Λs dBs . 0 ∇ 0 ∇ Z Xj=1 Z The inverse of the matrix Λt, denoted by Γt, exists and satisfies t m t j j Γt = I Γt b(Xs) ds Γt σ (Xs) dBs. − 0 ∇ − 0 ∇ Z Xj=1 Z 5 Rd Rd m It follows from Lemma 4.1 that if we assume that σ Cb ( ; × ) and b 4 Rd Rd ∈ n∈ Cb ( ; ), then the estimate in Lemma 4.1(ii) holds with the pair (V,V ) n n being replaced by (Γi, Γi ) or (Λi, Λi ), i = 1,...,d, where the subindex i denotes the ith column of each matrix. 5. Central limit theorem for weighted sums. Our goal in this section is to prove a central limit result for weighted sums (see Proposition 5.5 below) that will play a fundamental role in the proof of Theorem 6.1 in the next section. This result has an independent interest and we devote this entire section to it. EULER SCHEMES OF SDE DRIVEN BY FBM 29 We recall that B = Bt,t [0, T ] is an m-dimensional fBm, and we as- { ∈ } 1 3 sume that the Hurst parameter satisfies H ( 2 , 4 ]. For any n 1 we set jT ∈ ≥ tj = n , j = 0,...,n. Recall that η(s)= tk if tk s Proof. From inequality (A.8) in Lemma A.4 it follows that k j 2 (5.1) E( Ξn Ξn 4) C − , | tk − tj | ≤ n for any j k. This implies the tightness of (Ξn,B). Then it≤ remains to show the convergence of the finite dimensional dis- tributions of (Ξn,B) to that of (W, B). To do this, we fix a finite set of points r1, . . . , rL+1 [0, T ] such that 0= r1 < r2 < < rL+1 T and de- ∈ · · · ≤n n fine the random vectors BL = (Br2 Br1 ,...,BrL+1 BrL ),ΞL = (Ξr2 n n n − − − Ξr1 ,..., ΞrL+1 ΞrL ) and WL = (Wr2 Wr1 ,...,WrL+1 WrL ). We claim that as n tends− to infinity, the following− convergence in law− holds: (5.2) (Ξn ,B ) (W ,B ). L L ⇒ L L n For notational simplicity, we add one term to each component of ΞL, and we define r { l+1} (5.3) Θn(i, j):=Ξn,i,j Ξn,i,j + ζi,j = γ ζi,j , l rl+1 rl r ,n n k,n − { l} k= r X{ l} for 1 l L, 1 i, j d, where ≤ ≤ ≤ ≤ tk+1 ζi,j = (Bi Bi )δBj. k,n s − tk s Ztk Then Slutsky’s lemma implies that the convergence in law in (5.2) is equiv- alent to (Θn(i, j), 1 i, j d, 1 l L,B ) (W ,B ). l ≤ ≤ ≤ ≤ L ⇒ L L 30 Y. HU, Y. LIU AND D. NUALART According to Peccati and Tudor [27] (see also Theorem 6.2.3 in [21]), to n show the convergence in law of (ΘL,BL), it suffices to show the convergence n of each component of (ΘL,BL) to the correspondent component of (WL,BL) and the convergence of the covariance matrix. n The convergence of the covariance matrix of ΘL follows from Propositions 5.2 and 5.3 below. The convergence in law of each component to a Gaus- sian distribution follows from Proposition 5.4 below and the fourth moment theorem; see [24] and also Theorem 5.2.7 in [21]. This completes the proof. In order to show the convergence of the covariance matrix and the fourth moment of Θn we first introduce the following notation: k = (s, t, v, u) : tk v s tk+1, u, t [0, T ] , (5.4) D { ≤ ≤ ≤ ∈ } = (s, t, v, u) : t v s t ,t u t t . Dk1,k2 { k2 ≤ ≤ ≤ k2+1 k1 ≤ ≤ ≤ k1+1} The next two propositions provide the convergence of the covariance n n E[Θ ′ (i , j )Θ (i, j)] in the cases l = l and l = l , respectively. We denote l ′ ′ l ′ 6 ′ βk/n(s)= 1[tk,tk+1](s). n Proposition 5.2. Let Θl (i, j) be defined in (5.3). Then n n 2 rl+1 rl (5.5) E[Θ (i′, j′)Θ (i, j)] α − (Rδ ′ δ ′ + Qδ ′ δ ′ ), l l → H T ji ij jj ii as n + . Here δii′ is the Kronecker function, αH = H(2H 1) and Q and R→are∞ the constants defined in (2.12). − Proof. The proof will involve several steps. Step 1. Applying twice the integration by parts formula (2.7), we have E n n [Θl (i′, j′)Θl (i, j)] (5.6) r { l+1} 2 i j n = αH γn DuDt Θl (i′, j′)µ(dv du)µ(ds dt), k k= rl Z X{ } D nt where we recall that t = T for t [0, T ) and T = tn 1, and k is defined in (5.4). Since { } ⌊ ⌋ ∈ { } − D i j n DuDt Θl (i′, j′) (5.7) r { l+1} ′ ′ ′ ′ = γn (1[tk,t](u)βk/n(t)δjj δii + 1[tk,u](t)βk/n(u)δji δij ), k= r X{ l} EULER SCHEMES OF SDE DRIVEN BY FBM 31 the left-hand side of (5.5) equals r { l+1} 2 2 α γ 1 (u)β ′ (t)δ ′ δ ′ H n [tk′ ,t] k /n jj ii ′ k { k,k = rl Z X{ } D + 1 (t)β ′ (u)δji′ δij′ µ(dv du)µ(ds dt) [tk′ ,u] k /n } 2 2 := αH γn(G1δjj′ δii′ + G2δji′ δij′ ). 2 2 In the next two steps, we compute the limits of γnG1 and γnG2 as n tends to infinity in the case H ( 1 , 3 ) and in the case H = 3 separately. ∈ 2 4 4 Step 2. In this step, we consider the case H ( 1 , 3 ). Recall that ∈ 2 4 1 p+1 t s Q(p)= T 4H µ(dv du)µ(ds dt) Z0 Zp Z0 Zp = n4H µ(dv du)µ(ds dt), ′ ′ ZDk ,k +p which is independent of n, where the set k1,k2 is defined in (5.4). We can 2 D express γnG1 in terms of Q(p) as follows: r { l+1} 2 4H 1 γnG1 = n − µ(dv du)µ(ds dt) ′ k′,k k,k = rl Z X{ } D r r ( r p) r 1 { l+1}−{ l} { l+1}− ∧{ l+1} = Q(p) n p= r r k′=( r p) r { lX}−{ l+1} { lX}− ∨{ l} ∞ n = Ψl (p)Q(p), p= X−∞ where n ( rl+1 p) rl+1 ( rl p) rl Ψl (p)= { }− ∧ { }− { }− ∨ { }1[ r r , r r ](p). n { l}−{ l+1} { l+1}−{ l} n rl+1 rl The term Ψl (p) is uniformly bounded and converges to T− as n tends to infinity for any fixed p. Therefore, taking into account that p∞= Q(p)= Q< , the dominated convergence theorem implies −∞ ∞ P 2 rl+1 rl lim γ G1 = − Q. n n →∞ T Similarly, we can show that 2 rl+1 rl lim γ G2 = − R. n n →∞ T 32 Y. HU, Y. LIU AND D. NUALART 3 Step 3. In the case H = 4 , we can write r n2 { l+1} γ2G = µ(dv du)µ(ds dt) n 1 log n ′ k′,k k,k = rl Z X{ } D r r r 1 { l+1}−{ l} { l+1} = Q(p) n log n p= r r k′= r { lX}−{ l+1} X{ l} 0 rl p 1 rl+1 rl rl+1 1 { }− − { }−{ } { } + Q(p) − n log n (p= r r k′= r p=1 k′= r p+1) { lX}−{ l+1} X{ l} X { lX+1}− := G11 + G12. Taking into account that Q(p) behaves like 1/ p as p tends to infinity, it is | | | | then easy to see that G12 converges to zero. On the other hand, recall that P|p|≤n Q(p) Q Q = limn + . This implies that G11 converges to (rl+1 rl). → ∞ log n T 2 2 − This gives the limit of γnG1. The limit of γnG2 can be obtained similarly. Proposition 5.3. Let l, l 1,...,L be such that l = l . Let Θn be ′ ∈ { } 6 ′ defined as in (5.3). Then n n (5.8) lim E[Θ ′ (i′, j′)Θ (i, j)] = 0. n l l →∞ Proof. Without any loss of generality, we assume l′ < l. As in (5.6) we have r { l+1} E n n 2 i j n [Θl′ (i′, j′)Θl (i, j)] = αH γn DuDt Θl′ (i′, j′)µ(dv du)µ(ds dt). k k= rl Z X{ } D Taking into account (5.7), we can write E n n [Θl′ (i′, j′)Θl (i, j)] r r ′ { l+1} { l +1} 2 2 = α γ 1 (u)β ′ (t)δ ′ δ ′ H n [tk′ ,t] k /n jj ii ′ k { k= rl k = r ′ Z X{ } X{ l } D + 1 (t)β ′ (u)δji′ δij′ µ(dv du)µ(ds dt) [tk′ ,u] k /n } 2 2 := αH γn(G1δjj′ δii′ + G2δji′ δij′ ). e e EULER SCHEMES OF SDE DRIVEN BY FBM 33 In the case H ( 1 , 3 ) we have ∈ 2 4 r r ′ { l+1} { l +1} 2 4H 1 γ G = n 1 (u)β ′ (t)µ(dv du)µ(ds dt) n 1 − [tk′ ,t] k /n ′ k k= rl k = r ′ Z X{ } X{ l } D e r r ′ r ′ ( rl+1 p) 1 { l+1}−{ l } { l +1}∧ { }− = Q(p) n p= r r ′ k′=( r p) r ′ { l}−{X l +1} { lX}− ∨{ l } ∞ n = Φl (p)Q(p), p= X−∞ n where Φl (p) is equal to max ( rl′+1 p) rl+1 ( rl p) rl′ , 0 { { }− ∧ { }− { }− ∨ { } }1[ r r ′ , r r ′ ](p). n { l}−{ l +1} { l+1}−{ l } n The term Φl (p) is uniformly bounded and converges to 0 as n tends to infinity for any fixed p because l < l′. Therefore, taking into account that ∞ Q(p)= Q< , the dominated convergence theorem implies that p= ∞ 2 −∞ γPnG1 converges to zero as n tends to infinity. Similarly, we can show that 2 γnG2 converges to zero as n tends to infinity. 3 Ine the case H = 4 , since e r r ′ n2 { l+1} { l +1} γ2G = µ(dv du)µ(ds dt) n 1 ln n ′ k′,k k= rl k = r ′ Z X{ } X{ l } D e r r ′ r ′ ( rl+1 p) 1 { l+1}−{ l } { l +1}∧ { }− = Q(p), n lnn p= r r ′ k′=( r p) r ′ { l}−{X l +1} { lX}− ∨{ l } we have r r ′ r p 1 { l+1}−{ l } { l+1}− γ2G Q(p) n 1 ≤ n ln n p= r r ′ k′= r ′ { l}−{X l +1} X{ l } e 0 1 (p + 1)Q(p). ≤ n ln n p= n X− 1 2 C Noticing that Q(p)= O( p ), we conclude that γnG1 ln n . This shows | | ≤ 2 that γnG1 converges to zero as n tends to infinity. In the same way we can 2 e show that γnG2 converges to zero. e e 34 Y. HU, Y. LIU AND D. NUALART The following estimate is needed in the calculation of the fourth moment n of Θl (i, j) in Proposition 5.4. Lemma 5.1. Let H ( 1 , 3 ]. We have the following estimate: ∈ 2 4 n 1 − βk /n, βk /n βk /n, βk /n βk /n, βk /n βk /n, βk /n h 1 2 iHh 2 3 iHh 3 4 iHh 1 4 iH k1,k2X,k3,k4=0 2 2 Cn− γ− . ≤ n Proof. Since the indices k1, k2, k3, k4 are symmetric, it suffices to con- sider the case k k k k . By definition of the inner product we have 1 ≤ 2 ≤ 3 ≤ 4 βk /n, βk /n βk /n, βk /n βk /n, βk /n βk /n, βk /n h 1 2 iHh 2 3 iHh 3 4 iHh 1 4 iH 0 k1 k2 k3 k4 n 1 ≤ ≤ ≤X≤ ≤ − n 1 n 1 n 1 n 1 T 8H − − − − = ( k k + 1 2H + k k 1 2H 24n8H | 2 − 1 | | 2 − 1 − | kX1=0 kX2=k1 kX3=k2 kX4=k3 2 k k 2H ) − | 2 − 1| ( k k + 1 2H + k k 1 2H × | 3 − 2 | | 3 − 2 − | 2 k k 2H ) − | 3 − 2| ( k k + 1 2H + k k 1 2H × | 4 − 3 | | 4 − 3 − | 2 k k 2H ) − | 4 − 3| ( k k + 1 2H + k k 1 2H × | 4 − 1 | | 4 − 1 − | 2 k k 2H ). − | 4 − 1| Denote p = k k , i = 1, 2, 3. Then the above sum is bounded by i i+1 − i n 1 1 8H − 2H 2 2H 2 2H 2 2H 2 Cn − p1 − p2 − p3 − (p1 + p2 + p3) − , p ,p ,p =1 1 X2 3 which is again bounded by n 1 1 8H − 2H 2 2H 2 4H 4 Cn − p1 − p2 − p3 − . p ,p ,p =1 1 X2 3 1 3 n 1 4H 4 3 In the case H ( , ), the series − p − is convergent. When H = , ∈ 2 4 p3=1 3 4 it is bounded by C log n. So the above sum is bounded by Cn 4H 1 if 1 < P − − 2 3 4 3 H< 4 and bounded by Cn− log n if H = 4 . The proof is complete. The following proposition contains a result on the convergence of the l fourth moment of Θn(i, j). EULER SCHEMES OF SDE DRIVEN BY FBM 35 Proposition 5.4. The fourth moment of Θn(i, j) and 3E( Θn(i, j) 2)2 l | l | converge to the same limit as n . →∞ Proof. Applying the integration by parts formula (2.7) yields E n 4 [Θl (i, j) ] r { l+1} 2 E i j n 3 = αH γn [DuDt [Θl (i, j) ]]µ(dv du)µ(ds dt) k k= rl Z X{ } D r { l+1} 2 E n 2 i j n = αH γn [ 3Θl (i, j) DuDt [Θl (i, j)] k { k= rl Z X{ } D n j n + 6Θl (i, j)Dt [Θl (i, j)] Di [Θn(i, j)] ]µ(dv du)µ(ds dt) × u l } := G1 + G2. i j n E n Since DuDt [Θl (i, j)] is deterministic, it is easy to see that G1 = 3 ( Θl (i, j) 2)2. We have shown the convergence of E( Θl (i, j) 2) in Proposition| 5.2. | | n | It remains to show that G2 0 as n . Applying again the integration→ by→∞ parts formula (2.7) yields r { l+1} 4 2 i j j n i n G = 6α γ D ′ D D [Θ (i, j)]D [Θ (i, j)] 2 H n u t′ { t l u l } ′ k k′ k,k = rl Z X{ } D ×D µ(dv′ du′)µ(ds′ dt′)µ(dv du)µ(ds dt). × Using equation (5.7) we can derive the inequalities r { l+1} 4 4 G 6α γ β (t)β (t′)β ′ (u)β ′ (u′) 2 ≤ H n { h/n h/n h /n h /n ′ ′ k k′ k,k ,h,h = rl Z X { } D ×D + β (t)β (u′)β ′ (u)β ′ (t′) h/n h/n h /n h /n } µ(dv′du′)µ(ds′ dt′)µ(dv du)µ(ds dt) × r { l+1} 4 4 12αH γn βh/n, βk/n βh′/n, βk/n βh/n, βk′/n ≤ h iHh iHh iH k,k′,h,h′= r X { l} βh′/n, βk′/n . × h iH The convergence of G2 to zero now follows from Lemma 5.1. 36 Y. HU, Y. LIU AND D. NUALART We can now establish a central limit theorem for weighted sums based i,j tk+1 i i j on the previous proposition. Recall that ζ = (B B )δBs , k = k,n tk s tk i,j − 0,...,n 1 and ζn,n = 0. − R Proposition 5.5. Let f = ft,t [0, T ] be a stochastic process with values on the space of d d matrices{ and∈ with} H¨older continuous trajectories 1 × of index greater than 2 . Set, for i, j = 1,...,m, t { } Ψi,j(t)= f i,jζi,j . n tk k,n Xk=0 Then, the following stable convergence in the space D([0, T ]) holds as n tends to infinity: t i,j ij γnΨn(t),t [0, T ] fs dWs ,t [0, T ] , { ∈ }→ 0 1 i,j m ∈ Z ≤ ≤ where W is a matrix-valued Brownian motion independent of B with the covariance introduced in Section 2.4. Proof. This proposition is an immediate consequence of the central limit result for weighted random sums proved in [3]. In fact, the process i,j Ψn (t) satisfies the required conditions due to Proposition 5.1 and the esti- mate (5.1). 1 3 6. CLT for the modified Euler scheme in the case H ∈ ( 2 , 4 ]. The fol- 1 3 lowing central limit type result shows that in the case H ( 2 , 4 ], the process n ∈ γn(X X ) converges stably to the solution of a linear stochastic differen- tial equation− driven by a matrix-valued Brownian motion independent of B as n tends to infinity. Theorem 6.1. Let H ( 1 , 3 ], and let X, Xn be the solutions of the ∈ 2 4 SDE (1.1) and recursive scheme (1.3), respectively. Let W = Wt,t [0, T ] be the matrix-valued Brownian motion introduced in Section{ 2.4.∈ Assume} 5 Rd Rd m 4 Rd Rd σ Cb ( ; × ) and b Cb ( ; ). Then the following stable convergence in∈ the space C([0, T ]) holds∈ as n tends to infinity: (6.1) γ (X Xn),t [0, T ] U ,t [0, T ] , { n t − t ∈ } → { t ∈ } where U ,t [0, T ] is the solution of the linear d-dimensional SDE { t ∈ } t m t U = b(X )U ds + σj(X )U dBj t ∇ s s ∇ s s s Z0 j=1 Z0 (6.2) X m t j i ij + ( σ σ )(Xs) dWs . 0 ∇ i,jX=1 Z EULER SCHEMES OF SDE DRIVEN BY FBM 37 Remark 6.1. It follows from [13] that when B is replaced by a standard Brownian motion, the process √n(X Xn) converges in law to the unique solution of the d-dimensional SDE − m dU = b(X )U dt + σj(X )U dBj t ∇ t s ∇ t t t j=1 (6.3) X m T j i ij + ( σ σ )(Xt) dWt r 2 ∇ j,iX=1 ij with U0 = 0. Here W , i, j = 1,...,m are independent one-dimensional Brownian motions, independent of B. To compare our Theorem 6.1 with 1 this result, we let the Hurst parameter H converge to 2 . Then the con- stant R will converge to 0, and αH √Q R converges to T . This formally √T − 2 recovers equation (6.3). q Remark 6.2. The process U defined in (6.2) is given by m t j i ij (6.4) Ut = ΛtΓs( σ σ )(Xs) dWs , t [0, T ], 0 ∇ ∈ i,jX=1 Z where we recall that Λ is defined in (4.16) and Γ is its inverse. n Proof of Theorem 6.1. Recall that Yt = Xt Xt . We would like to − d+m show that the process γnYt,Bt,t [0, T ] converges weakly in C([0, T ]; R ) to U ,B ,t [0, T ] .{ To do this,∈ it suffices} to prove the following: { t t ∈ } (i) convergence of the finite dimensional distributions of γnYt,Bt,t [0, T ] ; { ∈ (ii)} tightness of the process γ Y ,B ,t [0, T ] . { n t t ∈ } We first show (i). Recall the decomposition of Yt given in (4.7) and (4.11), and recall the estimates obtained for each term in the decomposition of Yt. Since the other terms converge to zero in Lp for p 1, from the Slutsky theo- rem it suffices to consider the convergence of the finite≥ dimensional distribu- m tions of γn j=1 E2,j(t),Bt,t [0, T ] , where E2,j is defined in Theorem 4.1 step 3. Set{ ∈ } P (6.5) F i,j := ΛnΓn( σjσi)(Xn) Λ Γ ( σjσi)(X ). s t s ∇ s − t s ∇ s It follows from Lemma 4.1 and Remark 4.1 that i,j i,j i,j 1 2β sup ( Ft p DsFt p DrDsFt p) Cn − . r,s,t [0,T ] k k ∨ k k ∨ k k ≤ ∈ 38 Y. HU, Y. LIU AND D. NUALART Denote m nt/T ⌊ ⌋ tk+1 s j i i j (6.6) E2,j(t) = Λt Γtk ( σ σ )(Xtk ) δBuδBs , ∇ tk tk Xi=1 Xk=0 Z Z e for t [0, T ), and E (T )= E (T ). Then applying Lemma A.4 (A.9) with ∈ 2,j 2,j − F i,j defined by (6.5), we obtain that e e H 1 2β γ E (t) E (t) Cγ n− n − , nk 2,j − 2,j kp ≤ n which converges to zero as n e since β can be taken as close as possible to H. By Slutsky’s theorem again,→∞ it suffices to consider the convergence of the finite dimensional distributions of m (6.7) γn E2,j(t),Bt,t [0, T ] . ( ∈ ) Xj=1 e Applying Proposition 5.5 to the family of processes f i,j =Γ ( σjσi)(X ), t t ∇ t we obtain the convergence of the finite dimensional distributions of m γn ΓtE2,j(t),Bt,t [0, T ] ( ∈ ) Xj=1 e to those of ΓtUt,Bt,t [0, T ] . This implies the convergence of the finite dimensional{ distributions∈ of } m γn E2,j(t),Bt,t [0, T ] ( ∈ ) Xj=1 e to those of Ut,Bt,t [0, T ] . To show{ (ii), we prove∈ the} following tightness condition: E n n 4 2 (6.8) sup ( γn(Xt Xt ) γn(Xs Xs ) ) C(t s) . n 1 | − − − | ≤ − ≥ Taking into account (4.7) and (4.11), we only need to show the above in- equality for γnI11, γnI12,j, γnI13, γnI2,j, γnI4,j, γnE1,j, γnE2,j and γnE3,j. The tightness for the terms γnI11, γnI13 and γnE3,j is clear. Now we consider the tightness of the term I2,j. We write t I (t) I (s) = (Λn Λn) Γnbj (s)(s η(s)) dBj 2,j − 2,j t − s s 2 − s Z0 t + ΛnΓnbj (u)(u η(u)) dBj . s u 2 − u Zs EULER SCHEMES OF SDE DRIVEN BY FBM 39 Then it follows from Lemma A.3 (A.4) that t E( γ (Λn Λn) Γnbj (s)(s η(s)) dBj 4) C(t s)4β(E Λn 8 )1/2 | n t − s s 2 − s| ≤ − k kβ Z0 C(t s)4β. ≤ − Lemma A.3 (A.4) also implies that the fourth moment of the second term 4H is bounded by C(t s) . The tightness for γnI12,j , γnI4,j, γnE1,j, γnE2,j can be obtained in− a similar way by applying the estimates (A.5) and (A.4) from Lemma A.3, (A.15) from Lemma A.5, and (A.8) from Lemma A.4, respectively. 7. A limit theorem in Lp for weighted sums. Following the methodology used in [3], we can show the following limit result for random weighted sums. The proof uses the techniques of fractional calculus and the classical decompositions in large and small blocks. Consider a double sequence of random variables ζ = ζk,n,n N, k = 0, 1,...,n , and for each t [0, T ], we denote { ∈ } ∈ nt/T ⌊ ⌋ (7.1) gn(t) := ζk,n. Xk=0 Proposition 7.1. Fix λ> 1 β, where 0 <β< 1. Let p 1 and p′,q′ > 1 1 − 1 1 ≥ 1 such that p′ + q′ = 1 and pp′ > β , pq′ > λ . Let gn be the sequence of processes defined in (7.1). Suppose that the following conditions hold true: pq′ (i) for each t [0, T ], gn(t) z(t) in L ; (ii) for any j, k∈= 0, 1,...,n →we have ′ ′ E( g (kT/n) g (jT/n) pq ) C( k j /n)λpq . | n − n | ≤ | − | ′ ′ Let f = f(t),t [0, T ] be a process such that E( f pp ) C and E( f(0) pp ) { ∈ } k kβ ≤ | | ≤ C. Then for each t [0, T ], ∈ nt/T ⌊ ⌋ t n p (7.2) F (t) := f(tk)ζk,n f(s) dz(s) in L as n . → 0 →∞ Xk=0 Z t Remark 7.1. The integral 0 f(s) dz(s) is interpreted as a Young inte- gral in the sense of Proposition 2.2, which is well defined because f and z, ′ ′ as functions on [0, T ] with valuesR in Lpp and Lpq , are H¨older continuous [conditions (i) and (ii) together imply the H¨older continuity of z] of order β and λ, respectively. Recall that the H¨older continuity of a function with values in Lp is defined in (2.5). 40 Y. HU, Y. LIU AND D. NUALART Remark 7.2. Convergence (7.2) still holds true if the condition E pp′ ( f β ) C is weakened by assuming that f is H¨older continuous of order k k ′ ≤ β in Lpp . The proof will be similar to that of Proposition 7.1. Proof of Proposition 7.1. Given two natural numbers m [f(s) f(al)] dgn(s) (al,bl) − (7.5) Z bl α α 1 α = ( 1) D [f(s) f(al)]D − [gn(s) gn(bl )] ds, − al+ − bl − − Zal − EULER SCHEMES OF SDE DRIVEN BY FBM 41 where we take α (1 λ, β). By (2.2), it is easy to show that ∈ − α 1 β β α D [f(s) f(a )] f (s a ) − | al+ − l |≤ Γ(1 α) β αk kβ − l (7.6) − − α β C f m − . ≤ k kβ On the other hand, by (2.3) we have 1 α D − [g (s) g (b )] bl n n l (7.7) | − − − | 1 g (s) g (b ) bl g (s) g (u) = n − n l− + (1 α) n − n du . Γ(α) (b s)1 α − (u s)2 α l − Zs − − − We can calculate the integral in the above equation explicitly. bl g (s) g (u) n − n du (u s)2 α Zs − − bl gn(s) gn(u) = − 2 α du ε(s) (u s) − (7.8) Z − tk+1 α 2 = [gn(s) gn(tk)] (u s) − du − tk − k : tk [ε(s),bl) Z ∈X 1 α 1 α 1 = [g (s) g (t )] [(t s) − (t s) − ]. n − n k 1 α k − − k+1 − k : t [ε(s),b ) k∈X l − Substituting (7.6), (7.7) and (7.8) into (7.5), we obtain [f(s) f(a )] dg (s) − l n Z(al,bl) bl α β 1 α C f βm − D − [gn(s) gn(bl )] ds ≤ k k | bl − − | Zal − tk+1 α β 1 α C f m − D − [g (s) g (b )] ds β bl n n l ≤ k k tk | − − − | k : tk [η(al),bl) Z ∈X tk+1 α β α 1 C f βm − gn(tk) gn(bl ) (bl s) − ds ≤ k k | − − | tk − k : tk [η(al),bl) Z ∈X α β + C f m − g (t ) g (t ) k kβ | n k − n j | k,j : η(a ) t We denote the first term in the right-hand side of the above expression by A1,l and the second one by A2,l. Applying the Minkowski inequality, we see that the quantity mt/T p 1/p ⌊ ⌋ (7.9) E A1,l ! l=0 X is less than mt/T ⌊ ⌋ tk+1 α β E p p 1/p α 1 Cm − ( f β gn(tk) gn(bl ) ) (bl s) − ds , k k | − − | tk − l=0 k : tk [η(al),bl) Z X ∈X so by applying the H¨older inequality, condition (ii) and the assumption ′ E( f pp ) C to the above, we can show that quantity (7.9) is less than k kβ ≤ mt/T ⌊ ⌋ tk+1 α β λ α 1 (7.10) Cm − (bl tk) (bl s) − ds . − tk − l=0 k : tk [η(al),bl) Z X ∈X Since tk+1 λ α 1 (bl tk) (bl s) − ds − tk − k : tk [η(al),bl) Z ∈X λ+α tk+1 1 T λ α 1 = + (bl tk) (bl s) − ds α n − tk − k : tk [η(al),bl T/n) Z ∈ X − λ+α 1 T T λ α 1 + (b t ) (b t ) − ≤ α n n l − k l − k+1 k : tk [η(al),bl T/n) ∈ X − λ+α 1 T T n α+1 λ + C m− − ≤ α n n m α λ Cm− − , ≤ where in the second inequality we used the assumption that α> 1 λ and the − T fact that the number of partition points tk, k = 0, 1,...,n in [η(al), bl n ) n { } − is bounded by m , the estimate (7.10) of (7.9) implies that mt/T p 1/p mt/T ⌊ ⌋ ⌊ ⌋ α β α λ 1 β λ (7.11) E A1,l Cm − m− − Cm − − 0 ! ≤ ≤ → l=0 l=0 X X as m tends to . ∞ EULER SCHEMES OF SDE DRIVEN BY FBM 43 Using an argument similar to the estimate of quantity (7.9), it can be shown that the quantity mt/T p 1/p ⌊ ⌋ E A2,l ! l=0 X is less than mt/T ⌊ ⌋ α β λ C m − t t | k − j| l=0 k,j : η(a ) t The summand in the above can be estimated as follows: tk+1 λ α 1 α 1 tk tj [(tj s) − (tj+1 s) − ] ds | − | tk − − − k,j : η(al) tk This result has the following two consequences. 44 Y. HU, Y. LIU AND D. NUALART Corollary 7.1. Let B = B ,t [0, T ] be an m-dimensional fBm with { t ∈ } Hurst parameter H> 3/4. Define tk+1 ζij = n (Bi Bi )δBj, k,n s − η(s) s Ztk kT for i, j = 1,...,m and k = 0,...,n 1, where we recall that tk = n . Set also 1 − ζn,n = 0. Let λ = 2 , and β, p, p′, q′, f satisfy the assumptions in Proposi- tion 7.1. Then nt/T ⌊ ⌋ t ij ij p f(tk)ζk,n f(s) dZs in L , → 0 Xk=0 Z where Zij is the generalized Rosenblatt process defined in Section 2.5. Proof. To prove the corollary, it suffices to show that the conditions in Proposition 7.1 are all satisfied here. We have shown in Section 2.5 the L2 nt/T ij ij p convergence of gn(t)= ⌊k=0 ⌋ ζk,n to Zt . This convergence also holds in L due to the equivalence of all the Lp-norms in a finite Wiener chaos. Applying P (A.8) in Lemma A.4 with F 1 and taking into account that γn = n when 3 ≡ 1 H> 4 , we obtain condition (ii) in Proposition 7.1 with λ = 2 . The following result will also be useful later. Corollary 7.2. Let B = B ,t [0, T ] be one-dimensional fBm with { t ∈ } Hurst parameter H ( 1 , 1). Define ∈ 2 tk+1 (7.12) ζ = (s η(s)) dB , k,n − s Ztk for k = 0,...,n 1. Set also ζ = 0. Let λ = H, and β, p, p , q , f satisfy − n,n ′ ′ the assumptions in Proposition 7.1. Then for each t [0, T ], ∈ nt/T ⌊ ⌋ T t n f(tk)ζk,n f(s) dBs, → 2 0 Xk=0 Z in Lp, as n tends to infinity. This convergence still holds true when we replace the above ζk,n by tk+1 ζ˜ = (B B ) ds. k,n s − η(s) Ztk EULER SCHEMES OF SDE DRIVEN BY FBM 45 Proof. As before, to prove the corollary it suffices to show that the conditions in Proposition 7.1 are all satisfied here. Let us first consider the convergence for ζk,n. Set nt/T ⌊ ⌋ gn(t) := n ζk,n, Xk=0 where ζk,n is defined in (7.12). Condition (ii) follows from estimate (A.4) in Lemma A.3 by taking F 1 and ν = 1. The covariance of the process gn is given by ≡ t t E(g (t)g ′ (t)) = α nn′ (u η (u))(v η ′ (v))µ(du dv) n n H − n − n Z0 Z0 2 t t T 2H 2 α u v − du dv → 4 H | − | Z0 Z0 T 2 = t2H 4 2 as n,n′ , which implies that gn(t) is a Cauchy sequence in L . Here T→∞ T T T T T η (t)= i when i t< (i + 1) and η ′ (t)= i when i t< (i + 1). n n n ≤ n n n′ n′ ≤ n′ In fact, we can also calculate the kernel of the limit of zn(t). Suppose that φ satisfies g (t)= δ(φ (t)). Then for any ψ , n ∈H n n ∈H T η(t) 2H 2 nφn, ψ = nαH (u η(u))ψ(v) u v − du dv h iH − | − | Z0 Z0 T ψ, 1[0,t] , → 2 h iH T as n + . This implies that the kernel of the limit of gn(t) is 2 1[0,t]; in → ∞ 2 T other words, the random variable gn(t) converges in L to 2 Bt. The convergence result for ζ˜k,n can be shown by noticing that tk+1 T tk+1 ζ˜ = (B B ) ds = (B B ) (s η(s)) dB . k,n s − η(s) n tk+1 − tk − − s Ztk Ztk This completes the proof of the corollary. 3 8. Asymptotic error of the modified Euler scheme in case H ∈ ( 4 , 1). The limit theorems for weighted sums proved in the previous section allow us to derive the Lp-limit of the quantity n(X Xn) in the case H ( 3 , 1). t − t ∈ 4 Theorem 8.1. Let H ( 3 , 1). Suppose that X and Xn are defined by ∈ 4 (1.1) and (1.3), respectively. Let Zij, i, j = 1,...,m be the matrix-valued gen- eralized Rosenblatt process defined in Section 2.5. Assume σ C5(Rd; Rd m) ∈ b × 46 Y. HU, Y. LIU AND D. NUALART and b C4(Rd; Rd). Then ∈ b n(X Xn) U t − t → t p in L (Ω) as n tends to infinity, where U¯t,t [0, T ] is the solution of the following linear stochastic differential equation:{ ∈ } t m t j j U t = b(Xs)U s ds + σ (Xs)U s dBs 0 ∇ 0 ∇ Z Xj=1 Z m t + ( σjσi)(X ) dZij ∇ s s i,j=1 Z0 (8.1) X T t T t + ( bb)(X ) ds + ( bσ)(X ) dB 2 ∇ s 2 ∇ s s Z0 Z0 m t T j j + ( σ b)(Xs) dBs. 2 0 ∇ Xj=1 Z n Proof. Recall the decomposition Yt = Xt Xt given in (4.7) and (4.11). − p We have shown that nI13(t), nI4,j(t), nE1,j(t) and nE3,j(t) converge in L to zero for each t [0, T ]. It remains to show the Lp convergence of nI (t), ∈ 11 nI12,j(t), nI2,j(t) and nE2,j(t) and identify their limits. Step 1. Recall E2,j(t) is defined in (6.6). It has been shown in the proof of Theorem 6.1 that n(E (t) E (t)) converges to zero in Lp. On the other 2,j − 2,j hand, applying Corollarye 7.1 to nE2,j(t) yields m te j i ij p nE2,j(t) ΛtΓes( σ σ )(Xs) dZs in L . → 0 ∇ Xi=1 Z e p Therefore, nE2,j(t) converges in L , and the limit is the same as nE2,j(t). Step 2. Denote nt/T e ⌊ ⌋ tk+1 t ˜ j ∧ j I2,j(t) = Λt Γtk ( σ b)(Xtk ) (s η(s)) dBs, ∇ tk − Xk=0 Z for t [0, T ] [as before, we define t = T (n + 1)]. Applying Corollary 7.2 ∈ n+1 n to nI˜2,j(t) yields T t nI˜ (t) Λ Γ ( σjb)(X ) dBj in Lp. 2,j → 2 t s ∇ s s Z0 p We want to show that nI2,j(t) and nI˜2,j(t) have the same limit in L . Write n(I (t) I˜ (t)) 2,j − 2,j EULER SCHEMES OF SDE DRIVEN BY FBM 47 t (8.2) = n (ΛnΓnbj (s) Λ Γ ˜bj (s))(s η(s)) dBj t s 2 − t s 2 − s Z0 t + n Λ (Γ ˜bj (s) Γ ( σjb)(X ))(s η(s)) dBj, t s 2 − η(s) ∇ η(s) − s Z0 ˜j 1 j where b2(s)= 0 σ (θXs + (1 θ)Xη(s))b(Xη(s)) dθ. It suffices to show that the two terms on∇ the right-hand− side of (8.2) both converge to zero in Lp. The convergence ofR the second term follows from estimate (A.16) of Lemma A.5. p n n j ˜j Lemma 4.1 implies that the L -norms of [Λt Γs b2(s) ΛtΓsb2(s)] and its Malliavin derivative converge to zero as n . So applying− Lemma A.3 n n j ˜→∞j (A.4) with ν =1 and Fs = Λt Γs b2(s) ΛtΓsb2(s), we obtain the convergence of the first term. − Step 3. Following the lines in step 2 we can show that nI12,j(t) converges in Lp to T t Λ Γ ( bσj)(X ) dBj. 2 t s ∇ s s Z0 Instead of (A.4) and (A.16) in step 2, we need to use estimates (A.5) and (A.15) here. p Similarly, it can be shown that nI11 converges in L to T t Λ Γ ( bb)(X ) ds. 2 t s ∇ s Z0 n p Step 4. We have shown that n(Xt Xt ) converges in L to U t, where we define, for each t [0, T ], − ∈ m t t j i ij T U t = ΛtΓs( σ σ )(Xs) dZs + ΛtΓs( bb)(Xs) ds 0 ∇ 2 0 ∇ i,jX=1 Z Z t m t T T j j + ΛtΓs( bσ)(Xs) dBs + ΛtΓs( σ b)(Xs) dBs . 2 0 ∇ 2 0 ∇ Z Xj=1 Z The theorem follows from the fact that the process U satisfies equation (8.1). 9. Weak approximation of the modified Euler scheme. The next result provides the weak rate of convergence for the modified Euler scheme (1.3). Theorem 9.1. Let X and Xn be the solution to equations (1.1) and 3 Rd Rd 4 Rd Rd m (1.3), respectively. Suppose that b Cb ( ; ), σ Cb ( ; × ). Then 3 Rd ∈ ∈ for any function f Cb ( ) there exists a constant C independent of n such that ∈ E E n 1 (9.1) sup [f(Xt)] [f(Xt )] Cn− . 0 t T | − |≤ ≤ ≤ 48 Y. HU, Y. LIU AND D. NUALART If we further assume that b C4, σ C5 and f C4, then for each t [0, T ], the sequence ∈ ∈ ∈ ∈ n E[f(X )] E[f(Xn)] , n N, { t − t } ∈ converges as n tends to infinity, and the limit is equal to the sum of the following two quantities: m α2 T t t t H E Di Dj[ f(X )Λ Γ ( σjσi)(X )] 2 { u r ∇ t t s ∇ s } j,i=1 Z0 Z0 Z0 (9.2) X 2H 2 2H 2 u s − s r − du ds dr × | − | | − | and T t t E f(X )Λ Γ ( bb)(X ) ds + Γ ( bσ)(X ) dB 2 ∇ t t s ∇ s s ∇ s s ( "Z0 Z0 (9.3) m t j j + Γs( σ b)(Xs) dBs . 0 ∇ #) Xj=1 Z n Proof. We use again decompositions (4.7) and (4.11) of Yt = Xt Xt , t [0, T ], and we continue to use the notation there. Given a function− f ∈3 Rd ∈ Cb ( ), we can write 1 n E[f(X )] E[f(Xn)] = n E[ f(Zθ)Y ] dθ, { t − t } ∇ t t Z0 θ n where we denote Zt = θXt + (1 θ)Xt , 0 t T . − ≤ ≤E θ 1 Step 1. In this step, we show that sup0 t T [ f(Zt )Yt] Cn− , which implies (9.1). From estimates (4.9) and (4.10≤ ≤) it| follows∇ that this|≤ inequality is true when Y is replaced by I11, I13, I12,j, I2,j or I4,j. Therefore, it suffices to show that E[ f(Zθ)E (t)] Cn 1 for i = 1, 2, 3 and j = 1,...,m, where | ∇ t i,j |≤ − Eij(t) are defined in Theorem 4.1 step 3. Consider first the term i = 2. The use of expression (4.12) and an application of the integration by parts formula yield E[ f(Zθ)E (t)] ∇ t 2,j m nt/T ⌊ ⌋ tk+1 t s = E f(Zθ)Λn F n,i,j ∧ δBi δBj ∇ t t tk u s " i=1 k=0 Ztk Ztk # (9.4) X X m nt/T ⌊ ⌋ t tk+1 t t s = α2 E ∧ Di Dj[ f(Zθ)ΛnF n,i,j] H v r t t tk 0 tk 0 tk ∇ Xi=1 Xk=0 Z Z Z Z µ( du dv)µ(ds dr) , × EULER SCHEMES OF SDE DRIVEN BY FBM 49 n,i,j n j i n where we recall that Ft =Γt ( σ σ )(Xt ). [As before, in the above equa- T ∇ tion we set tn+1 = n (n + 1).] Therefore, n 1 tk+1 t tk+1 t E θ − [ f(Zt )E2,j(t)] C µ(du dv)µ(dr ds) | ∇ |≤ tk 0 tk 0 Xk=0 Z Z Z Z 1 Cn− . ≤ For the term containing E1,j we can write m t E θ E n,i,j i i j [ f(Zt )E1,j(t)] = Hs (Bs Bη(s)) dBs , ∇ 0 − Xi=1 Z n,i,j θ n n j,i n j i n where Hs = f(Z )Λ [Γ σ (s) Γ ( σ σ )(X )]. An application ∇ t t s 2 − η(s) ∇ η(s) of the relation between the Skorohod and path-wise integrals (2.9) yields t E Hn,i,j(Bi Bi ) dBj s s − η(s) s Z0 T t j n,i,j i i 2H 2 = α E[D (H (B B ))] s u − ds du H u s s − η(s) | − | Z0 Z0 T t j n,i,j i i 2H 2 = α E[D H (B B )] s u − ds du H u s s − η(s) | − | Z0 Z0 T t n,i,j 2H 2 + α E[H ]1 (u)δ s u − ds du H s [η(s),s] ij | − | Z0 Z0 := A1 + A2. By the integration by parts we see that A1 is equal to T t T s 2 i j n,i,j 2H 2 2H 2 α E[D D H ]1 (v) v r − s u − dv dr ds du. H r u s [η(s),s] | − | | − | Z0 Z0 Z0 Z0 i j n,i,j β 1 Using sup E[D DuHs ] Cn for any <β n,i,j θ n n n j n where Js = H f(Zt )Λt (Γη(s) Γs )σ0(s). By expressing the term (Γη(s) n ∇ − − Γs ) as the sum of integrals over the interval [η(s),s] and then applying (2.9) E n,i,j 1 and integration by parts, we can show that sups [0,T ] [Js ] Cn− . This ∈ ≤ implies θ 2H (9.7) E[ f(Z )E (t)] Cn− , | ∇ t 3,j |≤ which completes the proof of (9.1). Step 2. Now we show the second part of the theorem. From estimates (4.9), (4.10), (9.5), (9.6) and (9.7) we see that the expression n 1 E[ f(Zθ)Y ] dθ 0 ∇ t t converges to zero as n tends to infinity when Yt is replaced by I13(t), I4,j(t), 1 E R θ E1,j(t) or E3,j(t). Therefore, it suffices to consider n 0 [ f(Zt )Yt] dθ when Y is replaced by the remaining terms in the decomposition∇ of Y . t R t Consider first the term E2,j(t), and denote Gi,j = Di Dj[ f(X )Λ Γ ( σjσi)(X )]. s,r,v v r ∇ t t s ∇ s It is clear that m nt/T ⌊ ⌋ t tk+1 t t s nα2 ∧ Gi,j µ(du dv)µ(ds dr) H tk,r,v i,j=1 k=0 Z0 Ztk Z0 Ztk (9.8) X X 2 m t t t αH T i,j 2H 2 2H 2 Gs,r,v s v − r s − ds dv dr, → 2 0 0 0 | − | | − | i,jX=1 Z Z Z almost surely. Therefore, by the dominated convergence theorem, the ex- pectation of the left-hand side of the above expression converges to the expectation of the right-hand side, which is term (9.2). From Lemma 4.1, we have Di Dj[ f(Zθ)ΛnF n,i,j] Di Dj[ f(X )Λ Γ ( σjσi)(X )] v r t t tk v r t t tk tk p (9.9) k ∇ − ∇ ∇ k 1 2β Cn − , ≤ which, together with equation (9.4), implies that n m E[ f(Zθ)E (t)] j=1 ∇ t 2,j converges to the same limit as the expectation of the left-hand side of (9.8). P The results in steps 2 and 3 of the proof of Theorem 8.1 imply that the terms nE[ f(Zθ)I (t)], nE[ f(Zθ)I (t)] and nE[ f(Zθ) m I (t)] ∇ t 11 ∇ t 12,j ∇ t j=1 2,j converge to the second, third and fourth term in (9.3), respectively. For E θ m P example, let us consider n [ f(Zt ) j=1 I2,j(t)]. We have shown in Theo- rem 8.1 that ∇ P T t nI (t) Λ Γ ( σjb)(X ) dBj 2,j → 2 t s ∇ s s Z0 EULER SCHEMES OF SDE DRIVEN BY FBM 51 in Lp for any p 1. So it follows from the H¨older inequality that ≥ T t E n f(Zθ)I (t) f(X ) Λ Γ ( σjb)(X ) dBj 0 ∇ t 2,j −∇ t 2 t s ∇ s s → Z0 as n . The other two terms can be studied in similar way. This completes the proof→∞ of the theorem. Remark 9.1. Theorem 9.1 may be used to construct a Richard extrap- 1 olation scheme with error bound o(n− ). 10. Rate of convergence for the Euler scheme. In this section, we apply our approach based on Malliavin calculus developed in Section 4 to study the rate of convergence of the naive Euler scheme defined in (1.2). Our first result is the rate of the strong convergence of the naive Euler scheme. As we will see, the weak rate of convergence and the rate of strong convergence are the same for the naive Euler scheme. We still use Xn to represent the naive Euler scheme (1.2). This will not cause confusion since we will only deal with this scheme in this section. Theorem 10.1. Let X and Xn be the processes defined in (1.1) and 1 Rd Rd 2 Rd Rd m (1.2), respectively. Suppose that b Cb ( ; ) and σ Cb ( ; × ). Then for each p 1, we have ∈ ∈ ≥ 2H 1 E n p 1/p n − sup ( Xt Xt ) C. t [0,T ] | − | ≤ ∈ 3 Rd Rd 4 Rd Rd m If we assume b Cb ( ; ) and σ Cb ( ; × ), then as n tends to in- finity, ∈ ∈ 2H 1 m t 2H 1 n T − j j n − (Xt Xt ) ΛtΓs( σ σ )(Xs) ds, − → 2 0 ∇ Xj=1 Z 1 where Λ is the solution to linear equation (4.16) and Γt = Λt− , and the convergence holds in Lp for all p 1. ≥ Proof. We let Y = X Xn, t [0, T ]. Then as in the proof of Theo- t t − t ∈ rem 4.1, we can derive the decomposition of Yt t m n n n l n l l Yt = Λt Γs b3(s) b(Xη(s))(s η(s)) + σ (Xη(s))(Bs Bη(s)) ds 0 " − − # Z Xl=1 m t n n j j + Λt Γs b2(s)(s η(s)) dBs 0 − Xj=1 Z 52 Y. HU, Y. LIU AND D. NUALART m t n n j,i i i j + Λt Γs σ2 (s)(Bs Bη(s)) dBs 0 − i,jX=1 Z =: I1(t)+ I2(t)+ I3(t)+ I4(t), n n j j,i where Λ , Γ , b2(s), σ2 (s) and b3(s) are the same terms as those defined in the proof of Theorem 4.1 with the scheme Xn replaced by the classical Euler scheme (1.2). 1 It is clear that I1(t) p Cn− . On the other hand, estimates (A.4) and k k ≤ 1 1 (A.5) of Lemma A.3 imply that I2(t) p Cn− and I3(t) p Cn− . Fi- k k ≤ k k ≤ 1 2H nally, as in the proof of (A.15) in Lemma A.5 we obtain I4(t) p Cn − . This completes the proof of the first part of the theorem.k k ≤ Applying the integration by parts to I4(t) yields t ΛnΓnσj,i(s)(Bi Bi ) dBj t s 2 s − η(s) s Z0 t = ΛnΓnσj,i(s)(Bi Bi )δBj t s 2 s − η(s) s Z0 t t + α Dj[ΛnΓnσj,i(s)](Bi Bi )µ(ds dr) H r t s 2 s − η(s) Z0 Z0 t t n n j,i + δijαH Λt Γs σ2 (s)1[η(s),s](r)µ(ds dr) Z0 Z0 1 2 3 =: An(t)+ An(t)+ An(t). From (A.8) we have A1 (t) Cγ 1. Applying (A.5) with F replaced by k n kp ≤ n− u t j n n j,i 2H 2 D [Λ Γ σ (s)] r u − dr r t s 2 | − | Z0 2 1 2H 1 3 we obtain An(t) p Cn− . So it suffices to identify the limit of n − An(t) in Lp. It followsk fromk ≤ Lemma 4.1 and Remark 4.1 that n n j,j j j 1 2β Λ Γ σ (s) Λ Γ ( σ σ )(X ) Cn − . k t s 2 − t s ∇ s kp ≤ 2H 1 3 Therefore, n − An(t), and the quantity t t 2H 1 j j 2H 2 n − Λ Γ ( σ σ )(X )1 (r) r s − ds dr t s ∇ s [η(s),s] | − | Z0 Z0 converges to the same value in Lp. The theorem now follows by noticing that t t 2H 1 j j 2H 2 n − Λ Γ ( σ σ )(X )1 (r) r s − ds dr t s ∇ s [η(s),s] | − | Z0 Z0 EULER SCHEMES OF SDE DRIVEN BY FBM 53 t 2H 1 2H 1 j j (s η(s)) − = n − Λ Γ ( σ σ )(X ) − ds t s ∇ s 2H 1 Z0 − T 2H 1 t − Λ Γ ( σjσj)(X ) ds, → 2α t s ∇ s H Z0 in Lp for all p 1. ≥ As a consequence of the above theorem, we can deduce the following result. Corollary 10.1. Let X and Xn be the processes defined in (1.1) and 3 Rd Rd 4 Rd Rd m (1.2), respectively. Suppose that b Cb ( ; ), σ Cb ( ; × ) and f 2 Rd ∈ ∈ p ∈ Cb ( ). Let Λ be defined in (4.16). Then we have the following L -convergence as n for all p 1: →∞ ≥ 2H 1 m t 2H 1 n T − j j n − [f(Xt ) f(Xt)] f(Xt)ΛtΓs( σ σ )(Xs) ds. − → 2 0 ∇ ∇ Xj=1 Z Proof. We can write 1 2H 1 n 2H 1 θ n n − [f(X ) f(X )] = n − f(Z ) dθ (X X ), t − t ∇ t t − t Z0 θ n where we denote Zt = θXt + (1 θ)Xt , t [0, T ]. Then the result follows − n ∈ from Theorem 10.1, the convergence of Xt to Xt and the assumption on f. The above corollary implies the following weak approximation result: 2H 1 n lim n − E[f(Xt)] E[f(X )] n t →∞ { − } 2H 1 m t T − j j = E[ f(Xt)ΛtΓs( σ σ )(Xs)] ds. 2 0 ∇ ∇ Xj=1 Z APPENDIX A.1. Estimates of a Young integral. In this section, we give an estimate on the pathwise integral using fractional calculus. Lemma A.1. Let z = zt,t [0, T ] be a H¨older continuous function with index β (0, 1). Suppose{ that∈ f : R}l+m R is continuously differen- ∈ → tiable. We denote by f the l-dimensional vector with coordinates ∂f , x ∂xi ∇ ∂f i = 1,...,l, and by yf the m-dimensional vector with coordinates , ∇ ∂xl+i 54 Y. HU, Y. LIU AND D. NUALART i = 1,...,m. Consider processes x = x ,t [0, T ] and y = y ,t [0, T ] { t ∈ } { t ∈ } with dimensions l and m, respectively, such that x ′ and y ′ k k0,T,β k k0,T,β ,n are finite for each n 1, where β′ (0, 1) is such that β′ + β> 1. Then we have the following estimates:≥ ∈ (i) for any s,t [0, T ] such that s t and s = η(s), we have ∈ ≤ t f(x ,y ) dz K sup f(x ,y ) z (t s)β r η(r) r ≤ 1 | r η(r) |k kβ − Zs r [s,t] ∈ β+β′ ′ + K2 sup xf(xr1 ,yη(r2)) x s,t,β z β(t s) r1,r2 [s,t]|∇ |k k k k − ∈ β+β′ ′ + K3 sup yf(xr1 ,yr2 ) y s,t,β ,n z β(t s) , r1,r2 [s,t]|∇ |k k k k − ∈ where the Ki, i = 1, 2, 3, are constants depending on β and β′; (ii) if the function f only depends on the first l variables, then the above estimate holds for all 0 s t T . ≤ ≤ ≤ Proof. Take α such that β′ >α> 1 β. Let s,t [0, T ] be such that s = η(s) and s t. Applying the fractional− integration∈ by parts formula in Proposition 2.1≤, we obtain t t α 1 α (A.1) f(x ,y ) dz D f(x ,y ) D − (z z ) dr. r η(r) r ≤ | s+ r η(r) || t r − t | Zs Zs − By the definition of fractional differentiation in (2.3) and taking into account that α + β 1 > 0, we can show that − 1 α α+β 1 (A.2) Dt − (zr zt) K0 z β (t r) − , s r t, | − − |≤ k k − ≤ ≤ β where K0 = (β+α 1)Γ(α) . On the other hand, using (2.2) we obtain − Dα f(x ,y ) | s+ r η(r) | r 1 f(xr,yη(r)) f(xr,yη(r)) f(xu,yη(u)) | | + α | − | du ≤ Γ(1 α) (r s)α (r u)α+1 − − Zs − 1 ≤ Γ(1 α) (A.3) − α sup f(xr,yη(r)) (r s)− × r [s,t]| | − ∈ r β′ α 1 ′ + α sup xf(xr1 ,yη(r2)) x s,t,β (r u) − − du r1,r2 [s,t]|∇ |k k s − ∈ Z ′ r η(r) η(u) β ′ + α sup yf(xr1 ,yr2 ) y s,t,β ,n | − α+1| du . r1,r2 [s,t]|∇ |k k s (r u) ∈ Z − EULER SCHEMES OF SDE DRIVEN BY FBM 55 Inequalities (A.1), (A.3) and (A.2) together imply t f(xr,yη(r)) dzr Zs t 1 α sup f(xr,yη(r)) (r s)− ≤ Γ(1 α) s r [s,t]| | − − Z ∈ ′ + α sup xf(xr1 ,yη(r2)) x s,t,β r1,r2 [s,t]|∇ |k k ∈ r β′ α 1 (r u) − − du × − Zs ′ + α sup yf(xr1 ,yr2 ) y s,t,β ,n r1,r2 [s,t]|∇ |k k ∈ ′ r η(r) η(u) β | − | du × (r u)α+1 Zs − α+β 1 K z (t r) − dr × 0k kβ − β K1 sup f(xr,yη(r)) z β(t s) ≤ r [s,t]| |k k − ∈ β+β′ ′ + K2 sup xf(xr1 ,yη(r2)) x s,t,β z β (t s) r1,r2 [s,t]|∇ |k k k k − ∈ β+β′ ′ + K3 sup yf(xr1 ,yr2 ) y s,t,β ,n z β(t s) , r1,r2 [s,t]|∇ |k k k k − ∈ Γ(α+β) αΓ(α+β)Γ(β′ α+1) α where K1 = K0 Γ(β+1) , K2 = K0 Γ(1 α)Γ(β+β′+1)(− β′ α) , K3 = K0K4 Γ(1 α) and − − − K4 is the constant in Lemma A.2. This completes the proof. Lemma A.2. Let β, β′ and α be such that β′ >α> 1 β. Then for any s,t [0, T ] such that s t r β′ α+β 1 η(r) η(u) β+β′ (t r) − | − | du dr K (t s) . − (r u)α+1 ≤ 4 − Zs Zs − Proof. Without loss of generality, we let T = 1. Note that when η(s)= 1 s t η(r) β′ α+β 1 η(r) η(u) = (t r) − | − | du dr − (r u)α+1 Zη(s)+1/n Zη(s) − t η(r) η(r) 1/n β′ α+β 1 − η(r) η(u) = (t r) − + | − α+1| du dr η(s)+1/n − η(r) 1/n η(s) (r u) Z Z − Z − := J1 + J2. On one hand, notice that in the term J we always have r u> 1 , and 2 − n thus η(r) η(u) r u + 1 2(r u). Therefore, − ≤ − n ≤ − t η(r) 1/n β′ β′ α+β 1 − 2 (r u) J (t r) − − du dr 2 ≤ − (r u)α+1 Zη(s)+1/n Zη(s) − ′ K(t s)β+β . ≤ − On the other hand, t η(r) β′ α+β 1 η(r) η(u) J1 = (t r) − | − α+1| du dr η(s)+1/n − η(r) 1/n (r u) Z Z − − t β′ α+β 1 1 1 Kn− (t s) − dr ≤ − (r η(r))α − (r η(r) + 1/n)α Zη(s)+1/n − − t β′ α+β 1 1 Kn− (t s) − dr ≤ − (r η(r))α Zη(s)+1/n − β′ α+β 1 (η(t) + 1/n) (η(s) + 1/n) α 1 Kn− (t s) − − n − ≤ − 1/n ′ K(t s)β+β . ≤ − The lemma is now proved. A.2. Estimates for some special Young and Skorohod integrals. In this section we derive estimates for some specific Young and Skorohod integrals. We fix n N and consider the uniform partition on [0, T ]. ∈ Lemma A.3. Let B = Bt,t [0, T ] be a one-dimensional fBm with 1 { ∈ } 1 Hurst parameter H > 2 . Fix ν 0 and p H . Let F = Ft,t [0, T ] be a stochastic process whose trajectories≥ are≥ H¨older continuous{ of∈ order }γ> 1,q 1 H and such that Ft D , t [0, T ], for some q>p. For any ρ> 1 we set− ∈ ∈ F1,ρ = sup ( Ft ρ DsFt ρ). s,t [0,T ] k k ∨ k k ∈ EULER SCHEMES OF SDE DRIVEN BY FBM 57 Then there exists a constant C (independent of F ) such that the following inequalities hold for all 0 s Proof of ( A.4). Applying (2.9) we can decompose the Young integral as the sum of a Skorohod integral plus a complementary term, t F (u η(u))ν dB u − u Zs t (A.6) = F (u η(u))νδB u − u Zs t T ν 2H 2 + α (u η(u)) D F r u − dr du. H − r u| − | Zs Z0 It follows from (2.11) that the Lp-norm of the first integral of the right-hand ν H side of (A.6) is bounded by Cn− (t s) F1,p. On the other hand, from Minkowski’s inequality it follows that− the Lp-norm of the second integral is ν less than or equal to Cn− (t s)F1,p. These estimates imply (A.4) because (t s) (t s)H T 1 H . − − ≤ − − Proof of (A.5). If t s 1 , we can write − ≤ n t t F (B B ) du F (B B ) du u u − η(u) ≤ k u u − η(u) kp Zs p Zs H C sup Ft qn− (t s) ≤ t [0,T ] k k − ∈ 1 H C sup Ft qn− (t s) , ≤ t [0,T ] k k − ∈ where the first inequality follows from Minkowski’s inequality and the second 1 one from H¨older’s inequality. Suppose that t s n . Applying Fubini’s theorem for the Young integral, we obtain − ≥ t t ε(v) F (B B ) du = 1 (u)F du dB . u u − η(u) [s,t] u v Zs Zη(s)Zv Applying (A.4) with ν = 0 we obtain t ε(v) H 1 1 (u)F du dB C(t η(s)) n− F [s,t] u v ≤ − 1,p Zη(s)Zv p 58 Y. HU, Y. LIU AND D. NUALART (A.7) H 1 C(t s) n− F . ≤ − 1,p This completes the proof of (A.5). Lemma A.4. Let B = Bt,t [0, T ] be an m-dimensional fBm with 1 { ∈1 } Hurst parameter H > 2 . Fix p H . Let F = Ft,t [0, T ] be a stochastic process such that F D2,q, t ≥[0, T ], for some{ q>p∈. For any} ρ> 1 we set t ∈ ∈ F2,ρ = sup ( Ft ρ DsFt ρ DrDsFt ρ). r,s,t [0,T ] k k ∨ k k ∨ k k ∈ Set also F = sup ( Ft DsFt DrDsFt ). ∗ r,s,t [0,T ] | | ∨ | | ∨ | | ∈ Then there exists a constant C (independent of F ) such that the following holds for all 0 s Proof. Using (2.8), we can write nt/T ⌊ ⌋ tk+1 t u ∧ i j Ftk δBvδBu k= ns/T Ztk s Ztk X⌊ ⌋ ∨ t (A.10) = F (Bi Bi )δBj η(u) u − η(u) u Zs t T + α DjF (Bi Bi )µ(drdu). H r η(u) u − η(u) Zs Z0 Applying (A.5) to the second integral of the right-hand side of (A.10) with T j 2H 2 Fu replaced by 0 DrFη(u) r u − dr (notice that here we do not need the H¨older continuity of the| − integrand| for the Young integral to be well defined) yields R t T DjF (Bi Bi )µ(dr du) r η(u) u − η(u) Zs Z0 p T 1 H 2H 2 (A.11) Cn− (t s) F2,q sup r u − dr ≤ − u [0,T ] 0 | − | ∈ Z EULER SCHEMES OF SDE DRIVEN BY FBM 59 1 H Cn− (t s) F . ≤ − 2,q This implies both estimates (A.8) and (A.9). Applying (2.8) to the first summand on the right-hand side of (A.10) yields t i i j Fu(Bu Bη(u))δBu s − (A.12)Z t u t T u i j i j = FuδBvδBu + αH DvFuµ(drdv) δBu. Zs Zη(u) Zs Z0 Zη(u) Now we apply (2.11) to the second term of the right-hand side of (A.12), and we obtain t T u i j DvFuµ(dr dv) δBu Zs Z0 Zη(u) p T u (A.13) CF2,p 1[s,t](u) µ(dr dv) ≤ 1/H Z0 Zη(u) L ([0,T ]) 1 H CF n − (t s) . ≤ 2,p − Again, this inequality implies both estimates (A.8) and (A.9). t u i j It remains to estimate the term Is,t := s η(u) FuδBvδBu. It follows from (2.11) that R R I CF 1 (u)1 (v) 1/H 2 k s,tkp ≤ 2,pk [s,t] [η(u),u] kL ([0,T ] ) H H CF n− (t s) , ≤ 2,p − which completes the proof of (A.9). To derive (A.8) we need a more accurate estimate. Meyer’s inequality implies that Is,t p C[ 1[s,t](u)1[η(u),u](v)Fu ⊗2 p k k ≤ kk kH k + 1[s,t](u)1[η(u),u](v)DrFu ⊗3 p kk kH k + 1[s,t](u)1[η(u),u](v)Dr′ DrFu ⊗4 p] kk kH k C F p 1[s,t](u)1[η(u),u](v) ⊗2 . ≤ k ∗k k kH Therefore, to complete the proof, it suffices to show that 2 1[s,t](u)1[η(u),u](v) ⊗2 k kH t t u′ u 2 (A.14) = αH µ(dv dv′)µ(du du′) ′ Zs Zs Zη(u ) Zη(u) 2 (t s)γ− . ≤ − n 60 Y. HU, Y. LIU AND D. NUALART In the case t s 1 , − ≥ n t t u′ u µ(dv dv′ du du′) ′ Zs Zs Zη(u ) Zη(u) nt/T ′ ⌊ ⌋ tk′+1 tk+1 u u µ(dv dv′ du du′) ≤ k,k′= ns/T Ztk′ Ztk Ztk′ Ztk X⌊ ⌋ nt/T n 1 ′ ⌊ ⌋ − tk+p+1 tk+1 u u µ(dv dv′ du du′) ≤ k= ns/T p=1 n Ztk+p Ztk Ztk+p Ztk X⌊ ⌋ X− nt/T n 1 ⌊ ⌋ 4H − = n− Q(p) k= ns/T p=1 n X⌊ ⌋ X− 2 C(t s)γ− , ≤ − n where we recall that Q(p) is defined in Section 2.4, and inequality (A.14) follows. In the case t s 1 , we have the raw estimate − ≤ n t t u′ u t t 1 1 2H µ(dr dr′ du du′) 2H µ(du du′)= 2H (t s) ′ ≤ n n − Zs Zs Zη(u ) Zη(u) Zs Zs and 2H 2H 2 n− (t s) (t s)γ− . − ≤ − n So (A.14) is also true for this case. The proof of the lemma is now complete. Lemma A.5. Let B = Bt,t [0, T ] be a one-dimensional fBm with 1 { ∈ } Hurst parameter H> 2 . Suppose that F = Ft,t [0, T , G = Gt,t [0, T ] { ∈ } 1 { ∈ } are processes that are H¨older continuous of order β ( 2 ,H). Then there exists a constant C (not depending on F or G) such that∈ for all 0 s Proof of (A.15). We assume first that s,t [tk,tk+1] for some k = 0, 1,...,n 1. By Lemma A.1(ii), ∈ − t F (G G )(B B ) dB u u − η(u) u − η(u) u Zs β K1 sup Fu(Gu Gtk )(Bu B tk ) B β(t s) ≤ u [s,t]| − − |k k − ∈ 2 2β + K2 sup [ Fu(Gu Gtk ) B β(t s) u [s,t] | − |k k − (A.17) ∈ + F (B B ) G B (t s)2β | u u − tk |k kβk kβ − + (G G )(B B ) F B (t s)2β] | u − tk u − tk |k kβk kβ − 2β β Cκ (F, G)n− (t s) , ≤ β − 2 where κβ(F, G) = ( F + F β) G β B β . In the general case, we can write k k∞ k k k k k k t F (G G )(B B ) dB u u − η(u) u − η(u) u Zs ε(s) nt/T t t ⌊ ⌋ k+1 = + + Fu(Gu Gη(u))(Bu Bη(u)) dBu s tk η(t)! − − Z k= ns/T +1 Z Z ⌊ X ⌋ nt/T ⌊ ⌋ 2β β β β Cκ (F, G)n− (ε(s) s) + (t η(t)) + (T/n) ≤ β − − " k= ns/T +1 # ⌊ X ⌋ 2β β β 1 β Cκ (F, G)n− [(ε(s) s) + (t η(t)) + (η(t) ε(s))n − ] ≤ β − − − 1 3β β Cκ (F, G)n − (t s) , ≤ β − where the first inequality follows from (A.17). Proof of (A.16). 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