getdoc711c. 370KB Jun 04 2011 12:04:30 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b il i t y Vol. 15 (2010), Paper no. 70, pages 2117–.

Journal URL

http://www.math.washington.edu/~ejpecp/

The weak Stratonovich integral with respect to fractional

Brownian motion with Hurst parameter

1

/

6

Ivan Nourdin

∗

Université Nancy 1 - Institut de Mathématiques Élie Cartan B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France

nourdin@iecn.u-nancy.fr

Anthony Réveillac

† Institut für Mathematik Humboldt-Universität zu Berlin Unter des Linden 6, 10099 Berlin, Germany

areveill@mathematik.hu-berlin.de

Jason Swanson

‡ University of Central Florida Department of Mathematics 4000 Central Florida Blvd, P.O. Box 161364

Orlando, FL 32816-1364 swanson@mail.ucf.edu

Abstract

Let B be a fractional Brownian motion with Hurst parameter H =1/6. It is known that the symmetric Stratonovich-style Riemann sums forR g(B(s))d B(s)do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent ofB.

Key words: Stochastic integration; Stratonovich integral; fractional Brownian motion; weak convergence; Malliavin calculus.

AMS 2000 Subject Classification:Primary 60H05; Secondary: 60G15, 60G18, 60J05. Submitted to EJP on July 6, 2010, final version accepted December 6, 2010.

∗_{Supported in part by the (french) ANR grant ‘Exploration des Chemins Rugueux’.}

†_{Supported by DFG research center Matheon project E2.} ‡_{Supported in part by NSA grant H98230-09-1-0079.}

1

Introduction

The Stratonovich integral of X with respect to Y, denoted Rt

0X(s)◦d Y(s), can be defined as the limit in probability, if it exists, of

X tj≤t

X(t_j−1) +X(t_j)

2 (Y(tj)−Y(tj−1)), (1.1) as the mesh of the partition {tj} goes to zero. Typically, we regard (1.1) as a process in t, and

require that it converges uniformly on compacts in probability (ucp).

This is closely related to the so-called symmetric integral, denoted byR₀tX(s)d◦Y(s), which is the ucp limit, if it exists, of

1 ǫ

Z t

0

X(s) +X(s+ǫ)

2 (Y(s+ǫ)−Y(s))ds, (1.2) as ǫ → 0. The symmetric integral is an example of the regularization procedure, introduced by Russo and Vallois, and on which there is a wide body of literature. For further details on stochastic calculus via regularization, see the excellent survey article[13]and the many references therein. A special case of interest that has received considerable attention in the literature is whenY =BH, a fractional Brownian motion with Hurst parameterH. It has been shown independently in[2]and [5]that when Y =BH andX =g(BH)for a sufficiently differentiable function g(x), the symmetric integral exists for allH >1/6. Moreover, in this case, the symmetric integral satisfies the classical Stratonovich change of variable formula,

g(BH(t)) =g(BH(0)) +

Z t

0

g′(BH(s))d◦BH(s).

However, whenH =1/6, the symmetric integral does not, in general, exist. Specifically, in[2]and [5], it is shown that (1.2) does not converge in probability whenY =B1/6 andX = (B1/6)2. It can be similarly shown that, in this case, (1.1) also fails to converge in probability.

This brings us naturally to the notion which is the focus of this paper: the weak Stratonovich integral, which is the limit in law, if it exists, of (1.1). We focus exclusively on the caseY =B1/6. For simplicity, we omit the superscript and writeB=B1/6. Our integrands shall take the form g(B(t)), forg∈C∞(R), and we shall work only with the uniformly spaced partition, tj = j/n. In this case,

(1.1) becomes

I_n(g,B,t) =

⌊nt⌋

X j=1

g(B(t_j−1)) +g(B(t_j))

2 ∆Bj,

where_⌊x_⌋denotes the greatest integer less than or equal tox, and∆B_j=B(t_j)₋B(t_j−1). We show that the processes In(g,B) converge in law in DR[0,∞), the Skorohod space of càdlàg functions from[0,_∞)toR. We letR₀tg(B(s))d B(s) denote a process with this limiting law, and refer to this as the weak Stratonovich integral.

The weak Stratonovich integral with respect toBdoes not satisfy the classical Stratonovich change of variable formula. Rather, we show that it satisfies a change of variable formula with a correction term that is a classical Itô integral. Namely,

g(B(t)) =g(B(0)) +

Z t

0

g′(B(s))d B(s)₋ 1

12

Z t

0

where[[B]] is what we call the signed cubic variation ofB. That is,[[B]] is the limit in law of the sequence of processes V_n(B,t) = P⌊_jnt₌₁⌋∆B3_j. It is shown in [11] that [[B]] = κW, where W is a standard Brownian motion, independent ofB, andκ≃2.322. (See (2.5) for the exact definition of κ.) The correction term in (1.3) is then a standard Itô integral with respect to Brownian motion. Our precise results are actually somewhat stronger than this, in that we prove the joint convergence of the processesB,V_n(B), andI_n(g,B). (See Theorem 2.13.) We also discuss the joint convergence of multiple sequences of Riemann sums for different integrands. (See Theorem 2.14 and Remark 2.15.)

The work in this paper is a natural follow-up to[1]and[9]. There, analogous results were proven for B1/4 in the context of midpoint-style Riemann sums. The results in[1]and [9]were proven through different methods, and in the present work, we combine the two approaches to prove our main results.

Finally, let us stress the fact that, as a byproduct of the proof of (1.3), we show in the present paper that

n−1/2

⌊n·⌋

X j=1

g(B(tj−1))h3(n1/6∆Bj)→ −

1 8

Z · 0

g′′′(B(s))ds+

Z · 0

g(B(s))d[[B]]_s,

in the sense of finite-dimensional distributions on[0,∞), whereh3(x) = x3−3x denotes the third Hermite polynomial. (See more precisely Theorem 3.7 below. Also see Theorem 3.8.) From our point of view, this result has also its own interest, and should be compared with the recent results obtained in[7, 8], concerning the weighted Hermite variations of fractional Brownian motion.

2

Notation, preliminaries, and main result

LetB=B1/6be a fractional Brownian motion with Hurst parameterH =1/6. That is,Bis a centered Gaussian process, indexed byt _≥0, such that

R(s,t) =E[B(s)B(t)] =1

2(t

1/3_+s1/3

− |t₋s_|1/3).

Note that E|B(t)₋B(s)_|2 _{= |t−s|}1/3_{. For compactness of notation, we will sometimes write B}

t

instead ofB(t). Given a positive integern, let∆t =n−1 and t_j = t_j,n= j∆t. We shall frequently have occasion to deal with the quantity βj,n = βj = (B(tj−1) +B(tj))/2. In estimating this and

similar quantities, we shall adopt the notationr+=r∨1, which is typically applied to nonnegative integers r. We shall also make use of the Hermite polynomials,

h_n(x) = (₋1)nex2/2 d n d xn(e−

x2_/2

). (2.1)

Note that the first few Hermite polynomials areh0(x) =1,h1(x) = x,h2(x) =x2−1, andh3(x) = x3₋3x. The following orthogonality property is well-known: if U and V are jointly normal with

E(U) =E(V) =0 andE(U2) =E(V2) =1, then

E[hp(U)hq(V)] = (

q!(E[U V])q ifp=q,

If X is a càdlàg process, we writeX(t₋) =lim_s↑tX(s) and∆X(t) =X(t)₋X(t₋). The step func-tion approximafunc-tion to X will be denoted by Xn(t) = X(⌊nt⌋/n), where ⌊·⌋ is the greatest integer

function. In this case,∆X_n(t_j,n) =X(tj)−X(tj−1). We shall frequently use the shorthand notation

∆X_j = ∆X_j,n = ∆X_n(t_j,n). For simplicity, positive integer powers of∆X_j shall be written without parentheses, so that∆Xk_j = (∆X_j)k.

The discretep-th variation ofX is defined as

V_np(X,t) =

⌊nt⌋

X j=1

|∆X_j|p, and the discrete signedp-th variation ofX is

V_np±(X,t) =

⌊nt⌋

X j=1

|∆Xj|psgn(∆Xj).

For the discrete signed cubic variation, we shall omit the superscript, so that

V_n(X,t) =V_n3±(X,t) =

⌊nt⌋

X j=1

∆X3_j. (2.3)

When we omit the indext, we mean to refer to the entire process. So, for example,V_n(X) =V_n(X,_·)

refers to the càdlàg process which maps t _7→ V_n(X,t). We recall the following fact which will be extensevely used in this paper.

Remark 2.1. Let_{ρ(r)_}r∈Zbe the sequence defined by ρ(r) = 1

2(|r+1| 1/3₊

|r−1|1/3−2|r|1/3). (2.4)

It holds thatP_r∈Z|ρ(r)|<∞andE[∆Bi∆Bj] =n−1/3ρ(i−j)for alli,j∈N.

Letκ >0 be defined by κ2=6X

r∈Z

ρ3(r) =3

4

X r∈Z

(_|r+1_|1/3+_|r−1_|1/3₋2_|r|1/3)3_≃5.391, (2.5)

and letW be a standard Brownian motion, defined on the same probability space asB, and indepen-dent ofB. Define[[B]]_t =κW(t). We shall refer to the process[[B]]as the signed cubic variation ofB. The use of this term is justified by Theorem 2.12.

A function g : Rd _→R haspolynomial growth if there exist positive constants K and r such that |g(x)_{| ≤} K(1+_|x_|r) for all x _∈ Rd. If k is a nonnegative integer, we shall say that a function g

haspolynomial growth of order kif g∈Ck(Rd)and there exist positive constantsK andr such that |∂αg(x)_{| ≤}K(1+_|x_|r)for allx _∈Rdand all_|α| ≤k. (Here,α∈Nd₀ = (N_∪{0_})d is a multi-index, and we adopt the standard multi-index notation:∂j=∂ /∂xj,∂α=∂

α1

1 · · ·∂

αd

d , and|α|=α1+· · ·+αd.)

Given g :R _→R _{and a stochastic process{X}(t) : t ≥0}, the Stratonovich Riemann sum will be denoted by

I_n(g,X,t) =

⌊nt⌋

X j=1

g(X(tj−1)) +g(X(tj))

The phrase “uniformly on compacts in probability" will be abbreviated “ucp." IfX_nandY_nare càdlàg processes, we shall writeXn ≈Yn or Xn(t)≈ Yn(t)to mean thatXn−Yn→0 ucp. In the proofs in

this paper,C shall denote a positive, finite constant that may change value from line to line.

2.1

Conditions for relative compactness

The Skorohod space of càdlàg functions from [0,_∞) to Rd is denoted by DRd[0,∞). Note that

DRd[0,∞)and(DR[0,∞))d are not the same. In particular, the map(x,y)7→x +y is continuous fromDR2[0,∞)toDR[0,∞), but it is not continuous from(DR[0,∞))2 to DR[0,∞). Convergence inDRd[0,∞)implies convergence in(DR[0,∞))d, but the converse is not true.

Note that if the sequences _{X(_n1)_}, . . . ,_{X_n(d)_} are all relatively compact in DR[0,∞), then the se-quence ofd-tuples_{(X_n(1), . . . ,X_n(d))_}is relatively compact in(DR[0,∞))d. It may not, however, be relatively compact inDRd[0,∞). We will therefore need the following well-known result. (For more

details, see Section 2.1 of[1]and the references therein.)

Lemma 2.2. Suppose{(X_n(1), . . . ,X_n(d))_}∞_n=1is relatively compact in(DR[0,∞))d. If, for each j≥2, the sequence{X(nj)}∞n=1 converges in law in DR[0,∞)to a continuous process, then{(X_n(1), . . . ,X_n(d))}∞_n=1 is relatively compact in DRd[0,∞).

Proof.First note that ifxn→xinDR[0,∞), yn→ yinDR[0,∞), andxand yhave no simultaneous discontinuities, thenx_n+y_n→x+yinDR[0,∞). Thus, if two sequences{X_n}and{Y_n}are relatively compact inDR[0,∞)and every subsequential limit of{Yn}is continuous, then{Xn+Yn}is relatively

compact in DR[0,∞). The lemma now follows from the fact that a sequence {X_n(1), . . . ,X_n(d)} is relatively compact in DRd[0,∞) if and only if {X_n(k)} and {X(_nk)+X_n(ℓ)} are relatively compact in

DR[0,∞)for allkandℓ. (See, for example, Problem 3.22(c) in[4].) ƒ Our primary criterion for relative compactness is the following moment condition, which is a special case of Corollary 2.2 in[1].

Theorem 2.3. Let{X_n}be a sequence of processes in DRd[0,∞). Let q(x) =|x| ∧1. Suppose that for each T>0, there existsν >0,β >0, C>0, andθ >1such thatsup_nE[_|X_n(T)_|ν]<∞and

E[q(Xn(t)−Xn(s))β]≤C

⌊nt_{⌋ − ⌊}ns_⌋ n

θ

, (2.6)

for all n and all0_≤s_≤t_≤T . Then_{X_n}is relatively compact.

Of course, a sequence{Xn}converges in law inDRd[0,∞)to a processX if{Xn}is relatively compact

and X_{n →} X in the sense of finite-dimensional distributions on [0,_∞). We shall also need the analogous theorem for convergence in probability, which is Lemma A2.1 in [3]. Note that if x :

[0,∞)_→Rd _{is continuous, then xn→x inDR}d[0,∞)if and only ifx_n→x uniformly on compacts.

Lemma 2.4. Let{Xn},X be processes with sample paths in DRd[0,∞)defined on the same probability

space. Suppose that {Xn} is relatively compact in DRd[0,∞) and that for a dense set H ⊂ [0,∞), X_n(t)_→X(t)in probability for all t _∈H. Then X_n→X in probability in DRd[0,∞). In particular, if

We will also need the following lemma, which is easily proved using the Prohorov metric.

Lemma 2.5. Let (E,r) be a complete and separable metric space. Let X_n be a sequence of

E-valued random variables and suppose, for each k, there exists a sequence _{X_n,k}∞_n=1 such that

lim sup_n→∞E[r(Xn,Xn,k)]≤ δk, where δk →0 as k → ∞. Suppose also that for each k, there ex-ists Y_ksuch that X_n,k→Y_kin law as n_{→ ∞}. Then there exists X such that X_n→X in law and Y_k→X in law.

2.2

Elements of Malliavin calculus

In the sequel, we will need some elements of Malliavin calculus that we collect here. The reader is referred to[6]or[10]for any unexplained notion discussed in this section.

We denote by X = _{X(ϕ) : ϕ ∈H_} an isonormal Gaussian process over H, a real and separable Hilbert space. By definition,X is a centered Gaussian family indexed by the elements ofHand such that, for everyϕ,ψ∈H,

E[X(ϕ)X(ψ)] =_〈ϕ,ψ〉H.

We denote byH⊗q andH⊙q, respectively, the tensor space and the symmetric tensor space of order

q_≥1. Let_S be the set of cylindrical functionalsF of the form

F= f(X(ϕ1), . . . ,X(ϕn)), (2.7)

wheren≥1,ϕ_i∈Hand the functionf ∈C∞(Rn)is such that its partial derivatives have polynomial growth. The Malliavin derivativeDF of a functional F of the form (2.7) is the square integrableH -valued random variable defined as

DF=

n X i=1

∂f

∂xi

(X(ϕ1), . . . ,X(ϕn))ϕi.

In particular, DX(ϕ) = ϕ for every ϕ ∈H. By iteration, one can define the mth derivative DmF

(which is an element of L2(Ω,H⊙m)) for everym≥2, giving

DmF=

n X i1,...,im

∂m_f

∂x_i1· · ·∂x_im(X(ϕ1), . . . ,X(ϕn))ϕi1⊗ · · · ⊗ϕim.

As usual, form_≥1,Dm,2denotes the closure of_S with respect to the norm_{k · km,2}, defined by the relation

kFk2m,2=E F 2₊

m X

i=1

EkDiFk2_H⊗i.

The Malliavin derivative Dsatisfies the following chain rule: if f :Rn _→R is in C_b1 (that is, the collection of continuously differentiable functions with a bounded derivative) and if{Fi}i=1,...,n is a

vector of elements ofD1,2, then f(F₁, . . . ,F_n)_∈D1,2and

D f(F1, . . . ,Fn) = n X

i=1 ∂f

This formula can be extended to higher order derivatives as

Dmf(F₁, . . . ,F_n) = X

v∈Pm

C_v n X i1,...,ik=1

∂kf

∂x_i

1· · ·∂xik

(F₁, . . . ,F_n)Dv1_F

i1⊗ · · ·e ⊗eD

vk_F

ik, (2.9)

where _Pm is the set of vectors v = (v₁, . . . ,v_k) _∈ Nk such that k _≥ 1, v_{1 ≤ · · · ≤} v_k, and v₁+

· · ·+vk = m. The constants Cv can be written explicitly as Cv = m!( Qm

j=1mj!(j!)mj)−1, where m_j=_|{ℓ:vℓ= j}|.

Remark 2.6. In (2.9),a_⊗ebdenotes the symmetrization of the tensor producta_⊗b. Recall that, in

general, thesymmetrizationof a function f ofmvariables is the function ef defined by

e

f(t₁, . . . ,t_m) = 1

m!

X

σ∈Sm

f(t_σ(1), . . . ,t_σ(m)), (2.10)

whereS_mdenotes the set of all permutations of_{1, . . . ,m_}.

We denote byIthe adjoint of the operatorD, also called the divergence operator. A random element

u_∈L2(Ω,H)belongs to the domain ofI, noted Dom(I), if and only if it satisfies |E_〈DF,u_〉H| ≤cu

p

E F2 _{for anyF∈ S,}

wherecuis a constant depending only onu. Ifu∈Dom(I), then the random variableI(u)is defined

by the duality relationship (customarily called “integration by parts formula"):

E[F I(u)] =E〈DF,u〉H, (2.11)

which holds for everyF_∈D1,2.

For everyn≥1, letHnbe thenth Wiener chaos ofX, that is, the closed linear subspace of L2

gen-erated by the random variables_{h_n(X(ϕ)):ϕ∈H,_|ϕ|H=1}, wherehn is the Hermite polynomial

defined by (2.1). The mapping

In(ϕ⊗n) =hn(X(ϕ)) (2.12)

provides a linear isometry between the symmetric tensor productH⊙n (equipped with the modified norm p1

n!k · kH⊗n) andHn. We set In(f) := In(fe) when f ∈H

⊗n_{. The following duality formula}

holds:

E[F In(f)] =E〈DnF,f〉H⊗n, (2.13)

for any element f ∈ H⊙n and any random variable F ∈ Dn,2_{. We will also need the following}

particular case of the classical product formula between multiple integrals: ifϕ,ψ∈Handm,n_≥1, then

Im(ϕ⊗m)In(ψ⊗n) = m∧n X r=0

r!

_m r

_n

r

Im+n−2r(ϕ⊗(m−r)⊗eψ⊗(n−r))〈ϕ,ψ〉rH. (2.14)

Finally, we mention that the Gaussian space generated byB=B1/6can be identified with an isonor-mal Gaussian process of the typeB=_{B(h):h_∈H_}, where the real and separable Hilbert spaceH

is defined as follows: (i) denote by_E the set of allR-valued step functions on[0,_∞), (ii) defineH

as the Hilbert space obtained by closingE with respect to the scalar product 〈1_[0,t],1_[0,s]〉H=E[B(s)B(t)] =

1 2(t

1/3_+s1/3

In particular, note thatB(t) = B(1_[0,t]). To end up, let us stress that themth derivative Dm (with respect toB) verifies the Leibniz rule. That is, for anyF,G∈Dm,2such thatF G∈Dm,2, we have

Dm_t

1,...,tm(F G) =

X

D|_JJ|(F)Dm_Jc−|J|(G), t_i∈[0,T], i=1, . . . ,m, (2.15)

where the sum runs over all subsets J of{t1, . . . ,tm}, with |J| denoting the cardinality ofJ. Note

that we may also write this as

Dm(F G) =

m X k=0

_m k

(DkF)_⊗e(Dm−kG). (2.16)

2.3

Expansions and Gaussian estimates

A key tool of ours will be the following version of Taylor’s theorem with remainder.

Theorem 2.7. Let k be a nonnegative integer. If g ∈Ck(Rd), then g(b) = X

|α|≤k

∂αg(a)(b−a)

α

α! +Rk(a,b),

where

Rk(a,b) =k X

|α|=k

(b₋a)α

α!

Z 1

0

(1−u)k[∂αg(a+u(b−a))₋∂αg(a)]du

if k≥1, and R0(a,b) = g(b)−g(a). In particular, Rk(a,b) = P

|α|=khα(a,b)(b−a)α, where hα is a

continuous function with hα(a,a) =0for all a. Moreover,

|R_k(a,b)_{| ≤}(k_∨1) X

|α|=k

M_α|(b₋a)α_|,

where Mα=sup{|∂αg(a+u(b−a))−∂αg(a)|: 0≤u≤1}.

The following related expansion theorem is a slight modification of Corollary 4.2 in[1].

Theorem 2.8. Recall the Hermite polynomials hn(x) from (2.1). Let k be a nonnegative integer.

Suppose ϕ : R _→ R is measurable and has polynomial growth with constants K and r. Supposee f _∈Ck+1(Rd)has polynomial growth of order k+1, with constants K and r. Letξ∈Rd and Y _∈Rbe jointly normal with mean zero. Suppose that EY2=1and Eξ2_j ≤ν for someν >0. Defineη∈Rd _by

ηj=E[ξjY]. Then

E[f(ξ)ϕ(Y)] = X

|α|≤k

1 α!η

α_E[

∂αf(ξ)]E[h_|α|(Y)ϕ(Y)] +R,

where_|R_{| ≤}C K_|η|k+1 and C depends only onK, r,e ν, k, and d.

Proof. Although this theorem is very similar to Corollary 4.2 in[1], we provide here another proof

Observe first that, without loss of generality, we can assume that ξi = X(vi), i = 1, . . . ,d, and Y = X(vd+1), where X is an isonormal process over H = Rd+1 and where v1, . . . ,vd+1 are some adequate vectors belonging in H. Since ϕ has polynomial growth, we can expand it in terms of Hermite polynomials, that isϕ=P∞_q=0c_qh_q. Thanks to (2.2), note that q!c_q= E[ϕ(Y)h_q(Y)]. We set

b

ϕk= k X q=0

c_qh_q and ϕˇk=

∞

X q=k+1

c_qh_q. Of course, we have

E[f(ξ)ϕ(Y)] =E[f(ξ)ϕ_bk(Y)] +E[f(ξ)ϕˇ_k(Y)]. We obtain

E[f(ξ)ϕ_bk(Y)] = k X q=0

1

q!E[ϕ(Y)hq(Y)]E[f(ξ)hq(Y)]

=

k X q=0

1

q!E[ϕ(Y)hq(Y)]E[f(ξ)Iq(v ⊗q

d+1)] by (2.12)

=

k X q=0

1

q!E[ϕ(Y)hq(Y)]E[〈D

q_f(ξ),v⊗q

d+1〉H⊗q] by (2.13)

=

k X q=0

1

q!

d X i1,...,iq=1

E[ϕ(Y)hq(Y)]E 

 ∂qf

∂x_{i1· · ·}∂x_iq(ξ)  

q Y

ℓ=1

ηi_ℓ by (2.9).

Since the map Φ:{1, . . . ,d}q _{→ {α ∈N}d

0 :|α|= q} defined by(Φ(i1, . . . ,iq))j =|{ℓ: iℓ = j}| is a surjection with_|Φ−1(α)_|=q!/α!, this gives

E[f(ξ)ϕ_bk(Y)] =

k X q=0

1

q!

X

|α|=q q!

α!E[ϕ(Y)hq(Y)]E[∂

α_f(ξ)]ηα

= X

|α|≤k

1

α!E[ϕ(Y)h|α|(Y)]E[∂

α_f(ξ)]ηα_.

On the other hand, the identity (2.2), combined with the fact that each monomial xn can be ex-panded in terms of the first nHermite polynomials, implies that E[Y|α|ϕˇk(Y)] =0 for all |α| ≤k.

Now, let U = ξ−ηY and define g :Rd _→R by g(x) = E[f(U+x Y)ϕˇk(Y)]. Sinceϕ (and,

con-sequently, also ˇϕk) and f have polynomial growth, and all derivatives of f up to order k+1 have

polynomial growth, we may differentiate under the expectation and conclude that g ∈Ck+1(Rd). Hence, by Taylor’s theorem (more specifically, by the version of Taylor’s theorem which appears as Theorem 2.13 in[1]), and the fact thatU andY are independent,

E[f(ξ)ϕˇk(Y)] =g(η) = X

|α|≤k

1 α!η

α

∂αg(0) +R

= X

|α|≤k

1 α!η

α_E[∂α_f(U)]E[Y|α|_ϕˇ

where

|R| ≤ M d (k+1)/2

k! |η|

k+1_, andM=sup_{|∂αg(uη)_|: 0_≤u_≤1,_|α|=k+1_}. Note that

∂αg(uη) =E[∂αf(U+uηY)Y|α|ϕˇk(Y)] =E[∂αf(ξ−η(1−u)Y)Y|α|ϕˇk(Y)].

Hence,

|∂αg(uη)_{| ≤}KK Ee [(1+_|ξ−η(1−u)Y|r)_|Y||α|(1+_|Y|r)]

≤KK Ee [(1+2r_|ξ|r+2r_|η|r|Y_|r)(_|Y_||α|+_|Y_||α|+r).

Since_|η|2≤νd, this completes the proof. ƒ

The following special case will be used multiple times.

Corollary 2.9. Let X₁, . . . ,X_n be jointly normal, each with mean zero and variance bounded byν >0. Letηi j=E[XiXj]. If f ∈C1(Rn−1)has polynomial growth of order 1 with constants K and r, then

|E[f(X₁, . . . ,X_n−1)X_n3]_{| ≤}C Kσ3max

j<n |ηjn|, (2.17) whereσ= (EX2_n)1/2and C depends only on r,ν, and n.

Proof. Apply Theorem 2.8 withk=0. ƒ

Finally, the following covariance estimates will be critical.

Lemma 2.10. Recall the notationβj= (B(tj−1) +B(tj))/2and r+=r∨1. For any i,j,

(i) _|E[∆B_i∆B_j]_{| ≤}C∆t1/3_|j₋i_|−₊5/3, (ii) |E[B(ti)∆Bj]| ≤C∆t1/3(j−2/3+|j−i|−2

/3 + ),

(iii) _|E[βi∆Bj]| ≤C∆t1/3(j−2/3+|j−i|−

2/3 + ),

(iv) |E[βj∆Bj]| ≤C∆t1/3j−2/3, and (v) C1|t_j−t_i|1/3_≤E|β

j−βi|2≤C2|tj−ti|1/3,

where C₁,C₂ are positive, finite constants that do not depend on i or j.

Proof. (i) By symmetry, we may assumei≤ j. First, assume j−i≥2. Then

E[∆B_i∆B_j] =

Z ti

ti−1

Z tj

tj−1

∂_st2R(s,t)d t ds,

where∂_st2=∂1∂2. Note that fors<t,∂st2R(s,t) =−(1/9)(t−s)−5/3. Hence,

Now assume j−i≤1. By Hölder’s inequality,|E[∆B_i∆B_j]_{| ≤}∆t1/3= ∆t1/3|j−i|−+5/3. (ii) First note that by (i),

|E[B(t_i)∆B_j]_{| ≤}

i X k=1

|E[∆B_k∆B_j]_{| ≤}C∆t1/3 j X k=1

|k₋j_|+−5/3_≤C∆t1/3.

This proves the lemma when either j=1 or_|j−i|+=1. To complete the proof of (ii), suppose j>1 and|j−i|>1. Note that ift >0 ands6=t, then

∂2R(s,t) = 1 6t

−2/3 −1

6|t−s|

−2/3_sgn(t −s). We may therefore writeE[B(t_i)∆B_j] =R_ttj

j−1∂2R(ti,u)du, giving

|E[B(t_i)∆B_j]_{| ≤}∆t sup

u∈[tj−1,tj]

|∂2R(ti,u)| ≤C∆t1/3(j−2/3+|j−i|−

2/3 + ), which is (ii).

(iii) This follows immediately from (ii).

(iv) Note that 2βj∆Bj = B(tj)2−B(tj−1)2. Since EB(t)2 = t1/3, the mean value theorem gives |E[βj∆Bj]| ≤C(∆t)t−

2/3

j =C∆t

1/3_j−2/3_.

(v) Without loss of generality, we may assumei< j. The upper bound follows from 2(βj−βi) = (B(tj)−B(ti)) + (B(tj−1)−B(ti−1)),

and the fact that E_|B(t)₋B(s)_|2 =_|t₋s_|1/3. For the lower bound, we first assume i< j₋1 and write

2(βj−βi) =2(B(tj−1)−B(ti)) + ∆Bj+ ∆Bi.

For any random variablesa,b,c witha=b+crecall that₋E[ac]_≤(E[_|a_|2]E[_|c_|2])1/2leading to

E[_|b|2] =E[_|a−c|2]_≤((E[_|a|2])1/2+ (E[_|c|2])1/2)2. Taking the square root in both side of this inequality we get

(E[_|b|2])1/2_≤(E[_|a|2])1/2+ (E[_|c|2])1/2. Letting,a= (β_j−β_i), b= (B(t_j−1)−B(ti)), andc= (∆Bj+ ∆Bi)/2 yields

(E[_|βj−βi|2])1/2≥ |tj−1−ti|1/6−

1

2(E[|∆Bj+ ∆Bi| 2_])1/2_.

Since∆B_i and∆B_jare negatively correlated,

E[_|∆B_j+ ∆B_i|2]_≤E[_|∆B_j|2] +E[_|∆B_i|2] =2∆t1/3. Thus,

(E[_|βj−βi|2])1/2≥∆t1/6|j−1−i|1/6−2−1/2∆t1/6≥C∆t1/6|j−i|1/6,

for someC >0. This completes the proof wheni< j₋1.

2.4

Sextic and signed cubic variations

Theorem 2.11. For each T >0, we have E[sup_0≤t≤T|V_n6(B,t)₋15t_|2]_→0as n_{→ ∞}. In particular, V_n6(B,t)_→15t ucp.

Proof. SinceV_n6(B) is monotone, it will suffice to show thatV_n6(B,t)_→15t in L2 for each fixed t. Indeed, the uniform convergence will then be a direct consequence of Dini’s theorem. We write

V_n6(B,t)₋15t=

⌊nt⌋

X j=1

(∆B6_{j −}15∆t) +15(_⌊nt_⌋/n₋t).

Since |⌊nt⌋/n− t| ≤ ∆t, it will suffice to show that E|P⌊jnt=1⌋(∆B6j −15∆t)|2 → 0. For this, we

compute

E

⌊nt⌋

X j=1

(∆B6_{j −}15∆t)

2

=

⌊nt⌋

X i=1

⌊nt⌋

X j=1

E[(∆B6_{i −}15∆t)(∆B6_{j −}15∆t)]

=

⌊nt⌋

X i=1

⌊nt⌋

X j=1

(E[∆B6_i∆B6_j]₋225∆t2).

(2.18)

By Theorem 2.8, if ξ,Y are jointly Gaussian, standard normals, then E[ξ6Y6] = 225+R, where |R_{| ≤} C_|E[ξY]_|2. Applying this with ξ = ∆t−1/6∆B_i and Y = ∆t−1/6∆B_j, and using Lemma 2.10(i), gives|E[∆B_i6∆B6_j]₋225∆t2]_{| ≤}C∆t2|j−i|−+10/3. Substituting this into (2.18), we have

E

⌊nt⌋

X j=1

(∆B6_{j −}15∆t)

2

≤C_⌊nt_⌋∆t2_≤C t∆t_→0,

which completes the proof. ƒ

Theorem 2.12. As n_{→ ∞},(B,V_n(B))_→(B,[[B]])in law in DR2[0,∞).

Proof. By Theorem 10 in[11],(B,V_n(B))_→(B,κW) = (B,[[B]])in law in(DR[0,∞))2. By Lemma 2.2, this implies(B,V_n(B))_→(B,[[B]])inDR2[0,∞). ƒ

2.5

Main result

Giveng∈C∞(R), chooseGsuch thatG′= g. We then define

Z t

0

g(B(s))d B(s) =G(B(t))₋G(B(0)) + 1

12

Z t

0

G′′′(B(s))d[[B]]_s. (2.19) Note that, by definition, the change of variable formula (1.3) holds for allg∈C∞. We shall use the shorthand notationR g(B)d B to refer to the processt 7→R₀tg(B(s))d B(s). Similarly,R g(B)d[[B]]

andR g(B)dsshall refer to the processest_7→R₀tg(B(s))d[[B]]_sandt_7→R₀tg(B(s))ds, respectively. Our main result is the following.

Theorem 2.13. If g_∈C∞(R), then(B,V_n(B),I_n(g,B))_→(B,[[B]],R g(B)d B)in law in DR3[0,∞). We also have the following generalization concerning the joint convergence of multiple sequences of Riemann sums.

Theorem 2.14. Fix k _≥ 1. Let g_{j ∈} C∞(R) for 1_≤ j _≤ k. Let J_n be the Rk-valued process whose

j-th component is (J_n)_j = I_n(g_j,B). Similarly, define J by J_j = R g_j(B)d B. Then(B,V_n(B),J_n) _→ (B,[[B]],J)in law in DRk+2[0,∞).

Remark 2.15. In less formal language, Theorem 2.14 states that the Riemann sums I_n(g_j,B)

con-verge jointly, and the limiting stochastic integrals are all defined in terms of the same Brownian motion. In other words, the limiting Brownian motion remains unchanged under changes in the in-tegrand. In this sense, the limiting Brownian motion depends only onB, despite being independent ofBin the probabilistic sense.

The proofs of these two theorems are given in Section 5.

3

Finite-dimensional distributions

Theorem 3.1. If g_∈C∞(R)is bounded with bounded derivatives, then

 

B,Vn(B),

1 p

n

⌊n·⌋

X j=1

g(B(t_j−1)) +g(B(tj))

2 h3(n 1/6_∆B

j)   →

‚

B,[[B]],

Z

g(B)d[[B]]

Œ

,

in the sense of finite-dimensional distributions on[0,_∞).

The rest of this section is devoted to the proof of Theorem 3.1.

3.1

Some technical lemmas

During the proof of Theorem 3.1, we will need technical results that are collected here. Moreover, for notational convenience, we will make use of the following shorthand notation:

δ_j=1[tj−1,tj] and ǫj =1[0,tj].

For future reference, let us note that by (2.10), ǫ⊗_ta_⊗e_ǫ⊗(q−a)

s =

_q a

−1 _X

i1,...,iq∈{s,t}

|{j:ij=s}|=q−a

ǫi1⊗. . .⊗ǫiq. (3.1)

Lemma 3.2. We have

(i) _|E[B(r)(B(t)₋B(s))]_|=_|〈1_[0,r],1_[s,t]〉H| ≤ |t−s|1/3 for any r,s,t≥0;

(ii) sup 0≤s≤T

⌊nt⌋

X k=1

|E[B(s)∆B_k]_|= sup 0≤s≤T

⌊nt⌋

X k=1

|〈1_[0,s],δk〉H| =

(iii)

⌊nt⌋

X k,j=1

|E(B(t_j−1)∆B_k)_|=

⌊nt⌋

X k,j=1

|〈ǫj−1,δk〉H| =

n→∞O(n)for any fixed t>0;

(iv)

⌊nt⌋

X k=1

(E[B(tk−1)∆Bk])3+

1 8n =

⌊nt⌋

X k=1

〈ǫk−1,δk〉3H+

1 8n

n−→→∞0for any fixed t>0;

(v)

⌊nt⌋

X k=1

(E[B(tk)∆Bk] 3 − 1 8n =

⌊nt⌋

X k=1

〈ǫk,δk〉3H−

1 8n

n−→→∞0for any fixed t>0.

Proof.

(i) We have

E B(r)(B(t)₋B(s))= 1

2(t 1/3

−s1/3) +1

2

€

|s₋r_|1/3_{− |}t₋r_|1/3Š.

Using the classical inequality|b|1/3− |a|1/3≤ |b−a|1/3, the desired result follows.

(ii) Observe that

E(B(s)∆Bk) =

1 2n1/3

€

k1/3−(k−1)1/3_{− |}k−ns|1/3+_|k−ns−1_|1/3Š.

We deduce, for any fixeds_≤t: ⌊nt⌋

X k=1

|E(B(s)∆Bk)| ≤

1 2t

1/3₊ 1 2n1/3

(_⌊ns⌋ −ns+1)1/3₋(ns− ⌊ns⌋)1/3

+

⌊ns⌋

X k=1

(ns+1₋k)1/3₋(ns−k)1/3

+

⌊nt⌋

X k=⌊ns⌋+2

(k−ns)1/3₋(k−ns−1)1/3

= 1

2(t

1/3_+s1/3₊

|t₋s_|1/3) +R_n,

where_|R_{n| ≤}C n−1/3, andC does not depend onsor t. The case wheres>t can be obtained similarly. Taking the supremum overs∈[0,T]gives us(ii).

(iii) is a direct consequence of(ii).

(i v) We have

E(B(tk−1)∆Bk) 3 + 1 8n = 1 8n €

k1/3−(k−1)1/3Š

×

€k1/3−(k−1)1/3Š2₋3(k1/3−(k−1)1/3) +3

. Thus, the desired convergence is immediately checked by combining the bound 0_≤ k1/3−

(v) The proof is very similar to the proof of(i v). ƒ Lemma 3.3. Let s_≥1, and suppose thatφ∈C6(Rs)and g₁,g_2∈C6(R)have polynomial growth of order6, all with constants K and r. Fix a,b_∈[0,T]. Then

sup

u1,...,us∈[0,T]

sup

n≥1 ⌊_Xna⌋

i1=1

⌊nb⌋

X i2=1

E φ(B(u₁), . . . ,B(u_s))g₁(B(t_i1−1))g₂(B(t_i2−1))I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )

is finite.

Proof. LetC denote a constant depending only onT,s,K, andr, and whose value can change from

one line to another. Define f :Rs+3_→Rby

f(x) =φ(x1, . . . ,xs)g1(xs+1)g2(xs+2)h3(xs+3). Let ξi = B(ui), i = 1, . . . ,s; ξs+1 = B(ti1−1), ξs+2 = B(ti2−1), ξs+3 = n

1/6_∆B

i1, and ηi =

n1/6E[ξi∆Bi2]. Applying Theorem 2.8 withk=5, we obtain

E φ(B(u₁), . . . ,B(u_s))g₁(B(t_i1−1))g₂(B(t_i2−1))I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )

= 1

nE φ(B(u1), . . . ,B(us))g1(B(ti1−1))g2(B(ti2−1))h3(n 1 6∆B_i

1)h3(n 1 6∆B_i

2)

= 1

n X

|α|=3 6 α!E[∂

α_f(ξ)]ηα₊R

n,

where_|R| ≤C|η|6.

By Lemma 3.2 (i), we have |ηi| ≤ n−1/6 for any i ≤ s+2, and |ηs+3| ≤ 1 by Lemma 2.10 (i). Moreover, we have

1

n

⌊_Xna⌋

i1=1

⌊nb⌋

X i2=1

|ηs+3|= 1 2n

⌊na⌋

X i1=1

⌊_Xnb⌋

i2=1

_|i₁₋i₂+1_|1/3+_|i₁₋i₂₋1_|1/3₋2_|i₁₋i_2|1/3_≤C.

Therefore, by taking into account these two facts, we deduce 1_nP⌊_ina⌋

1=1

P_⌊nb⌋

i2=1|R| ≤C.

On the other hand, ifα∈Ns+3

0 is such that|α|=3 withαs+36=0, we have 1

n

⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

6 α!

E[∂αf(ξ)]_|ηα|

≤ C n

⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

_|i1−i2+1|1/3+|i1−i2−1|1/3−2|i1−i2|1/3

_≤C. Note thatE[∂αf(ξ)]<∞since the function∂αf has polynomial growth by our assumptions onφ,

g₁, andg₂, and sinceξis a Gaussian vector. Finally, if α∈Ns+3

0 is such that |α| = 3 with αs+3 = 0 then ∂αf = ∂αbf ⊗h3 with fb: Rs+2 → R defined by fb(x) =φ(x1, . . . ,xs)g1(xs+1)g2(xs+2). Hence, applying Theorem 2.8 to f =∂αbf and ϕ=h₃ withk=2, we deduce, forηb∈N₀s+2defined byηbi=ηi,

E[∂αf(ξ)]=E[∂αbf(ξ)h3(n1/6∆Bi1)]

so that

1

n

⌊na⌋

X i1=1

⌊_Xnb⌋

i2=1

6 α!

E[∂αf(ξ)]_|ηα|= 1

n

⌊na⌋

X i1=1

⌊_Xnb⌋

i2=1

6 α!

E[∂αf(ξ)]_|η_bα| ≤C.

The proof of Lemma 3.3 is done. ƒ

Lemma 3.4. Let g,h∈Cq(R), q≥1, having bounded derivatives, and fix s,t ≥0. Setǫ_t=1[0,t]and ǫs=1[0,s]. Then g(B(t))h(B(s))belongs inDq,2and we have

Dq g(B(t))h(B(s))=

q X a=0

_q a

g(a)(B(t))h(q−a)(B(s))ǫ⊗_ta_⊗e_ǫ⊗(q−a)

s . (3.2)

Proof. This follows immediately from (2.16). ƒ

Lemma 3.5. Fix an integer r ≥ 1, and some real numbers s1, . . . ,sr ≥ 0. Suppose ϕ ∈ C∞(Rr)

and g_{j ∈} C∞(R), j = 1, 2, 3, 4, are bounded with bounded partial derivatives. For i₁,i₂,i₃,i_{4 ∈}N, setΦ(i₁,i₂,i₃,i₄) :=ϕ(B_s1, . . . ,B_sr)Q4_j=1g_j(B_t

i j−1). Then, for any fixed a,b,c,d >0, the following

estimate is in order:

sup

n≥1 ⌊na⌋

X i1=1

⌊_Xnb⌋

i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

E

Φ(i1,i2,i3,i4)I3(δ⊗i13)I3(δ

⊗3

i2 )I3(δ

⊗3

i3 )I3(δ

⊗3

i4 )

<∞. (3.3)

Proof. Using the product formula (2.14), we have thatI3(δ⊗i33)I3(δ

⊗3

i4 )equals

I6(δ⊗i33⊗δ

⊗3

i4 ) +9I4(δ

⊗2

i3 ⊗δ

⊗2

i4 )〈δi3,δi4〉H+18I2(δi3⊗δi4)〈δi3,δi4〉

2

H+6〈δi3,δi4〉

3

H.

As a consequence, we get ⌊_Xna⌋

i1=1

⌊nb⌋

X i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

EΦ(i₁,i₂,i₃,i₄)I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )I3(δ

⊗3

i3 )I3(δ

⊗3

i4 )

≤ ⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

E

Φ(i1,i2,i3,i4)I3(δ⊗i13)I3(δ

⊗3

i2 )I6(δ

⊗3

i3 ⊗δ

⊗3

i4 )

+9 ⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

E

Φ(i₁,i₂,i₃,i₄)I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )I4(δ

⊗2

i3 ⊗δ

⊗2

i4 )

〈δi3,δi4〉H

+18 ⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

EΦ(i₁,i₂,i₃,i₄)I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )I2(δi3⊗δi4)

〈δi3,δi4〉

2

H

+6 ⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

E

Φ(i1,i2,i3,i4)I3(δ⊗i13)I3(δ

⊗3

i2 )

〈δ_i3,δ_i4〉H

3

(1)First, we deal with the termA(₁n).

A(₁n)=

⌊_Xna⌋

i1=1

⌊nb⌋

X i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

E

Φ(i1,i2,i3,i4)I3(δ⊗i13)I3(δ

⊗3

i2 )I6(δ

⊗3

i3 ⊗δ

⊗3

i4 )

=

⌊_Xna⌋

i1=1

⌊nb⌋

X i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

E

〈D6

Φ(i₁,i₂,i₃,i₄)I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )

,δ⊗_i 3

3 ⊗δ

⊗3

i4 〉H⊗6

When computing the sixth Malliavin derivative D6 Φ(i₁,i₂,i₃,i₄)I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )

(using Lemma 3.4), there are three types of terms:

(1a)The first type consists in terms arising when oneonlydifferentiatesΦ(i1,i2,i3,i4). By Lemma 3.2(i), these terms are all bounded by

n−2

⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

E

e

Φ(i1,i2,i3,i4)I3(δ⊗i13)I3(δ

⊗3

i2 )

, which is less than

cd sup

i3=1,...,⌊nc⌋

sup

i4=1,...,⌊nd⌋

⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

EΦ(e i₁,i₂,i₃,i₄)I₃(δ⊗_i 3

1 )I3(δ

⊗3

i2 )

.

(Here,Φ(e i₁,i₂,i₃,i₄) means a quantity having a similar form asΦ(i₁,i₂,i₃,i₄).) Therefore, Lemma 3.3 shows that the terms of the first type inA(₁n)well agree with the desired conclusion (3.3).

(1b) The second type consists in terms arising when one differentiatesΦ(i1,i2,i3,i4) andI3(δ⊗i13),

but not I3(δ⊗_i23) (the case where one differentiates Φ(i1,i2,i3,i4) and I3(δ⊗_i23) but not I3(δ⊗_i13) is, of course, completely similar). In this case, withρ defined by (2.4), the corresponding terms are bounded either by

C n−2

⌊_Xna⌋

i1=1

⌊_Xnb⌋

i2=1

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

2

X

α=0

EΦ(e i₁,i₂,i₃,i₄)Iα(δ⊗_i1α)I3(δ⊗i23)

|ρ(i₃₋i₁)_|,

or by the same quantity withρ(i₄₋i₁)instead ofρ(i₃₋i₁). In order to get the previous estimate, we have used Lemma 3.2(i)plus the fact that the sequence_{ρ(r)_}r∈Z, introduced in (2.4), is bounded (see Remark 2.1). Moreover, by (2.13) and Lemma 3.2(i), observe that

EΦ(e i₁,i₂,i₃,i₄)Iα(δ⊗_i1α)I3(δ⊗i23)

=

E_〈D3 Φ(e i₁,i₂,i₃,i₄)Iα(δ⊗_i1α)

,δ⊗_i 3

2 〉H⊗3

≤C n−1,

for anyα=0, 1, 2. Finally, since

sup

i1=1,...,⌊na⌋

⌊nc⌋

X i3=1

⌊_Xnd⌋

i4=1

|ρ(i₃₋i₁)_{| ≤}nd sup

i1=1,...,⌊na⌋

X r∈Z

|ρ(r)_|=C n

(and similarly forρ(i₄₋i₁)instead ofρ(i₃₋i₁)), we deduce that the terms of the second type in

Proof. Fixa,b_∈R. Let x= (a+b)/2 andh= (b₋a)/2. By Theorem 2.7, g(b)₋g(a) = (g(x+h)₋g(x))₋(g(x₋h)₋g(x))

=

6

X

j=1

g(j)(x)h

j

j! −

6

X

j=1

g(j)(x)(−h)

j

j! +R1(x,h)−R1(x,−h) =

6

X

j=1 jodd

1 j!2j−1g

(j)_(x)(b

−a)j+R₁(x,h)₋R₁(x,₋h)

=g′(x)(b−a) + 1 24g

′′′_(x)(b−a)3₊ 1

5!24g

(5)_(x)(b

−a)5+R2(a,b),

whereR₂(a,b) =R₁(x,h)₋R₁(x,₋h)and R1(x,h) =

h6 5!

Z 1 0

(1₋u)6[g(6)(x+uh)₋g(6)(x)]du. Similarly,

g′(a) +g′(b)

2 −g

′_{(x) =} 1 2(g

′_(x+h)−g′_{(x)) +}1 2(g

′_(x−h)−g′_(x)) = 1

2

5

X

j=1

g(j+1)(x)h

j

j! + 1 2

5

X

j=1

g(j+1)(x)(−h)

j

j! +R4(a,b) = 1

8g

′′′_(x)(b−a)2₊ 1

4!24g

(5)_(x)(b

−a)4+R₄(a,b), whereR₄(a,b) =R₃(x,h) +R₃(x,₋h)and

R3(x,h) =

h5 4!

Z 1 0

(1₋u)5[g(6)(x+uh)₋g(6)(x)]du. Combining these two expansions gives

g(b)₋g(a) = g

′_{(a) +g}′_(b)

2 (b−a)− 1 12g

′′′_(x)(b−a)3_+γg(5)_(x)(b

−a)5+R6(a,b),

whereγ= (5!24₎−1_−(4!24₎−1 _and

R₆(a,b) =R₂(a,b)₋R₄(a,b)(b₋a). Note thatR₆(a,b) =h(a,b)(b₋a)6, where

|h(a,b)_{| ≤}C sup

0≤u≤1|

g(6)(x+uh)₋g(6)(x)_|. Takinga=B(t_j−1)andb=B(tj)gives

g(B(t_j))₋g(B(tj−1)) =

g′(B(t_j−1)) +g′(B(t_j))

2 ∆Bj−

1 12g

′′′_(β

j)∆B3j +γg

(5)_(β

j)∆B5j

Recall thatB_n(t) =B(_⌊nt_⌋/n), so that

g(B(t))₋g(B(0)) =I_n(g′,B,t)₋ 1 12

⌊nt⌋ X

j=1

g′′′(βj)∆B3j +ǫn(g,t),

where

ǫn(g,t) =γ

⌊nt⌋ X

j=1

g(5)(βj)∆B5j +

⌊nt⌋ X

j=1

h(B(t_j−1),B(t_j))∆B_j6+g(B(t))₋g(B_n(t)). It will therefore suffice to show thatǫn(g,t)→0 ucp.

By the continuity of gandB, g(B(t))₋g(B_n(t))_→0 uniformly on compacts, with probability one. By Lemma 5.1, since g(5)_∈C1(R),γP⌊_jnt₌₁⌋g(5)(βj)∆B5j →0 ucp. It remains only to show that

⌊nt⌋ X

j=1

h(B(t_j−1),B(t_j))∆B6_{j →}0 ucp. (5.1) FixT >0. Let_{n(k)_}∞_k=1 be an arbitrary sequence of positive integers. By Theorem 2.11, we may find a subsequence_{m(k)_}∞_k=1and a measurable subsetΩ∗_⊂Ωsuch thatP(Ω∗) =1, t_7→B(t,ω)is continuous for allω∈Ω∗, and

⌊m_X(k)t⌋

j=1

∆B_j,m(k)(ω)6_→15t, (5.2)

ask_{→ ∞}uniformly on[0,T]for allω∈Ω∗. Fixω∈Ω∗. We will show that ⌊m_X(k)t⌋

j=1

h(B(tm_j−(₁k),ω),B(tm_j(k),ω))∆B_j,m(k)(ω)6_→0, ask_{→ ∞}uniformly on[0,T], which will complete the proof.

For this, it will suffice to show that ⌊m_X(k)T⌋

j=1

|h(B(tm_j−(₁k),ω),B(tm_j (k),ω))_|∆B_j,m(k)(ω)6_→0,

as k _{→ ∞}. We begin by observing that, by (5.2), there exists a constant L such that P_⌊m(k)T⌋

j=1 ∆Bj,m(k)(ω)6 < L for all k. Now let ǫ > 0. Since g has compact support, g(6) is uni-formly continuous. Hence, there existsδ >0 such that_|b₋a_| < δ implies_|h(a,b)_| < ǫ/L for all t. Moreover, there exists k0 such that k ≥ k0 implies |∆Bj,m(k)(ω)| < δfor all 1 ≤ j ≤ ⌊m(k)T⌋. Hence, ifk_≥k₀, then

⌊m_X(k)T⌋

j=1

|h(B(tm_j−(₁k),ω),B(tm_j(k),ω))_|∆B_j,m(k)(ω)6< ǫ

L ⌊m_X(k)T⌋

j=1

∆B_j,m(k)(ω)6< ǫ,

which completes the proof. ƒ

As a corollary to this result, we find that although we do not have a bound on the second moments of In(g,B), we can instead approximateIn(g,B), in the ucp sense, by processes whose second moments

Corollary 5.3. If g_∈C6(R)has compact support, then there exists a sequence of càdlàg processes X_n such that In(g′,B,t)≈Xn(t)and, for any T >0,

sup

n

sup

t∈[0,T]

E_|X_n(t)_|2<∞.

Proof. This follows immediately from Lemma 5.2 and Theorem 4.3 by taking X_n(t) = g(B(t))₋

g(B(0)) +₁₂1 P⌊_j=nt₁⌋g′′′(βj)∆B3j. ƒ

Lemma 5.4. If g∈C6(R)has compact support, then{In(g′,B)}is relatively compact in DR[0,∞).

Proof. Define

X_n(t):= 1 12

⌊nt⌋ X

j=1

g′′′(βj)∆B3j,

Y(t):=g(B(t))₋g(B(0))

ǫn(t):=In(g′,B,t)−Y(t)−Xn(t).

Since (x,y,z) _7→ x + y +z is a continuous function from DR3[0,∞) to DR[0,∞), it will suffice

to show that _{(X_n,Y,ǫn)} is relatively compact in DR3[0,∞). By Lemma 5.2, ǫ_n → 0 ucp, and

therefore inDR[0,_∞). Hence, by Lemma 2.2, it will suffice to show that_{Xn}is relatively compact

inDR[0,_∞).

For this, we apply Theorem 2.3 withβ =4. Fix T >0 and let 0 ≤s < t ≤ T. Let c = _⌊ns⌋and d=_⌊nt_⌋. Note thatq(a+b)4_≤C(_|a_|2+_|b_|4). Hence, since ghas compact support and, therefore, g′′′is bounded,

E[q(X_n(t)₋X_n(s))4] =E q ₁ 12 d X

j=c+1

g′′′(β_j)∆B3_j 4

≤C E d X

j=c+1

(g′′′(βj)−g′′′(βc))∆B3j

2

+C E d X

j=c+1

g′′′(βc)∆B3j

4

≤C E d X

j=c+1

(g′′′(βj)−g′′′(βc))∆B3j

2

+C E d X

j=c+1

∆B3_j 4 . Sinceg′′′_∈C3(R), we may apply Theorems 4.4 and 4.1, which give

E[q(X_n(t)₋X_n(s))4]_≤C_|t_d−t_c|4/3+C_|t_d−t_c|2_≤C

⌊nt_{⌋ − ⌊}ns_⌋ n

4/3

, which verifies condition (2.6) of Theorem 2.3. As above,

E|Xn(T)|2≤C E

⌊_XnT⌋

j=1

(g′′′(βj)−g′′′(βc))∆B3j

2

+C E

⌊_XnT⌋

j=1

∆B3_j

2

≤C E

⌊_XnT⌋

j=1

(g′′′(βj)−g′′′(βc))∆B3j

2 +C E

⌊_XnT⌋

j=1

∆B3_j

41/2

≤C T4/3+C T.

Lemma 5.5. If g_∈C9(R)has compact support, then

I_n(g′,B,t)_≈ g(B(t))₋g(B(0)) + 1 12pn

⌊nt⌋ X

j=1

g′′′(B(t_j−1)) +g′′′(B(t_j))

2 h3(n

1/6_∆B j).

Proof. Using the Taylor expansions in the proof of Lemma 5.2, together with Lemma 5.1, we have ⌊nt⌋

X

j=1

g′′′(βj)∆B3j ≈

⌊nt⌋ X

j=1

g′′′(B(t_j−1)) +g′′′(B(t_j))

2 ∆B

3 j.

By Lemma 5.2, sinceh3(x) =x3−3x, it therefore suffices to show that

n−1/3 ⌊nt⌋ X

j=1

g′′′(B(t_j−1)) +g′′′(B(t_j))

2 ∆Bj=n

−1/3_I

n(g′′′,B,t)≈0.

Sinceg′′′∈C6(R), this follows from Lemma 5.4, Corollary 5.3, and Lemma 2.4. ƒ

Proof of Theorem 2.13. We first assume thatg (and alsoG) has compact support. By Lemma 5.5 and Theorem 3.1, we need only show that_{(B,V_n(B),I_n(g,B))_}is relatively compact inDR3[0,∞).

By Lemma 2.2, it will suffice to show that _{I_n(g,B)_} is relatively compact in DR[0,_∞). But this follows from Lemma 5.4, completing the proof whenghas compact support.

Now consider general g. Let

Ξ_n= (B,V_n(B),I_n(g,B)) and Ξ = (B,[[B]],R g(B)d B). ForT >0, defineΞT

n(t) = Ξn(t)1{t<T}andΞT(t) = Ξ(t)1{t<T}. Using the definition of the Skorohod metricronDRd[0,∞)(formally given by (3.5.2) in[4]that we will not recall for simplicity), it holds

that

r(x,y)_≤ Z _∞

0

e−u(sup

t≥0|

x(t_∧u)₋y(t_∧u)_{| ∧}1)du,

for any elementsx andy inDRd[0,∞). Hence, if two càdlàg functionsx andy agree on the interval

[0,T), thenr(x,y)_≤R∞

T e

−u_du=e−T_{. Hence, by Lemma 2.5, it will suffice to show thatΞ}T n →Ξ

T

in law, where T>0 is fixed.

LetH : DR3[0,∞)→R be continuous and bounded, with M =sup|H(x)|. Define X_n =H(ΞT

n), so

that it will suffice to show thatXn→H(ΞT)in law. For eachk>0, chooseGk∈C6(R)with compact

support such thatG_k=Gon[₋k,k]. Letg_k=G_k′, e

Ξ_n,k= (B,Vn(B),In(gk,B)), Ξek= (B,[[B]],

R

gk(B)d B),

X_n,k = H(eΞ_n,kT ) and Y_k = H(eΞ_kT). Note that since I_n(g_k,B) = I_n(g,B) on _{ω ∈ Ω : sup_0≤t≤T|B(t)(ω)_|<k}, we have E|Xn−Xn,k| ≤δk, where

δk=2M P

‚ sup

0≤t≤T|

B(t)_{| ≥}k Œ

Also note that thatδk →0 ask→ ∞. SinceGk has compact support, we have already proven that

Xn,k→Yk in law. Hence, by Lemma 2.5, the sequences{Xn}and{Yk}are both convergent in law,

and they have the same limit. Thus, to show thatX_n →H(ΞT_{)in law, it will suffice to show that}

Y_k→H(ΞT) in law. However, it is an immediate consequence of (2.19) thatΞeT_{k →}ΞT ucp, which

completes the proof. ƒ

Proof of Theorem 2.14.As in the proof of Theorem 2.13,_{(B,V_n(B),J_n)_}is relatively compact. Let (B,X,Y)be any subsequential limit. By Theorem 2.12,X =κW, whereW is a standard Brownian motion, independent of B. Hence,(B,X,Y) = (B,[[B]],Y). Fix j _{∈ {}1, . . . ,k_}. By Theorem 2.13, (B,[[B]],Y_j) has the same law as(B,[[B]],R g_j(B)d B). Using the general fact we observed at the beginning of the proof of Theorem 3.1, together with (2.19) and the definition of[[B]], this implies (R g_j(B)d B,Y_j) and(R g_j(B)d B,R g_j(B)d B) have the same law. Hence, Y_j = R g_j(B)d B a.s., so

(B,X,Y) = (B,[[B]],J). ƒ

Acknowledgments

We would like to thank the referee for his or her careful and thorough reading, and the many comments and suggestions which helped us to improve the presentation of the paper.

References

[1] K. Burdzy and J. Swanson. A change of variable formula with Itô correction term.Ann. Probab., 38(5):1817–1869, 2010.http://www.ams.org/mathscinet-getitem?mr=MR2722787

[2] P. Cheridito and D. Nualart. Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H ∈ (0,1

2). Ann. Inst. H. Poincaré Probab. Statist.,

41(6):1049–1081, 2005.http://www.ams.org/mathscinet-getitem?mr=MR2172209

[3] P. Donnelly and T. G. Kurtz. A countable representation of the Fleming-Viot measure-valued dif-fusion. Ann. Probab., 24(2):698–742, 1996. http://www.ams.org/mathscinet-getitem? mr=MR1419488

[4] S. N. Ethier and T. G. Kurtz. Markov processes: Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986.http://www.ams.org/mathscinet-getitem?mr=MR0838085

[5] M. Gradinaru, I. Nourdin, F. Russo, and P. Vallois. m-order integrals and generalized Itô’s for-mula: the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist., 41(4):781–806, 2005. http://www.ams.org/mathscinet-getitem?mr= MR2144234

[6] P. Malliavin. Stochastic analysis, volume 313 of Fundamental Principles of Mathematical Sciences. Springer-Verlag, Berlin, 1997. http://www.ams.org/mathscinet-getitem?mr= MR1450093

[7] I. Nourdin and D. Nualart. Central limit theorems for multiple Skorohod integrals. J. Theoret. Probab., 23(1):39–64, 2010.http://www.ams.org/mathscinet-getitem?mr=MR2591903

[8] I. Nourdin, D. Nualart, and C. A. Tudor. Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist., 46(4), 1055–1079, 2010. Math. Review not available.

[9] I. Nourdin and A. Réveillac. Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H =1/4. Ann. Probab., 37(6):2200–2230, 2009.http: //www.ams.org/mathscinet-getitem?mr=MR2573556

[10] D. Nualart. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. http://www.ams.org/mathscinet-getitem? mr=MR2200233

[11] D. Nualart and S. Ortiz-Latorre. Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl., 118(4):614–628, 2008.http://www.ams.org/ mathscinet-getitem?mr=MR2394845

[12] A. Réveillac. Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets. Stoch. Anal. Appl., 27(1):51–73, 2009. http: //www.ams.org/mathscinet-getitem?mr=MR2473140

[13] F. Russo and P. Vallois. Elements of stochastic calculus via regularization. In Séminaire de Probabilités XL, volume 1899 ofLecture Notes in Math., pages 147–185. Springer, Berlin, 2007.