arXiv:math/0703794v3 [math.PR] 24 Oct 2008

Bernoulli 14(3), 2008, 822–837 DOI: 10.3150/08-BEJ124

Asymptotic expansions at any time for scalar fractional SDEs with Hurst index H > 1/2 ´ SEBASTIEN DARSES1 and IVAN NOURDIN2 1

Boston University, Department of Mathematics and Statistics, 111 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] 2 Universit´e Pierre et Marie Curie, Laboratoire de Probabilit´es et Mod`eles Al´eatoires, 4 place Jussieu, Boˆıte courrier 188, 75252 Paris Cedex 5, France. E-mail: [email protected] We study the asymptotic expansions with respect to h of E[∆h f (Xt )],

E[∆h f (Xt )|FtX ]

and

E[∆h f (Xt )|Xt ],

where ∆h f (Xt ) = f (Xt+h ) − f (Xt ), when f : R → R is a smooth real function, t ≥ 0 is a fixed time, X is the solution of a one-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst index H > 1/2 and F X is its natural filtration. Keywords: asymptotic expansion; fractional Brownian motion; Malliavin calculus; stochastic differential equation

1. Introduction We study the asymptotic expansions with respect to h of Pt f (h) , E[∆h f (Xt )], Pbt f (h) , E[∆h f (Xt )|FtX ],

(1)

Pet f (h) , E[∆h f (Xt )|Xt ],

with ∆h f (Xt ) , f (Xt+h ) − f (Xt ), when f : R → R is a smooth real function, t ≥ 0 is a fixed time, X is the solution to the fractional stochastic differential equation Z t Z t σ(Xs ) dBs , t ∈ [0, T ], (2) b(Xs ) ds + Xt = x + 0

0

and F X is its natural filtration. Here, b, σ : R → R are real functions belonging to the space C∞ b of all bounded continuous functions having bounded derivatives of all order, This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 3, 822–837. This reprint differs from the original in pagination and typographic detail. 1350-7265

c

2008 ISI/BS

Asymptotic expansions for fractional SDEs

823

while B is a one-dimensional fractional Brownian motion with Hurst index H ∈ (1/2, 1). When the integral with respect to B is understood in the Young sense, equation (2) has a unique pathwise solution X in the set of processes whose paths are H¨ older continuous of index α ∈ (1 − H, H). Moreover, by, for example, [12], Theorem 4.3, we have, for any g : R → R ∈ C∞ b , g(Xt ) = g(x) +

Z

t

g ′ (Xs )σ(Xs ) dBs +

Z

t

g ′ (Xs )b(Xs ) ds,

0

0

t ∈ [0, T ].

(3)

The asymptotic expansion of E[f (Xh )] with respect to h has been recently studied in [1], [7]. In our framework, it turns out to be the case where t = 0 since we obviously have E[f (Xh )] − f (x) = P0 f (h) = Pb0 f (h) = Pe0 f (h).

In these last references, the authors work in a multidimensional setting and under the weaker assumption that the Hurst index H of the fractional Brownian motion B is greater than 1/3 (the integral with respect to B is then understood in the rough paths sense of Lyons’ type for [1] and of Gubinelli’s type for [7]). In particular, it is proved in [1], [7] that there exists a family Γ = {Γ2kH+ℓ : (k, ℓ) ∈ N2 , (k, ℓ) 6= (0, 0)} of differential operators such that for any smooth f : R → R, we have the following asymptotic expansion: X h2kH+ℓ Γ2kH+ℓ (f, σ, b)(x). (4) P0 f (h) ∼ h→0

Moreover, in [7], operators Γ2kH+ℓ are expressed using trees. A natural question now arises. Can we also get an expansion of Pt f (h) when t 6= 0? Let us first consider the case where B is the standard Brownian motion (which corresponds to the case where H = 1/2). By the Markov property on one hand, we have Pbt f (h) = Pet f (h). On the other hand, we always have Pt f (h) = E[Pbt f (h)]. Thus, there exist relations between Pt f (h), Pbt f (h) and Pet f (h). Moreover, the asymptotic expansion of Pt f (h) can be obtained as a corollary to that of P0 f (h) using the conditional expectation either with respect to the past FtX of X, or with respect to Xt only and the strong Markov property. When H > 1/2, B is not Markovian. The situation regarding Pt f (h), Pbt f (h) and e Pt f (h) is then completely different and actually more complicated. In particular, we no longer have Pbt f (h) = Pet f (h) and we cannot deduce the asymptotic expansion of Pt f (h) from that of P0 f (h). The current paper is concerned with the study of possible asymptotic expansions of the various quantities Pt f (h), Pbt f (h) and Pet f (h) when H > 1/2. We will see that some nontrivial phenomena appear. More precisely, we will show in Section 3 that Pbt f (h) does not admit an asymptotic expansion in the scale of the fractional powers of h when t 6= 0. Regarding Pbt f (h), the situations when t = 0 and t > 0 are thus really different. On

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S. Darses and I. Nourdin

the other hand, unlike Pbt f (h), the quantities Pt f (h) and Pet f (h) admit, when t 6= 0, an asymptotic expansion in the scale of the fractional powers of h. However, the computation of this expansion is more difficult than in the case where t = 0 (as carried out in [1], [7]). That is why we prefer to consider only the one-dimensional case. As an illustration, let us consider the trivial equation dXt = dBt , t ∈ [0, T ], X0 = 0. That is, Xt = Bt for every t ∈ [0, T ]. We have, thanks to a Taylor expansion, Pe0 f (h) =

n X f (k) (0) k=1

k!

⌊n/2⌋

E[(Bh )k ] + · · · =

X f (2k) (0) h2Hk + · · · , 2k k! k=1

while by a linear Gaussian regression, when t 6= 0,      h2H H 2H e Pt f (h) = E f 1 + h − 2H + · · · Bt + (h + · · ·)N − f (Bt ) Bt t 2t =

Bt f ′ (Bt ) 2H HBt f ′ (Bt ) h +··· h− t 2t2H

with N ∼ N (0, 1) a random variable independent of Bt . One of the key points of our strategy relies on the use of a Girsanov transformation and the Malliavin calculus for fractional Brownian motion. We refer to [3], [10] for a deep insight of this topic. We will restrict the exposition of our asymptotic expansions to the case when σ = 1. Indeed, under the assumption (A)

the function σ is elliptic on R, that is, it satisfies inf R |σ| > 0

and using the change of variable formula (3), equation (2) can be reduced to a diffusion Y with a constant diffusion coefficient, Yt =

Z

0



Xt

dz . σ(z)

dz Moreover, since 0 σ(z) is strictly monotone from R to R under assumption (A), the σ-fields generated by Xt (resp. by Xs , s ≤ t) and Yt (resp. by Ys , s ≤ t) are the same. Consequently, assuming that σ = 1 is not at all restrictive since it allows the recovery of the general case under assumption (A). We therefore consider in the sequel that X is the unique solution of Z t Xt = x + b(Xs ) ds + Bt , t ∈ [0, T ], (5) 0

with b ∈ C∞ b and x ∈ R. The paper is organized as follows. In Section 2, we recall some basic facts about fractional Brownian motion, the Malliavin calculus and fractional stochastic differential equations. In Section 3, we prove that Pbt f (h) does not admit an asymptotic expansion

Asymptotic expansions for fractional SDEs

825

with respect to the scale of fractional powers of h, up to order n ∈ N. We eventually show, in Section 4, that Pet f (h) admits an asymptotic expansion.

2. Preliminaries

We first briefly recall some basic facts about stochastic calculus with respect to a fractional Brownian motion. One may refer to [9], [10] for further details. Let B = (Bt )t∈[0,T ] be a fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) defined on a probability space (Ω, A , P). We mean that B is a centered Gaussian process with the covariance function E(Bs Bt ) = RH (s, t), where RH (s, t) = 21 (t2H + s2H − |t − s|2H ).

(6)

We denote by E the set of step R-valued functions on [0, T ]. Let H be the Hilbert space defined as the closure of E with respect to the scalar product h1[0,t] , 1[0,s] iH = RH (t, s). We denote by | · |H the associate norm. The mapping 1[0,t] 7→ Bt can be extended to an isometry between H and the Gaussian space H1 (B) associated with B. We denote this isometry ϕ 7→ B(ϕ). The covariance kernel RH (t, s) introduced in (6) can be written as RH (t, s) =

Z

s∧t

KH (s, u)KH (t, u) du,

0

where KH (t, s) is the square-integrable kernel defined, for s < t, by KH (t, s) = cH s1/2−H

Z

s

t

(u − s)H−3/2 uH−1/2 du

(7)

H(2H−1) and β the Beta function. By convention, we set KH (t, s) = 0 if with c2H = β(2−2H,H−1/2) s ≥ t. We define the operator KH on L2 ([0, T ]) by

(KH h)(t) =

Z

t

KH (t, s)h(s) ds.

0

∗ Let KH : E → L2 ([0, T ]) be the linear operator defined by ∗ KH (1[0,t] ) = KH (t, ·).

The following equality holds for any φ, ψ ∈ E: ∗ ∗ ψiL2 ([0,T ]) = E(B(φ)B(ψ)). hφ, ψiH = hKH φ, KH

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S. Darses and I. Nourdin

∗ KH then provides an isometry between the Hilbert space H and a closed subspace of 2 L ([0, T ]). The process W = (Wt )t∈[0,T ] defined by ∗ −1 Wt = B((KH ) (1[0,t] ))

(8)

is a Wiener process and the process B has the following integral representation: Bt =

Z

t

KH (t, s) dWs . 0

Hence, for any φ ∈ H, we have ∗ B(φ) = W (KH φ).

If b, σ ∈ C∞ b , then (2) admits a unique solution X in the set of processes whose paths are H¨ older continuous of index α ∈ (1 − H, H). Moreover, X has the Doss–Sussman’s-type representation (see, e.g., [5]) t ∈ [0, T ],

Xt = φ(At , Bt ),

(9)

with φ and A given, respectively, by ∂φ (x1 , x2 ) = σ(φ(x1 , x2 )), ∂x2

x1 , x2 ∈ R

φ(x1 , 0) = x1 ,

and  Z A′t = exp −

Bt

0

 σ ′ (φ(At , s)) ds b(φ(At , Bt )),

t ∈ [0, T ].

A0 = x0 ,

Let b ∈ C∞ b and X be the solution of (5). Following [11], the fractional version of the Girsanov theorem applies and ensures that X is a fractional Brownian motion with Hurst parameter H under the new probability Q defined by dQ = η −1 dP, where Z η = exp

0

T

−1 KH

Z

·

0

 Z b(Xr ) dr (s) dWs + 12

0

T

−1 KH

Z

0

·

b(Xr ) dr

2

 (s) ds .

(10)

Let S be the set of all smooth cylindrical random variables, that is, of the form F = f (B(φ1 ), . . . , B(φn )), where n ≥ 1, f : Rn → R is a smooth function with compact support and φi ∈ H. The Malliavin derivative of F with respect to B is the element of L2 (Ω, H) defined by Ds F =

n X ∂f (B(φ1 ), . . . , B(φn ))φi (s), ∂x i i=1

s ∈ [0, T ].

Asymptotic expansions for fractional SDEs

827

In particular, Ds Bt = 1[0,t] (s). As usual, D1,2 denotes the closure of the set of smooth random variables with respect to the norm kF k21,2 = E[F 2 ] + E[|D· F |2H ]. The Malliavin derivative D verifies the following chain rule. If ϕ : Rn → R is C1b and if (Fi )i=1,...,n is a sequence of elements of D1,2 , then ϕ(F1 , . . . , Fn ) ∈ D1,2 and we have, for any s ∈ [0, T ], Ds ϕ(F1 , . . . , Fn ) =

n X ∂ϕ (F1 , . . . , Fn )Ds Fi . ∂x i i=1

The divergence operator δ is the adjoint of the derivative operator D. If a random variable u ∈ L2 (Ω, H) belongs to the domain of the divergence operator, that is, if there exists cu > 0 such that |EhDF, uiH | ≤ cu kF kL2

for any F ∈ S ,

then δ(u) is defined by the duality relationship E(F δ(u)) = EhDF, uiH for all F ∈ D1,2 . c 3. Study of the asymptotic expansion of P t f (h)

Recall that Pbt f (h) is defined by (1), where X is given by (5).

Definition 1. We say that Pbt f (h) admits an asymptotic expansion with respect to the scale of fractional powers of h, up to order n ∈ N, if there exist some real numbers 0 < α1 < · · · < αn and some random variables C1 , . . . , Cn ∈ L2 (Ω, FtX ), not identicallyzero, such that Pbt f (h) = C1 hα1 + · · · + Cn hαn + o(hαn )

as h → 0,

where o(hα ) stands for a random variable of the form hα φh , with E[φ2h ] → 0 as h → 0. If Pbt f (h) admits an asymptotic expansion in the sense of Definition 1, we must, in particular, have the existence of α > 0 verifying the following condition: lim h−α Pbt f (h) exists in L2 (Ω) and is not identically zero.

h→0

However, we have the following.

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S. Darses and I. Nourdin

Theorem 1. Let f : R → R ∈ C∞ b and t ∈ (0, T ]. Assume, moreover, that Leb({x ∈ R : f ′ (x) = 0}) = 0.

(11)

Then, as h → 0, h−α Pbt f (h) converges in L2 (Ω) if and only if α < H. In this case, the limit is zero. Remark 1. Since Pb0 f (h) = Pe0 f (h), we refer to Theorem 2 for the case where t = 0.

Proof of Theorem 1. The proof is divided into two cases. (i) First case: α ∈ (0, 1]. Since H > 1/2, let us first remark that h−α Pbt f (h) converges in L2 (Ω) if and only if h−α f ′ (Xt )E[Xt+h − Xt |FtX ] converges in L2 (Ω). Indeed, we use a Taylor expansion: |f (Xt+h ) − f (Xt ) − f ′ (Xt )(Xt+h − Xt )| ≤ 12 |f ′′ |∞ |Xt+h − Xt |2 , so |Pbt f (h) − f ′ (Xt )E[Xt+h − Xt |FtX ]| ≤ 21 |f ′′ |∞ E[|Xt+h − Xt |2 |FtX ].

Thus, applying Jensen’s formula:

2

h−2α E[|Pbt f (h) − f ′ (Xt )E[Xt+h − Xt |FtX ]| ] 2

≤ 41 |f ′′ |2∞ h−2α E[E[|Xt+h − Xt |2 |FtX ]]

≤ 14 |f ′′ |2∞ h−2α E[|Xt+h − Xt |4 ] = O(h4H−2α ). Since α ≤ 1 < 2H, we can conclude. By (11) and the fact that Xt has a positive density on R (see, e.g., [8] Theorem A), we have that h−α f ′ (Xt )E[Xt+h − Xt |FtX ] converges in L2 (Ω) if and only if h−α E[Xt+h − Xt |FtX ] converges in L2 (Ω). For X given by (5), we have that F X = F B . Indeed, one inclusion is obvious, while the other can be proven using (9). Moreover, since b ∈ Cb∞ and α ≤ 1, the term R t+h h−α E[ t b(Xs ) ds|FtX ] converges in L2 (Ω) when h ↓ 0. Therefore, we have, due to (5) that h−α E[Xt+h − Xt |FtX ] converges in L2 (Ω) if and only if h−α E[Bt+h − Bt |FtB ] converges in L2 (Ω). Set Z t (t) −α B −α Zh = h E[Bt+h − Bt |Ft ] = h (KH (t + h, s) − KH (t, s)) dWs , 0

where the kernel KH is given by (7) and the Wiener process W is defined by (8). We have Z t 2 (t) Var(Zh ) = h−2α (KH (t + h, s) − KH (t, s)) ds 0

Asymptotic expansions for fractional SDEs = h−2α c2H

Z

t

s1−2H

0

829

Z

t+h

t

(u − s)H−3/2 uH−1/2 du

2

ds.

We deduce (t)

Var(Zh ) ≥ h−2α ×

Z

 t

0

cH H − 1/2

2

t2H−1 2

s1−2H ((t + h − s)H−1/2 − (t − s)H−1/2 ) ds

(12) 2 Z t  1−2H c s H 2 = h−2α 1− ((s + h)H−1/2 − sH−1/2 ) ds H − 1/2 t 0 2 Z t/h  1−2H  hs cH 2(H−α) 1− g 2 (s) ds =h H − 1/2 t 0 

with g(s) = (s + 1)H−1/2 − sH−1/2 . Similarly, (t) Var(Zh )

≤h

−2α

×

Z

0



t

cH H − 1/2

2

(t + h)2H−1 2

s1−2H ((t + h − s)H−1/2 − (t − s)H−1/2 ) ds

 2H−1  2 h cH 1+ =h (13) t H − 1/2 1−2H Z t s 2 × 1− ((s + h)H−1/2 − sH−1/2 ) ds t 0 2H−1  2 Z t/h  1−2H  hs cH h 2(H−α) 1− g 2 (s) ds. 1+ =h t H − 1/2 t 0 −2α

Note that g 2 (s) ∼ (H − 12 )2 s2H−3 as s → +∞. So, sg 2 (s) −→ 0 as s → +∞ and R +∞ 2 |g (s)| ds < +∞ since 2H − 3 < −1. Since s 7→ sg 2 (s) is bounded on R+ , we have, 0 by the dominated convergence theorem, that Z

0

t/h 

hs 1− t

1−2H

 Z − 1 g 2 (s) ds =

1 0

  (1 − u)1−2H − 1 2 tu tu g du u h h

tends to zero as h → 0. Thus, lim

Z

h→0 0

t/h 

1−

hs t

1−2H

g 2 (s) ds =

Z

0



g 2 (s) ds < +∞.

830

S. Darses and I. Nourdin

Combined with (12)–(13), we now deduce that 2 Z ∞  cH (t) 2(H−α) g 2 (s) ds Var(Zh ) ∼ h H − 1/2 0 (t)

as h → 0.

(14)

(t)

If Zh converges in L2 (Ω) as h → 0, then limh→0 Var(Zh ) exists and is finite. But, thanks (t) (t) to (14), we have that limh→0 Var(Zh ) = +∞ when α > H. Consequently, Zh does not 2 converge in L (Ω) as h → 0 when α > H. (t)

(t) L2

Conversely, when α < H, we have, from (14), that limh→0 Var(Zh ) = 0. Then Zh −→0 when α < H. In order to complete the proof of the first case, it remains to consider the critical case (t) Law 2 ), as h → 0, with where α = H. We first deduce from (14) that Zh −→N (0, σH 2 Z ∞  cH 2 g 2 (s) ds. σH = H − 1/2 0 Let us finally show that the previous limit does not hold in L2 . Assume for a moment (t) (t) that Zh converges in L2 (Ω) as h → 0. In particular, {Zh }h>0 is Cauchy in L2 (Ω). So, (t) (t) denoting by Z (t) the limit in L2 (Ω), we have E[Zε Zδ ] → E[|Z (t) |2 ] when ε, δ → 0. But, for any fixed x > 0, we can show by using the same transformations as above that as h → 0,  2 cH 2 (t) (t) E(Zhx Zh/x ) −→ r(x) = E(|Z (t) | ), H − 1/2 where

r(x) =

Z



((s + x)

0

=x

H−1/2

Z

−s

H−1/2



  H−1/2 1 H−1/2 ) s+ ds −s x

g(x2 u)g(u) du.

0

Consequently, the function r is constant on ]0, +∞[. The Cauchy–Schwarz inequality yields √ √ √ |g|2L2 = r(1) = r( 2) = h 2g(2·), giL2 ≤ 2|g(2·)|L2 |g|L2 = |g|2L2 . We thus have an equality in the previous inequality. We deduce that there exists λ ∈ R such that g(2u) = λg(u) for all u ≥ 0. Since g(0) = 1, we have λ = 1. Consequently, for any u ≥ 0 and any integer n, we get   u g(u) = g n −→ g(0) = 1, n→∞ 2 (t)

which is absurd. Therefore, when α = H, Zh does not converge in L2 (Ω) as h → 0. This concludes the proof of the first case.

Asymptotic expansions for fractional SDEs

831

(ii) Second case: α ∈ (1, +∞). If h−α Pbt f (h) converges in L2 (Ω), then h−1 Pbt f (h) converges in L2 (Ω) toward zero. This contradicts the first case, which concludes the proof  of Theorem 1. f 4. Study of the asymptotic expansion of P t f (h)

Recall that Pet f (h) is defined by (1), where X is given by (5). The main result of this section is the first point of the following theorem. 2 Theorem 2. Let t ∈ [0, T ] and f : R → R ∈ C∞ b . We write N for N \ {(0, 0)}. For (p, q) ∈ N , set

J2pH+q = {(m, n) ∈ N : 2mH + n ≤ 2pH + q}. (t)

1. If t 6= 0, there exists a family {Z2mH+n }(m,n)∈N of random variables measurable with respect to Xt such that for any (p, q) ∈ N , Pet f (h) =

X

(t)

Z2mH+n h2mH+n + o(h2pH+q ).

(15)

(m,n)∈J2pH+q

2. If t = 0, for any (p, q) ∈ N , we have P0 f (h) = Pe0 f (h) = Pb0 f (h) =

X

(m,n)∈J2pH+q

X

!

cI ΓI (f, b)(x) h2mH+n + o(h2pH+q )

I∈{0,1}2m+n ,|I|=2m

with cI and ΓI defined, respectively, by (17) and (19) below. (t)

Remark 2. 1. In (15), Z0H+1 coincides with the stochastic derivative of X with respect to its present t, as defined in [2]. 2. The expansion (15) allows the expansion of Pt f (h) to obtained for t 6= 0: Pt f (h) = E[Pet f (h)] =

X

(t)

E[Z2mH+n ]h2mH+n + o(h2pH+q ).

(m,n)∈J2pH+q

The following subsections are devoted to the proof of Theorem 2. Note that a quicker proof of the first assertion seems to be as follows. Once t > 0 is fixed, we write Xt+h = Xt +

Z

t

t+h

(t)

e , b(Xs ) ds + B h

h ≥ 0,

(16)

832

S. Darses and I. Nourdin (t)

e = Bt+h − Bt is again a fractional Brownian motion. We could then think that where B h an expansion for Pet f (h) directly follows from the one for Pe0 f (h), simply by a shift. This is unfortunately not the case, due to the fact that the initial value in (16) is not just a real number as in the case t = 0, but a random variable. Consequently, the computation e (t) |Xt ] is not trivial since B e (t) and Xt are not independent. of E[B h h

4.1. Proof of Theorem 2, part (2)

The proof of this part is actually a direct consequence of Theorem 2.4 in [7]. But, for the sake of completeness on one hand and taking into account that we are dealing with the one-dimensional case on the other hand, we give all the details here. Indeed, contrary to the multidimensional case, it is easy to compute explicitly the coefficients which appear (see Lemma 1 below and compare with [1] Theorem 31 or [7] Proposition 5.4) which also has its own interest from our point of view. The differential operators ΓI appearing in Theorem 2 are recursively* defined by Γ(0) (f, b) = bf ′ ,

Γ(1) (f, b) = f ′

and, for I ∈ {0, 1}k , Γ(I,0) (f, b) = b(ΓI (f, b))′ ,

Γ(I,1) (f, b) = (ΓI (f, b))′

(17)

with (I, 0), (I, 1) ∈ {0, 1}k+1 . The constants cI are explained as follows. Set  dBt , if i = 1, (i) dBt = dt, if i = 0.

(18)

Then for a sequence I = (i1 , . . . , ik ) ∈ {0, 1}k , we define cI = E

Z

∆k [0,1]

dB

I



=E

Z

0

1

(i ) dBtkk

Z

tk

0

(ik−1 ) dBtk−1 ···

Z

0

t2

(i ) dBt11



.

(19)

P Set |I| = 1≤j≤k ij . Equivalently, |I| denotes the number of integrals with respect to ‘dB’. Since B and −B have the same law, note that we have cI = cI (−1)|I| . Thus, cI vanishes when |I| is odd. In general, the computation of the coefficients cI can be made as follows. * We can also use a rooted trees approach in order to define the Γ ’s. See [7] for a thorough study, I even in the multidimensional case and where H > 1/3.

Asymptotic expansions for fractional SDEs

833

Lemma 1. Let I ∈ {0, 1}k . We denote by J = {j1 < · · · < jm } the set of indices j ∈ (j) {1, . . . , k} such that dBt = dt. We then have that cI is given by # " Z tj2 Z 1 jk −jk−1 m (B (B1 − Btjm )k−jm (Btj1 )j1 −1 Y tjk − Btjk−1 ) . dtj1 E dtjm · · · (k − jm )!(j1 − 1)! (jk − jk−1 )! 0 0 k=2

The expectation appearing in the above formula can always be computed using the moment generating function of an m-dimensional Gaussian random variable. For instance, we have Z 1 Z t3 Z t2  1 E dt1 dBt2 dBt3 = , 2(2H + 1) 0 0 0 Z 1 Z t3 Z t2  2H − 1 , E dBt1 dt2 dBt3 = 2(2H + 1) 0 0 0  Z 1 Z t3 Z t2 1 E dBt1 dBt2 dt3 = , 2(2H + 1) 0 0 0 Z 1 Z t4 Z t3 Z t2  1 , E dt1 dt2 dBt3 dBt4 = 2(2H + 1)(2H + 2) 0 0 0 0  Z 1 Z t4 Z t3 Z t2 H E dt1 dBt2 dt3 dBt4 = , (20) (2H + 1)(2H + 2) 0 0 0 0  Z 1 Z t4 Z t3 Z t2 1 E dt1 dBt2 dBt3 dt4 = , 2(2H + 1)(2H + 2) 0 0 0 0  Z 1 Z t4 Z t3 Z t2 H , dBt1 dt2 dBt3 dt4 = E (2H + 1)(2H + 2) 0 0 0 0  Z 1 Z t4 Z t3 Z t2 H(2H − 1) dBt1 dt2 dt3 dBt4 = E , 2(2H + 1)(2H + 2) 0 0 0 0  Z 1 Z t4 Z t3 Z t2 1 dBt1 dBt2 dt3 dt4 = E . 2(2H + 1)(2H + 2) 0 0 0 0 Lemma 2. When f : R → R ∈ C∞ b , we have f (Xh ) = f (x) +

n−1 X

k=1 Ik

+

X

In ∈{0,1}n

X

∈{0,1}k

Z

∆n [0,h]

ΓIk (f, b)(x)

Z

dB Ik (t1 , . . . , tk )

∆k [0,h]

(21) In

ΓIn (f, b)(Xt1 ) dB (t1 , . . . , tn ),

834

S. Darses and I. Nourdin

where, again using the convention (18), for g : R → R ∈ C∞ b , Z

g(Xt1 ) dB Ik (t1 , . . . , tk ) ,

∆k [0,h]

Z

h 0

(i )

dBtkk

Z

tk

(i

)

k−1 dBtk−1 ···

0

Z

0

t2

(i )

dBt11 g(Xt1 ).

Proof. Applying (3) twice, we can write f (Xh ) = f (x) +

Z

h



f (Xs ) dBs +

0

Z

h

(bf ′ )(Xs ) ds

0

= f (x) + Γ(1) (f, b)(x) Bh + Γ(0) (f, b)(x)h X Z ΓI2 (f, b)(Xt1 ) dB I2 (t1 , t2 ). + I2 ∈{0,1}2

∆2 [0,h]



Applying (3) repeatedly, we finally obtain (21). The remainder can be bounded by the following lemma. Lemma 3. If n ≥ 2, ε > 0 (small enough) and g : R → R ∈ C∞ b are fixed, we have Z X E g(Xt1 ) dB In (t1 , . . . , tn ) = O(hnH−ε ). ∆n [0,h]

In ∈{0,1}n

Proof. This involves a direct application of Theorem 2.2 in [6], combined with the Garsia, Rodemich and Rumsey Lemma [4].  Thus, in order to obtain the asymptotic expansion of Pet f (h), Lemmas 2 and 3 say that it is sufficient to compute Z  E dB Ik (t1 , . . . , tk ) ∆k [0,h]

for any Ik ∈ {0, 1}k , with 1 ≤ k ≤ n − 1. By the self-similarity and the stationarity of fractional Brownian motion, we have that Z Z L H|Ik |+k−|Ik | Ik dB (t1 , . . . , tk ) = h dB Ik (t1 , . . . , tk ). ∆k [0,h]

∆k [0,1]

Hence, it follows that E

Z

∆k [0,h]

 dB (t1 , . . . , tk ) = hH|Ik |+k−|Ik | cIk Ik

and the proof of point (2) of Theorem 2 is a consequence of Lemmas 2 and 3 above. 

Asymptotic expansions for fractional SDEs

835

4.2. Proof of Theorem 2, point (1) Let f : R → R ∈ C∞ b and X be the solution of (5). We then know (see Section 2) that X is a fractional Brownian motion with Hurst index H under the new probability Q defined 1,2 . by dQ = η −1 dP, with η given by (10). Since b : R → R ∈ C∞ b , we observe that η ∈ D 2 2 Moreover, the following well-known formula holds for any ξ ∈ L (P) ∩ L (Q): E[ξ|Xt ] =

EQ [ηξ|Xt ] . EQ [η|Xt ]

In particular, EQ [η(f (Xt+h ) − f (Xt ))|Xt ] . EQ [η|Xt ]

E[f (Xt+h ) − f (Xt )|Xt ] =

We now need the following technical lemma. Lemma 4. Let ζ ∈ D1,2 (H) be a random variable. Then for any h > 0, the conditional expectation E[ζ(f (Bt+h ) − f (Bt ))|Bt ] is equal to H(2H − 1)f ′ (Bt )

Z

T

du E[Du ζ|Bt ]

0

Z

t

t+h

|v − u|2H−2 dv

1 + t−2H f ′ (Bt )(Bt E[ζ|Bt ] − E[hDζ, 1[0,t] iH |Bt ])(h2H − (t + h)2H + t2H ) 2 H − t−2H f ′′ (Bt )E[ζ|Bt ] 2 Z t+h ((v − t)2H−1 − v 2H−1 )(t2H + v 2H − (v − t)2H ) dv × t

1 + f ′′ (Bt )E[ζ|Bt ]((t + h)2H − t2H ) 2 Z T Z t+h + H(2H − 1) du |v − u|2H−2 E[Du ζ(f ′ (Bv ) − f ′ (Bt ))|Bt ] dv 0

+ Ht−2H Bt

Z

t+h

t

− Ht−2H −

Z

H −2H t 2

t+h

t

Z

t

t

((v − t)2H−1 − v 2H−1 )E[ζ(f ′ (Bv ) − f ′ (Bt ))|Bt ] dv

((v − t)2H−1 − v 2H−1 )E[hDζ, 1[0,t] iH (f ′ (Bv ) − f ′ (Bt ))|Bt ] dv

t+h

((v − t)2H−1 − v 2H−1 )(t2H + v 2H − (v − t)2H )

× E[ζ(f ′′ (Bv ) − f ′′ (Bt ))|Bt ] dv

(22)

836

S. Darses and I. Nourdin +H

Z

t

t+h

E[ζ(f ′′ (Bv ) − f ′′ (Bt ))|Bt ]v 2H−1 dv.

Proof. Let g : R → R ∈ C1b . We can write, using the Itˆ o formula ([9], (5.44), page 294) and basics identities of Malliavin calculus, E[ζg(Bt )(f (Bt+h ) − f (Bt ))] = E[ζg(Bt )δ(f ′ (B· )1[t,t+h] )] + H

Z

t+h

E[ζg(Bt )f ′′ (Bv )]v 2H−1 dv

t

= E[g(Bt )hDζ, 1[t,t+h] f ′ (B· )iH ] + E[ζg ′ (Bt )h1[0,t] , 1[t,t+h] f ′ (B· )iH ] Z t+h E[ζg(Bt )f ′′ (Bv )]v 2H−1 dv +H t

= H(2H − 1)

Z

T

0

+ H(2H − 1) +H

Z

t+h

du

Z

t+h

|v − u|2H−2 E[f ′ (Bv )g(Bt )Du ζ] dv

t

Z

t

du

0

Z

t+h

t

(v − u)2H−2 E[ζf ′ (Bv )g ′ (Bt )] dv

E[ζg(Bt )f ′′ (Bv )]v 2H−1 dv.

t

But, E[ζg(Bt )f ′ (Bv )Bt ] = E[hD(ζg(Bt )f ′ (Bv )), 1[0,t] iH ] = E[g(Bt )f ′ (Bv )hDζ, 1[0,t] iH ] + E[ζg ′ (Bt )f ′ (Bv )]t2H + 12 E[ζg(Bt )f ′′ (Bv )](t2H + v 2H − (v − t)2H ). Consequently, E[ζg(Bt )(f (Bt+h ) − f (Bt ))] Z T Z t+h = H(2H − 1) du |v − u|2H−2 E[f ′ (Bv )g(Bt )Du ζ] dv 0

+ H(2H − 1)t−2H − H(2H − 1)t−2H

t

Z

t

du

0

Z

1 − H(2H − 1)t−2H 2

Z

t+h

t

t

du

Z

t+h

t

0

Z

0

t

du

Z

t

(v − u)2H−2 E[ζg(Bt )f ′ (Bv )Bt ] dv (v − u)2H−2 E[g(Bt )f ′ (Bv )hDζ, 1[0,t] iH ] dv

t+h

(v − u)2H−2 E[ζg(Bt )f ′′ (Bv )]

× (t2H + v 2H − (v − t)2H ) dv

Asymptotic expansions for fractional SDEs +H

Z

t+h

837

E[ζg(Bt )f ′′ (Bv )]v 2H−1 dv.

t

We deduce that E[ζ(f (Bt+h ) − f (Bt ))|Bt ] Z T Z t+h = H(2H − 1) du |v − u|2H−2 E[f ′ (Bv )Du ζ|Bt ] dv 0

+ Ht−2H Bt

t

Z

t+h

t

− Ht

−2H

Z

t+h

t

((v − t)2H−1 − v 2H−1 )E[ζf ′ (Bv )|Bt ] dv

((v − t)2H−1 − v 2H−1 )E[f ′ (Bv )hDζ, 1[0,t] iH |Bt ] dv

Z H −2H t+h ((v − t)2H−1 − v 2H−1 )(t2H + v 2H − (v − t)2H )E[ζf ′′ (Bv )|Bt ] dv − t 2 t Z t+h +H E[ζf ′′ (Bv )|Bt ]v 2H−1 dv. t



Finally, (22) follows.

First, we apply the previous lemma with ζ = η, E = EQ and B = X, with η given by (10), dQ = η −1 dP and X given by (5). We note that η ∈ D∞,2 (see, e.g., [9] Lemma 6.3.1 and [2] for the expression of Malliavin derivatives via the transfer principle). We can particularly deduce that each random variable Vk , recursively defined by V0 = η and Vk+1 = hDVk , 1[0,t] iH for k ≥ 0, belongs to D1,2 . R t+h In (22), the deterministic terms t |v − u|2H−2 dv, (t + h)2H − t2H and Z

t+h

t

((v − t)2H−1 − v 2H−1 )(t2H + v 2H − (v − t)2H ) dv

have a Taylor expansion in h of the type (15). Lemma 4 allows the first term of the R t+h asymptotic expansion to be obtained using the fact that t φ(s) ds = hφ(t) + o(h) for any continuous function φ. By a recursive argument, again using Lemma 4, we finally  deduce that (15) holds. This concludes the proof of Theorem 2.

Acknowledgements The computations (20) were carried at by A. Neuenkirch. We would like to thank him.

838

S. Darses and I. Nourdin

References [1] Baudoin, F. and Coutin, L. (2007). Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 550–574. MR2320949 [2] Darses, S. and Nourdin, I. (2007). Stochastic derivatives for fractional diffusions. Ann. Probab. 35 1998–2020. MR2349582 ¨ unel, A.S. (1999). Stochastic analysis of the fractional Brownian [3] Decreusefond, L. and Ust¨ motion. Potential Anal. 10 177–214. MR1677455 [4] Garsia, A., Rodemich, E. and Rumsey, H. (1970). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 565–578. MR0267632 [5] Klingenh¨ ofer, F. and Z¨ ahle, M. (1999). Ordinary differential equations with fractal noise. Proc. Amer. Math. Soc. 127 1021–1028. MR1486738 [6] Lyons, T. (1994). Differential equations driven by rough signals. I. An extension of an inequality of L.C.Young. Math. Res. Lett. 1 451–464. MR1302388 [7] Neuenkirch, A., Nourdin, I., R¨ oßler, A. and Tindel, S. (2008). Trees and asymptotic expansions for fractional diffusion processes. Ann. Inst. H. Poincar´e Probab. Statist. To appear. [8] Nourdin, I. and Simon, T. (2006). On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. Statist. Probab. Lett. 76 907–912. MR2268434 [9] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Berlin: Springer. MR2200233 [10] Nualart, D. (2003). Stochastic integration with respect to the fractional Brownian motion and applications. In Stochastic Models (Mexico City, 2002). Contemp. Math. 336 3–39. Providence, RI: Amer. Math. Soc. MR2037156 [11] Nualart, D. and Ouknine, Y. (2002). Regularization of differential equations by fractional noise. Stoch. Process. Appl. 102 103–116. MR1934157 [12] Z¨ ahle, M. (1998). Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 333–374. MR1640795 Received September 2007 and revised September 2007

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