Proof of Theorem 3.1 getdoc711c. 370KB Jun 04 2011 12:04:30 AM

ii and Lemma 3.3, we also have ⌊nu p ⌋ X j=1 E 〈D〈λ, V n 〉, δ j 〉 2 H ≤ C n −13 sup k=1,...,m ⌊nu k ⌋ X i, ℓ=1 Eg ′ k Bt i −1 g ′ ℓ Bt ℓ−1 I 3 δ ⊗3 i I 3 δ ⊗3 l × sup i=1,..., ⌊nu k ⌋ ⌊nu p ⌋ X j=1 〈ǫ i −1 , δ j 〉 H + Cn −13 X r ∈Z |ρr| 2 ≤ C n −13 , which is 3.6. The proof of 3.7 follows the same lines, and is left to the reader. ƒ

3.2 Proof of Theorem 3.1

We are now in position to prove Theorem 3.1. For g : R → R, let G − n g, B, t := 1 p n ⌊nt⌋ X j=1 gBt j −1 h 3 n 1 6 ∆B j , t ≥ 0, n ≥ 1. We recall that h 3 x = x 3 − 3x, see 2.1, and the definition 2.3 of V n B, t. In particular, observe that V n B, t = G − n 1, B, t + 3n −13 B ⌊nt⌋n. 3.9 Our main theorem which will lead us toward the proof of Theorem 3.1 is the following. Theorem 3.7. If g ∈ C ∞ R is bounded with bounded derivatives, then the sequence B, G − n 1, B, G − n g, B converges to B, [[B]], −18 R g ′′′ B ds + R gB d[[B]] in the sense of finite-dimensional distributions on [0, ∞. Proof. We have to prove that, for any ℓ + m ≥ 1 and any u 1 , . . . , u ℓ+m ≥ 0: B, G − n 1, B, u 1 , . . . , G − n 1, B, u ℓ , G − n g, B, u ℓ+1 , . . . , G − n g, B, u ℓ+m Law −−−→ n →∞ B, [[B]] u 1 , . . . , [[B]] u ℓ , − 1 8 Z u ℓ+1 g ′′′ Bs ds + Z u ℓ+1 gBs d[[B]] s , . . . , − 1 8 Z u ℓ+m g ′′′ Bs ds + Z u ℓ+m gBs d[[B]] s . Actually, we will prove the following slightly stronger convergence. For any m ≥ 1, any u 1 , . . . , u m ≥ 0 and all bounded functions g 1 , . . . , g m ∈ C ∞ R with bounded derivatives, we have B, G − n g 1 , B, u 1 , . . . , G − n g m , B, u m Law −−−→ n →∞ B, − 1 8 Z u 1 g ′′′ 1 Bs ds + Z u 1 g 1 Bs d[[B]] s , . . . , − 1 8 Z u m g ′′′ m Bs ds + Z u m g m Bs d[[B]] s . 3.10 2137 Using 2.12, observe that G − n g, B, t = ⌊nt⌋ X j=1 gBt j −1 I 3 δ ⊗3 j . 3.11 The proof of 3.10 is divided into several steps, and follows the methodology introduced in [7]. Step 1.- We first prove that: lim n →∞ E € G − n g 1 , B, u 1 , . . . , G − n g m , B, u m Š = ‚ − 1 8 Z u 1 Eg ′′′ 1 Bs ds, . . . , − 1 8 Z u m Eg ′′′ m Bs ds Œ , lim n →∞ E G − n g 1 , B, u 1 , . . . , G − n g m , B, u m 2 R m = m X i=1 κ 2 Z u i Eg 2 i Bs ds + 1 64 E ‚Z u i g ′′′ i Bs ds Œ 2 . 3.12 For g as in the statement of the theorem, we can write, for any fixed t ≥ 0: E € G − n g, B, t Š = ⌊nt⌋ X j=1 E gBt j −1 I 3 δ ⊗3 j by 3.11 = ⌊nt⌋ X j=1 E 〈D 3 gBt j −1 , δ ⊗3 j 〉 H ⊗3 by 2.13 = ⌊nt⌋ X j=1 Eg ′′′ Bt j −1 〈ǫ j −1 , δ j 〉 3 H by 2.8 = − 1 8n ⌊nt⌋ X j=1 Eg ′′′ Bt j −1 + ⌊nt⌋ X j=1 Eg ′′′ Bt j −1 〈ǫ j −1 , δ j 〉 3 H + 1 8n −−−→ n →∞ − 1 8 Z t Eg ′′′ Bs ds by Lemma 3.2 i v. Now, let us turn to the second part of 3.12. We have E kG − n g 1 , B, u 1 , . . . , G − n g m , B, u m k 2 R m = m X i=1 EG − n g i , B, u i 2 . By the product formula 2.14, we have I 3 δ ⊗3 j I 3 δ ⊗3 k = I 6 δ ⊗3 j ⊗ δ ⊗3 k + 9I 4 δ ⊗2 j ⊗ δ ⊗2 k 〈δ j , δ k 〉 H + 18I 2 δ j ⊗ δ k 〈δ j , δ k 〉 2 H + 6〈δ j , δ k 〉 3 H . 2138 Thus, for any fixed t ≥ 0, EG − n

g, B, t

2 = ⌊nt⌋ X j,k=1 E gBt j −1 gBt k −1 I 3 δ ⊗3 j I 3 δ ⊗3 k = ⌊nt⌋ X j,k=1 E gBt j −1 gBt k −1 I 6 δ ⊗3 j ⊗ δ ⊗3 k + 9 ⌊nt⌋ X j,k=1 E gBt j −1 gBt k −1 I 4 δ ⊗2 j ⊗ δ ⊗2 k 〈δ j , δ k 〉 H + 18 ⌊nt⌋ X j,k=1 E € gBt j −1 gBt k −1 I 2 δ j ⊗ δ k Š 〈δ j , δ k 〉 2 H + 6 ⌊nt⌋ X j,k=1 E € gBt j −1 gBt k −1 Š 〈δ j , δ k 〉 3 H =: A n + B n + C n + D n . We will estimate each of these four terms using the Malliavin integration by parts formula 2.13. For that purpose, we use Lemma 3.4 and the notation of Remark 2.6. First, we have A n = ⌊nt⌋ X j,k=1 E 〈D 6 [gBt j −1 gBt k −1 ], δ ⊗3 j ⊗ δ ⊗3 k 〉 H ⊗6 3.2 = ⌊nt⌋ X j,k=1 6 X a=0 6 a E € g a Bt j −1 g 6−a Bt k −1 Š 〈ǫ ⊗a j −1 e ⊗ ǫ ⊗6−a k −1 , δ ⊗3 j ⊗ δ ⊗3 k 〉 H ⊗6 3.1 = ⌊nt⌋ X j,k=1 6 X a=0 E € g a Bt j −1 g 6−a Bt k −1 Š × X i 1 ,...,i 6 ∈{ j−1,k−1} |{ℓ:i ℓ = j−1}|=a 〈ǫ i 1 ⊗ . . . ⊗ ǫ i 6 , δ ⊗3 j ⊗ δ ⊗3 k 〉 H ⊗6 . Actually, in the previous double sum with respect to a and i 1 , . . . , i 6 , only the following term is 2139 non-negligible: ⌊nt⌋ X j,k=1 Eg ′′′ Bt j −1 g ′′′ Bt k −1 〈ǫ j −1 , δ j 〉 3 H 〈ǫ k −1 , δ k 〉 3 H = E ⌊nt⌋ X j=1 g ′′′ Bt j −1 〈ǫ j −1 , δ j 〉 3 H 2 = E − 1 8n ⌊nt⌋ X j=1 g ′′′ Bt j −1 + ⌊nt⌋ X j=1 g ′′′ Bt j −1 〈ǫ j −1 , δ j 〉 3 H + 1 8n 2 −−−→ n →∞ 1 64 E Z t g ′′′ Bs ds 2 by Lemma 3.2 i v. Indeed, the other terms in A n are all of the form ⌊nt⌋ X j,k=1 Eg a Bt j −1 g 6−a Bt k −1 〈ǫ j −1 , δ k 〉 H 5 Y i=1 〈ǫ x i −1 , δ y i 〉 H , 3.13 where x i and y i are for j or k. By Lemma 3.2 iii, we have P ⌊nt⌋ j,k=1 |〈ǫ j −1 , δ k 〉 H | = On as n → ∞. By Lemma 3.2 i, sup j,k=1,...,[nt] Q 5 i=1 |〈ǫ x i −1 , δ y i 〉 H | = On −53 as n → ∞. Hence, the quantity in 3.13 tends to zero as n → ∞. We have proved A n −−−→ n →∞ 1 64 E Z t g ′′′ Bs ds 2 . Using the integration by parts formula 2.13 as well as Lemma 3.4, we have similarly that |B n | ≤ 9 ⌊nt⌋ X j,k=1 4 X a=0 4 a |Eg a Bt j −1 g 4−a Bt k −1 〈ǫ ⊗a j −1 e ⊗ ǫ ⊗4−a k −1 , δ ⊗2 j ⊗ δ ⊗2 k 〉 H ⊗4 〈δ j , δ k 〉 H | ≤ C n −43 ⌊nt⌋ X j,k=1 |〈δ j , δ k 〉 H | by Lemma 3.2 i = C n −53 ⌊nt⌋ X j,k=1 |ρ j − k| ≤ C n −23 X r ∈Z |ρr| = C n −23 −−−→ n →∞ 0, with ρ defined by 2.4. Using similar computations, we also have |C n | ≤ C n −13 ∞ X r= −∞ ρ 2 r = C n −13 −−−→ n →∞ 0, 2140 while D n = 6 n ⌊nt⌋ X j,k=1 EgBt j −1 gBt k −1 ρ 3 j − k = 6 n X r ∈Z ⌊nt⌋∧⌊nt⌋−r X j=1 ∨1−r EgBt j −1 , gBt j+r −1 ρ 3 r −−−→ n →∞ 6 X r ∈Z ρ 3 r Z t Eg 2 Bs ds = κ 2 Z t Eg 2 Bs ds, the previous convergence being obtained as in the proof of 3.27 below. Finally, we have obtained E G − n

g, B, t

2 −−−→ n →∞ κ 2 Z t E g 2 Bs ds + 1 64 E Z t g ′′′ Bs ds 2 , 3.14 and the proof of 3.12 is done. Step 2.- By Step 1, the sequence B, G − n g 1 , B, u 1 , . . . , G − n g m , B, u m is tight in D R [0, ∞ × R m . Consider a subsequence converging in law to some limit denoted by B, G − ∞ g 1 , B, u 1 , . . . , G − ∞ g m , B, u m for convenience, we keep the same notation for this subsequence and for the sequence itself. Recall V n , defined in Lemma 3.6, and note that by 3.11, we have V n := G − n g 1 , B, u 1 , . . . , G − n g m , B, u m , n ∈ N ∪ {∞}. 3.15 Let us also define W := − 1 8 Z u 1 g ′′′ 1 Bs ds + Z u 1 g 1 Bs d[[B]] s , . . . , − 1 8 Z u m g ′′′ m Bs ds + Z u m g m Bs d[[B]] s . We have to show that, conditioned on B, the laws of V ∞ and W are the same. Let λ = λ 1 , . . . , λ m denote a generic element of R m and, for λ, µ ∈ R m , write 〈λ, µ〉 for P m i=1 λ i µ i . We consider the conditional characteristic function of W given B: Φλ := E € e i 〈λ,W〉 B Š . 3.16 Recall that [[B]] = κW , where W is a Brownian motion independent of B. Hence, if we condition on B, then W has the same law as µ + N , where µ ∈ R m has components µ k := −18 R u k g ′′′ k Bs ds, and N is a centered Gaussian random vector in R m with covariance matrix Q = q i j given by q i j := κ 2 Z u i ∧u j g i Bsg j Bs ds. 2141 Thus, Φ λ = e i 〈λ,µ〉− 1 2 〈λ,Qλ〉 . The point is that Φ is the unique solution of the following system of PDEs see [12]: ∂ ϕ ∂ λ p λ = ϕλ i µ p − m X k=1 λ k q pk , p = 1, . . . , m, 3.17 where the unknown function ϕ : R m → C satisfies the initial condition ϕ0 = 1. Hence, we have to show that, for every random variable ξ of the form ψBs 1 , . . . , Bs r , with ψ : R r → R belonging to C ∞ b R r and s 1 , . . . , s r ≥ 0, we have ∂ ∂ λ p E € e i 〈λ,V ∞ 〉 ξ Š = − i 8 Z u p E g ′′′ p Bsξe i 〈λ,V ∞ 〉 ds − κ 2 m X k=1 λ k Z u p ∧u k E € g p Bsg k Bsξe i 〈λ,V ∞ 〉 Š ds 3.18 for all p ∈ {1, . . . , m}. Step 3.- Since V ∞ , B is defined as the limit in law of V n , B on one hand, and V n is bounded in L 2 on the other hand, note that ∂ ∂ λ p E € e i 〈λ,V ∞ 〉 ξ Š = lim n →∞ ∂ ∂ λ p E € e i 〈λ,V n 〉 ξ Š . Let us compute ∂ ∂ λ p E € e i 〈λ,V n 〉 ξ Š . We have ∂ ∂ λ p E € e i 〈λ,V n 〉 ξ Š = i E G − n g p , B, u p e i 〈λ,V n 〉 ξ . 3.19 Moreover, see 3.11 and use 2.13, for any t ≥ 0: E G − n g, B, te i 〈λ,V n 〉 ξ = ⌊nt⌋ X j=1 E gBt j −1 I 3 δ ⊗3 j e i 〈λ,V n 〉 ξ = ⌊nt⌋ X j=1 E 〈D 3 € gBt j −1 e i 〈λ,V n 〉 ξ Š , δ ⊗3 j 〉 H ⊗3 . 3.20 The first three Malliavin derivatives of gBt j −1 e i 〈λ,V n 〉 ξ are respectively given by DgBt j −1 e i 〈λ,V n 〉 ξ = g ′ Bt j −1 e i 〈λ,V n 〉 ξ ǫ j −1 + i gBt j −1 e i 〈λ,V n 〉 ξD〈λ, V n 〉 + gBt j −1 e i 〈λ,V n 〉 D ξ, D 2 gBt j −1 e i 〈λ,V n 〉 ξ = g ′′ Bt j −1 e i 〈λ,V n 〉 ξ ǫ ⊗2 j −1 + 2i g ′ Bt j −1 e i 〈λ,V n 〉 ξ D〈λ, V n 〉 e ⊗ ǫ j −1 + 2g ′ Bt j −1 e i 〈λ,V n 〉 D ξ e ⊗ ǫ j −1 − gBt j −1 e i 〈λ,V n 〉 ξ D〈λ, V n 〉 ⊗2 + 2i gBt j −1 e i 〈λ,V n 〉 D ξ e ⊗ D〈λ, V n 〉 + i gBt j −1 e i 〈λ,V n 〉 ξ D 2 〈λ, V n 〉 + gBt j −1 e i 〈λ,V n 〉 D 2 ξ, 2142 and D 3 gBt j −1 e i 〈λ,V n 〉 ξ = g ′′′ Bt j −1 e i 〈λ,V n 〉 ξ ǫ ⊗3 j −1 + 3i g ′′ Bt j −1 e i 〈λ,V n 〉 ξ ǫ ⊗2 j −1 e ⊗ D〈λ, V n 〉 + 3g ′′ Bt j −1 e i 〈λ,V n 〉 ǫ ⊗2 j −1 e ⊗ Dξ − 3g ′ Bt j −1 e i 〈λ,V n 〉 ξ D〈λ, V n 〉 ⊗2 e ⊗ ǫ j −1 + 6i g ′ Bt j −1 e i 〈λ,V n 〉 D ξ e ⊗ D〈λ, V n 〉 e ⊗ ǫ j −1 − i gBt j −1 e i 〈λ,V n 〉 ξ D〈λ, V n 〉 ⊗3 − 3gBt j −1 e i 〈λ,V n 〉 D 〈λ, V n 〉 ⊗2 e ⊗ Dξ + i gBt j −1 e i 〈λ,V n 〉 ξ D 3 〈λ, V n 〉 + gBt j −1 e i 〈λ,V n 〉 D 3 ξ + 3i gBt j −1 e i 〈λ,V n 〉 D 2 ξ e ⊗ D〈λ, V n 〉 + 3i gBt j −1 e i 〈λ,V n 〉 D ξ e ⊗ D 2 〈λ, V n 〉 + 3g ′ Bt j −1 e i 〈λ,V n 〉 D 2 ξ e ⊗ ǫ j −1 + 3i g ′ Bt j −1 e i 〈λ,V n 〉 ξ ǫ j −1 e ⊗ D 2 〈λ, V n 〉 − 3gBt j −1 e i 〈λ,V n 〉 ξ D〈λ, V n 〉 e ⊗ D 2 〈λ, V n 〉. 3.21 Let us compute the term D 3 〈λ, V n 〉. Recall that 〈λ, V n 〉 = m X k=1 λ k G − n g k , B, u k = m X k=1 λ k ⌊nu k ⌋ X ℓ=1 g k Bt ℓ−1 I 3 δ ⊗3 ℓ . Combining the Leibniz rule 2.15 with D I q f ⊗q = qI q −1 f ⊗q−1 f for any f ∈ H, we have D 3 〈λ, V n 〉 = m X k=1 λ k ⌊nu k ⌋ X ℓ=1 g ′′′ k Bt ℓ−1 I 3 δ ⊗3 ℓ ǫ ⊗3 ℓ−1 + 9g ′′ k Bt ℓ−1 I 2 δ ⊗2 ℓ ǫ ⊗2 ℓ−1 e ⊗ δ ℓ + 18g ′ k Bt ℓ−1 I 1 δ ℓ ǫ ℓ−1 e ⊗ δ ⊗2 ℓ + 6g k Bt ℓ−1 δ ⊗3 ℓ . 3.22 Combining relations 3.19, 3.20, 3.21, and 3.22 we obtain the following expression: ∂ ∂ λ p E € e i 〈λ,V n 〉 ξ Š = i E e i 〈λ,V n 〉 ξ ⌊nu p ⌋ X j=1 g ′′′ p Bt j −1 〈ǫ j −1 , δ j 〉 3 H − 6 E e i 〈λ,V n 〉 ξ ⌊nu p ⌋ X j=1 m X k=1 λ k ⌊nu k ⌋ X ℓ=1 g p Bt j −1 g k Bt ℓ−1 〈δ ℓ , δ j 〉 3 H + i ⌊nu p ⌋ X j=1 r j,n , 3.23 with r j,n = i m X k=1 λ k ⌊nu k ⌋ X ℓ=1 E € g p Bt j −1 g ′′′ k Bt ℓ−1 I 3 δ ⊗3 ℓ e i 〈λ,V n 〉 ξ Š 〈ǫ ℓ−1 , δ j 〉 3 H + 9i m X k=1 λ k ⌊nu k ⌋ X ℓ=1 E € g p Bt j −1 g ′′ k Bt ℓ−1 I 2 δ ⊗2 ℓ e i 〈λ,V n 〉 ξ Š 〈ǫ ℓ−1 , δ j 〉 2 H 〈δ ℓ , δ j 〉 H + 18i m X k=1 λ k ⌊nu k ⌋ X ℓ=1 E € g p Bt j −1 g ′ k Bt ℓ−1 I 1 δ ℓ e i 〈λ,V n 〉 ξ Š 〈δ ℓ , δ j 〉 2 H 〈ǫ ℓ−1 , δ j 〉 H 2143 + 3i E g ′′ p Bt j −1 e i 〈λ,V n 〉 ξ〈D〈λ, V n 〉, δ j 〉 H 〈ǫ j −1 , δ j 〉 2 H + 3E g ′′ p Bt j −1 e i 〈λ,V n 〉 〈Dξ, δ j 〉 H 〈ǫ j −1 , δ j 〉 2 H − 3E g ′ p Bt j −1 e i 〈λ,V n 〉 ξ〈D〈λ, V n 〉, δ j 〉 2 H 〈ǫ j −1 , δ j 〉 H + 6i E g ′ p Bt j −1 e i 〈λ,V n 〉 〈D〈λ, V n 〉, δ j 〉 H 〈Dξ, δ j 〉 H 〈ǫ j −1 , δ j 〉 H + 3E g ′ p Bt j −1 e i 〈λ,V n 〉 〈D 2 ξ, δ ⊗2 j 〉 H ⊗2 〈ǫ j −1 , δ j 〉 H − iE € g p Bt j −1 e i 〈λ,V n 〉 ξ〈D〈λ, V n 〉, δ j 〉 3 H Š − 3E € g p Bt j −1 e i 〈λ,V n 〉 〈D〈λ, V n 〉, δ j 〉 2 H 〈Dξ, δ j 〉 H Š + 3i E g ′ p Bt j −1 e i 〈λ,V n 〉 ξ〈D 2 〈λ, V n 〉, δ ⊗2 j 〉 H ⊗2 〈ǫ j −1 , δ j 〉 H − 3E g p Bt j −1 e i 〈λ,V n 〉 ξ〈D〈λ, V n 〉, δ j 〉 H 〈D 2 〈λ, V n 〉, δ ⊗2 j 〉 H ⊗2 + 3i E g p Bt j −1 e i 〈λ,V n 〉 〈Dξ, δ j 〉 H 〈D 2 〈λ, V n 〉, δ ⊗2 j 〉 H ⊗2 + 3i E g p Bt j −1 e i 〈λ,V n 〉 〈D〈λ, V n 〉, δ j 〉 H 〈D 2 ξ, δ ⊗2 j 〉 H ⊗2 + E g p Bt j −1 e i 〈λ,V n 〉 〈D 3 ξ, δ ⊗3 j 〉 H ⊗3 = 15 X a=1 R a j,n . 3.24 Assume for a moment see Steps 4 to 8 below that ⌊nu p ⌋ X j=1 r j,n −−−→ n →∞ 0. 3.25 By Lemma 3.2 i v and since e i 〈λ,V n 〉 , ξ and g ′′′ p are bounded, we have E e i 〈λ,V n 〉 ξ ⌊nu p ⌋ X j=1 g ′′′ p Bt j −1 〈ǫ j −1 , δ j 〉 3 H − E e i 〈λ,V n 〉 ξ × −1 8n ⌊nu p ⌋ X j=1 g ′′′ p Bt j −1 → 0. Moreover, by Lebesgue bounded convergence, we have that E e i 〈λ,V n 〉 ξ × −1 8n ⌊nu p ⌋ X j=1 g ′′′ p Bt j −1 − E e i 〈λ,V n 〉 ξ × −1 8 Z u p g ′′′ p Bs ds → 0. Finally, note that B, V n → B, V ∞ in law in D R [0, ∞ × R m . By the Skorohod representation theorem see, for example, Theorem 2.1.8 in [4], we may construct a version of this sequence which converges a.s., so that again by Lebesgue bounded convergence, we have E e i 〈λ,V n 〉 ξ × −1 8 Z u p g ′′′ p Bs ds → E e i 〈λ,V ∞ 〉 ξ × −1 8 Z u p g ′′′ p Bs ds . 2144 Putting these convergences together, we obtain: E e i 〈λ,V n 〉 ξ ⌊nu p ⌋ X j=1 g ′′′ p Bt j −1 〈ǫ j −1 , δ j 〉 3 H → E e i 〈λ,V ∞ 〉 ξ × −1 8 Z u p g ′′′ p Bs ds . 3.26 Similarly, let us show that 6E e i 〈λ,V n 〉 ξ ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 g p Bt j −1 g k Bt ℓ−1 〈δ ℓ , δ j 〉 3 H → κ 2 E ‚ e i 〈λ,V ∞ 〉 ξ × Z u p ∧u k g p Bsg k Bs ds Œ . 3.27 We have, see 2.4 for the definition of ρ: 6 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 g p Bt j −1 g k Bt ℓ−1 〈δ ℓ , δ j 〉 3 H = 6 n ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 g p Bt j −1 g k Bt ℓ−1 ρ 3 ℓ − j = 6 n ⌊nu k ⌋−1 X r=1 −⌊nu p ⌋ ρ 3 r ⌊nu p ⌋∧⌊nu k ⌋−r X j=1 ∨1−r g p Bt j −1 g k Bt r+ j −1 . 3.28 For each fixed integer r 0 the case r ≤ 0 being similar, we have 1 n ⌊nu p ⌋∧⌊nu k ⌋−r X j=1 ∨1−r g p Bt j −1 g k Bt r+ j −1 − 1 n ⌊nu p ⌋∧⌊nu k ⌋−r X j=1 ∨1−r g p Bt j −1 g k Bt j −1 ≤ Ckg p k ∞ sup 1 ≤ j≤⌊nu p ⌋ g k Bt r+ j −1 − g k Bt j −1 ≤ Ckg p k ∞ kg ′ k k ∞ sup s ∈[0,u p ] |Bs + rn − Bs| a.s. −−−→ n →∞ 0, since the Brownian paths are uniformly continuous on compact intervals. Hence, for all fixed r ∈ Z, 1 n ⌊nu p ⌋∧⌊nu k ⌋−r X j=1 ∨1−r g p Bt j −1 g k Bt r+ j −1 a.s. −−−→ n →∞ Z u p ∧u k g p Bsg k Bsds. By combining a bounded convergence argument with 3.28 observe in particular that κ 2 = 6 P r ∈Z ρ 3 r ∞, we deduce that 6 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 g p Bt j −1 g k Bt ℓ−1 〈δ ℓ , δ j 〉 3 H a.s. −−−→ n →∞ κ 2 Z u p ∧u k g p Bsg k Bs ds. 2145 Since B, V n → B, V ∞ in D R [0, ∞ × R m , we deduce that V n , ξ, 6 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 g p Bt j −1 g k Bt ℓ−1 〈δ ℓ , δ j 〉 3 H k=1,...,m Law −−→ V ∞ , ξ, κ 2 Z u p ∧u k g p Bsg k Bs ds k=1,...,m in R d × R × R m . By boundedness of e i 〈λ,V n 〉 , ξ and g i , we have that 3.27 follows. Putting 3.25, 3.26, and 3.27 into 3.23, we deduce 3.18. Now, it remains to prove 3.25. Step 4.- Study of R 5 j,n , R 8 j,n and R 15 j,n in 3.24. Let k ∈ {1, 2, 3}. Since D k ξ = r X i 1 ,...,i k =1 ∂ k ψ ∂ s i 1 · · · ∂ s i k B s 1 , . . . , B s r 1 [0,s i1 ] ⊗ . . . ⊗ 1 [0,s ik ] , with ψ ∈ C ∞ b R r , we have P ⌊nt⌋ j=1 |〈D k ξ, δ ⊗k j 〉 H | ≤ C n −k−13 by Lemma 3.2 i and ii. Moreover, |〈ǫ j −1 , δ j 〉 H | ≤ n −13 by Lemma 3.2 i. Hence, P ⌊nt⌋ j=1 |R p j,n | = On −23 −−−→ n →∞ 0 for p ∈ {5, 8, 15}. Step 5.- Study of R 2 j,n and R 3 j,n in 3.24. We can write, using Lemma 3.2 i, Cauchy-Schwarz inequality and the definition 2.4 of ρ: ⌊nu p ⌋ X j=1 R 3 j,n ≤ 18 m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 E € g p Bt j −1 g ′ k Bt ℓ−1 I 1 δ ℓ e i 〈λ,V n 〉 ξ Š 〈δ ℓ , δ j 〉 2 H 〈ǫ ℓ−1 , δ j 〉 H ≤ C n −76 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 ρℓ − j 2 ≤ C n −16 X r ∈Z ρr 2 = C n −16 −−−→ n →∞ 0. Concerning R 2 j,n , we can write similarly: ⌊nu p ⌋ X j=1 R 2 j,n ≤ 9 m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 E € g p Bt j −1 g ′′ k Bt ℓ−1 I 2 δ ⊗2 ℓ e i 〈λ,V n 〉 ξ Š 〈δ ℓ , δ j 〉 H 〈ǫ ℓ−1 , δ j 〉 2 H ≤ C n −43 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 ρℓ − j ≤ Cn −13 X r ∈Z ρr = Cn −13 −−−→ n →∞ 0. 2146 Step 6.- Study of R 1 j,n , R 6 j,n , R 10 j,n , and R 12 j,n . First, let us deal with R 1 j,n . In order to lighten the notation, we set e ξ j, ℓ = g p Bt j −1 g ′′′ k Bt ℓ−1 ξ. Using I 3 δ ⊗3 ℓ = I 2 δ ⊗2 l I 1 δ ℓ − 2n −13 I 1 δ ℓ see e.g. Proposition 1.1.2 from [10] and then integrating by parts through 2.11, we get E € e i 〈λ,V n 〉 e ξ j, ℓ I 3 δ ⊗3 ℓ Š = E € e i 〈λ,V n 〉 e ξ j, ℓ I 2 δ ⊗2 ℓ I 1 δ ℓ Š − 2n −13 E € e i 〈λ,V n 〉 e ξ j, ℓ I 1 δ ℓ Š = E € e i 〈λ,V n 〉 I 2 δ ⊗2 ℓ 〈De ξ j, ℓ , δ ℓ 〉 H Š + i E € e i 〈λ,V n 〉 I 2 δ ⊗2 ℓ e ξ j, ℓ 〈D〈λ, V n 〉, δ ℓ 〉 H Š . Due to Lemma 3.2 i and Cauchy-Schwarz inequality, we have sup ℓ=1,...,⌊nu k ⌋ sup j=1,..., ⌊nu p ⌋ E € e i 〈λ,V n 〉 I 2 δ ⊗2 ℓ 〈De ξ j, ℓ , δ ℓ 〉 H Š ≤ C n −23 . By 3.5 and Cauchy-Schwarz inequality, we also have sup ℓ=1,...,⌊nu k ⌋ sup j=1,..., ⌊nu p ⌋ E € e i 〈λ,V n 〉 I 2 δ ⊗2 ℓ e ξ j, ℓ 〈D〈λ, V n 〉, δ ℓ 〉 H Š ≤ C n −23 . Hence, combined with Lemma 3.2 i and ii, we get: ⌊nu p ⌋ X j=1 R 1 j,n ≤ m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X ℓ=1 E € g p Bt j −1 g ′′′ k Bt ℓ−1 I 3 δ ⊗3 ℓ e i 〈λ,V n 〉 ξ Š 〈ǫ ℓ−1 , δ j 〉 H 3 ≤ C n −13 sup ℓ=1,...,⌊nu k ⌋ ⌊nu p ⌋ X j=1 〈ǫ ℓ−1 , δ j 〉 H ≤ Cn −13 −−−→ n →∞ 0. Now, let us concentrate on R 6 j,n . Since e i 〈λ,V n 〉 , ξ, and g ′ p are bounded, we have that ⌊nu p ⌋ X j=1 |R 6 j,n | ≤ C ⌊nu p ⌋ X j=1 E € 〈D〈λ, V n 〉, δ j 〉 2 H Š |〈ǫ j −1 , δ j 〉 H | ≤ C n −13 ⌊nu p ⌋ X j=1 E € 〈D〈λ, V n 〉, δ j 〉 2 H Š by Lemma 3.2 i ≤ C n −23 −−−→ n →∞ by 3.6. Similarly, ⌊nu p ⌋ X j=1 |R 12 j,n | ≤ 3 ⌊nu p ⌋ X j=1 E g p Bt j −1 e i 〈λ,V n 〉 ξ〈D〈λ, V n 〉, δ j 〉 H 〈D 2 〈λ, V n 〉, δ ⊗2 j 〉 H ⊗2 ≤ C ⌊nu p ⌋ X j=1 E € 〈D〈λ, V n 〉, δ j 〉 2 H Š + E 〈D 2 〈λ, V n 〉, δ ⊗2 j 〉 2 H ⊗2 ≤ C n −13 −−−→ n →∞ by 3.6 and 3.7. 2147 For R 10 j,n , we can write: ⌊nu p ⌋ X j=1 |R 10 j,n | ≤ 3 ⌊nu p ⌋ X j=1 E € g p Bt j −1 e i 〈λ,V n 〉 〈D〈λ, V n 〉, δ j 〉 2 H 〈Dξ, δ j 〉 H Š ≤ C n −13 ⌊nu p ⌋ X j=1 E € 〈D〈λ, V n 〉, δ j 〉 2 H Š by Lemma 3.2 i ≤ C n −23 −−−→ n →∞ by 3.6. Step 7.- Study of R 4 j,n , R 7 j,n , R 11 j,n , R 13 j,n , and R 14 j,n . Using 3.8, and then Cauchy-Schwarz inequality and Lemma 3.2 i, we can write ⌊nu p ⌋ X j=1 R 4 j,n ≤ 3 ⌊nu p ⌋ X j=1 E g ′′ p Bt j −1 e i 〈λ,V n 〉 ξ〈D〈λ, V n 〉, δ j 〉 H 〈ǫ j −1 , δ j 〉 2 H ≤ 3 m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 E g ′′ p Bt j −1 g ′ k Bt i −1 e i 〈λ,V n 〉 ξI 3 δ ⊗3 i 〈ǫ i −1 , δ j 〉 H 〈ǫ j −1 , δ j 〉 2 H + 9 m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 E g ′′ p Bt j −1 g k Bt i −1 e i 〈λ,V n 〉 ξI 2 δ ⊗2 i 〈δ i , δ j 〉 H 〈ǫ j −1 , δ j 〉 2 H ≤ C n −16 sup 1 ≤i≤⌊nu k ⌋ ⌊nu p ⌋ X j=1 〈ǫ i −1 , δ j 〉 H + Cn −43 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 ρi − j ≤ Cn −16 −−−→ n →∞ 0. Using the same arguments, we show that P ⌊nu p ⌋ j=1 R 7 j,n −−−→ n →∞ 0 and P ⌊nu p ⌋ j=1 R 14 j,n −−−→ n →∞ 0. Differentiating in 3.8, we get 〈D 2 〈λ, V n 〉, δ ⊗2 j 〉 H ⊗2 = m X k=1 λ k ⌊nu k ⌋ X i=1 g ′′ k Bt i −1 I 3 δ ⊗3 i 〈ǫ i −1 , δ j 〉 2 H + 6 m X k=1 λ k ⌊nu k ⌋ X i=1 g ′ k Bt i −1 I 2 δ ⊗2 i 〈ǫ i −1 , δ j 〉 H 〈δ i , δ j 〉 H + 6 m X k=1 λ k ⌊nu k ⌋ X i=1 g k Bt i −1 I 1 δ i 〈δ i , δ j 〉 2 H . 2148 Hence, using Cauchy-Schwarz inequality and Lemma 3.2 i-ii, we can write ⌊nu p ⌋ X j=1 R 11 j,n ≤ 3 ⌊nu p ⌋ X j=1 E g ′ p Bt j −1 e i 〈λ,V n 〉 ξ〈D 2 〈λ, V n 〉, δ ⊗2 j 〉 H ⊗2 〈ǫ j −1 , δ j 〉 H ≤ 3 m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 E g ′ p Bt j −1 g ′′ k Bt i −1 e i 〈λ,V n 〉 ξI 3 δ ⊗3 i 〈ǫ i −1 , δ j 〉 2 H 〈ǫ j −1 , δ j 〉 H + 18 m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 E g ′ p Bt j −1 g ′ k Bt i −1 e i 〈λ,V n 〉 ξI 2 δ ⊗2 i × 〈ǫ i −1 , δ j 〉 H 〈ǫ j −1 , δ j 〉 H 〈δ i , δ j 〉 H + 18 m X k=1 |λ k | ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 E g ′ p Bt j −1 g k Bt i −1 e i 〈λ,V n 〉 ξI 1 δ i 〈ǫ j −1 , δ j 〉 H 〈δ i , δ j 〉 2 H ≤ C n −16 sup 1 ≤i≤⌊nu k ⌋ ⌊nu p ⌋ X j=1 〈ǫ i −1 , δ j 〉 H + Cn −43 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 ρi − j + C n −56 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i=1 ρi − j ≤ C n −16 −−−→ n →∞ 0. Using the same arguments, we show that P ⌊nu p ⌋ j=1 R 13 j,n −−−→ n →∞ 0. Step 8.- Now, we consider the last term in 3.24, that is R 9 j,n . Since e i 〈λ,V n 〉 , ξ, and g p are bounded, we can write ⌊nu p ⌋ X j=1 R 9 j,n ≤ C ⌊nu p ⌋ X j=1 E |〈D〈λ, V n 〉, δ j 〉 H | 3 ≤ C ⌊nu p ⌋ X j=1 E 〈D〈λ, V n 〉, δ j 〉 2 H + E 〈D〈λ, V n 〉, δ j 〉 4 H . 2149 In addition we have, see 3.8, that ⌊nu p ⌋ X j=1 E 〈D〈λ, V n 〉, δ j 〉 4 H ≤ 8m 3 ⌊nu p ⌋ X j=1 m X k=1 λ 4 k ⌊nu k ⌋ X i 1 =1 ⌊nu k ⌋ X i 2 =1 ⌊nu k ⌋ X i 3 =1 ⌊nu k ⌋ X i 4 =1 E 4 Y a=1 〈ǫ t ia , δ j 〉 H g ′ k Bt i a −1 I 3 δ ⊗3 i a + 648m 3 ⌊nu p ⌋ X j=1 m X k=1 λ 4 k ⌊nu k ⌋ X i 1 =1 ⌊nu k ⌋ X i 2 =1 ⌊nu k ⌋ X i 3 =1 ⌊nu k ⌋ X i 4 =1 E 4 Y a=1 〈δ i a , δ j 〉 H g k Bt i a −1 I 2 δ ⊗2 i a ≤ C m X k=1 λ 4 k n −43 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i 1 =1 ⌊nu k ⌋ X i 2 =1 ⌊nu k ⌋ X i 3 =1 ⌊nu k ⌋ X i 4 =1 E 4 Y a=1 g ′ k Bt i a −1 I 3 δ ⊗3 i a + ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i 1 =1 ⌊nu k ⌋ X i 2 =1 ⌊nu k ⌋ X i 3 =1 ⌊nu k ⌋ X i 4 =1 4 Y a=1 |〈δ i a , δ j 〉 H | E 4 Y a=1 g k Bt i a −1 I 2 δ ⊗2 i a . By Lemma 3.5 we have that ⌊nu k ⌋ X i 1 =1 ⌊nu k ⌋ X i 2 =1 ⌊nu k ⌋ X i 3 =1 ⌊nu k ⌋ X i 4 =1 E 4 Y a=1 g ′ k Bt i a −1 I 3 δ ⊗3 i a ≤ C, so that n −43 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i 1 =1 ⌊nu k ⌋ X i 2 =1 ⌊nu k ⌋ X i 3 =1 ⌊nu k ⌋ X i 4 =1 E 4 Y a=1 g ′ k Bt i 1 −1 I 3 δ ⊗3 i a ≤ C n −13 . On the other hand, by Cauchy-Schwarz inequality, we have E 4 Y a=1 g k Bt i a −1 I 2 δ ⊗2 i a ≤ C, so that, with ρ defined by 2.4, ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i 1 =1 ⌊nu k ⌋ X i 2 =1 ⌊nu k ⌋ X i 3 =1 ⌊nu k ⌋ X i 4 =1 4 Y a=1 |〈δ i a , δ j 〉 H | E 4 Y a=1 g k Bt i a −1 I 2 δ ⊗2 i a ≤ C n −43 ⌊nu p ⌋ X j=1 ⌊nu k ⌋ X i 1 =1 |ρi 1 − j| × ⌊nu k ⌋ X i 2 =1 |ρi 2 − j| × ⌊nu k ⌋ X i 3 =1 |ρi 3 − j| × ⌊nu k ⌋ X i 4 =1 |ρi 4 − j| ≤ C n −13 X r ∈Z |ρr| 4 = C n −13 . As a consequence, combining the previous estimates with 3.6, we have shown that ⌊nu p ⌋ X j=1 R 9 j,n ≤ C n −13 −−−→ n →∞ 0, and the proof of Theorem 3.7 is done. ƒ 2150 Theorem 3.8. If g ∈ C ∞ R is bounded with bounded derivatives, then the sequence B, G + n 1, B, G + n g, B converges to B, [[B]], 18 R g ′′′ B ds + R gBd[[B]] in the sense of finite- dimensional distributions on [0, ∞, where G + n g, B, t := 1 p n ⌊nt⌋ X j=1 gBt j h 3 n 1 6 ∆B j , t ≥ 0, n ≥ 1. Proof. The proof is exactly the same as the proof of Theorem 3.7, except ǫ j −1 must be everywhere replaced ǫ j , and Lemma 3.2 v must used instead of Lemma 3.2 i v. ƒ Proof of Theorem 3.1. We begin by observing the following general fact. Suppose U and V are càdlàg processes adapted to a filtration under which V is a semimartingale. Similarly, suppose e U and e V are càdlàg processes adapted to a filtration under which e V is a semimartingale. If the processes U, V and e U, e V have the same law, then U0V 0 + R · Us − dV s and e U0e V 0 + R · e Us − d e V s have the same law. This is easily seen by observing that these integrals are the limit in probability of left-endpoint Riemann sums. Now, let G − n and G + n be as defined previously in this section. Define G − g, B, t = − 1 8 Z t g ′′′ Bs ds + Z t gBs d[[B]] s , G + g, B, t = 1 8 Z t g ′′′ Bs ds + Z t gBs d[[B]] s . Let t = t 1 , . . . , t d , where 0 ≤ t 1 · · · t d . Let G − n

g, B, t = G