Approximation of the fractional brownian sheet via Ornstein-Uhlenbeck sheet

March 2007, Vol. 11, p. 115–146 www.edpsciences.org/ps

DOI: 10.1051/ps:2007010

APPROXIMATION OF THE FRACTIONAL BROWNIAN SHEET VIA ORNSTEIN-UHLENBECK SHEET

Laure Coutin

¹

and Monique Pontier

¹

Abstract. A stochastic “Fubini” lemma and an approximation theorem for integrals on the plane are used to produce a simulation algorithm for an anisotropic fractional Brownian sheet. The convergence rate is given. These results are valuable for any value of the Hurst parameters (α1, α2)∈]0,1[², αi=¹₂. Finally, the approximation process is iterative on the quarter plane

R

²+.A sample of such simulations can be used to test estimators of the parametersαi, i= 1,2.

Mathematics Subject Classification. 60G60, 60G15, 62M40.

Received November 16, 2005. Revised April 21 and July 22, 2006.

1. Introduction

The aim of this paper is to produce and to study a simulation algorithm for an anisotropic fractional Brownian sheet. An important application of such a simulation is to supply a sample of fractional Brownian sheet almost sure approximations: thus estimators like these defined in [13] can be tested. Recall that the non necessarily Gaussian random fields were started by Samorodnitsky and Taqqu [17] and developed by Cohen [8]. For the 1−dimensional fractional Brownian motion, Meyeret al. [15], Ayache and Taqqu [3] study an approximation of the fractional Brownian motion with Hurst parameterα, using a wavelet decomposition. Ayacheet al.[2] also use a wavelet decomposition but for the anisotropic fractional Brownian sheet. Besides, Bardetet al. [4] did a careful comparison between the different algorithms. Here we recall and develop the results of [7], but there all the proofs are omitted. For the sake of completeness, and because these proofs enlighten the more general cases, we give them in Section 3. The scheme of [7] is as follows: Fubini’s Lemma allows to get a representation of the fractional Brownian motion as a deterministic integral of Ornstein-Ulhenbeck processes; this integral is approximated by a finite sum over a geometric subdivision; besides, [7] introduce some operators on H¨older functions which – applied to the Brownian motion – give the integrands under the deterministic integral. This step allows them to obtain some fine results about these integrands regularity; they obtain a time iterative algorithm using Markov properties of Ornstein-Ulhenbeck processes. Gathering this algorithm and the deterministic integral approximation, they produce an approximation of the fractional Brownian motion. The rate convergence of this algorithm is studied in [7] (cf. p. 162) but without a temporal approximation nor an accuracy evaluation. Here we add the temporal approximation. Their rate convergence is aboutO(logN N^1+1+ββ ) whereβ=α∧(¹₂−α).

More precisely concerning our algorithm, if we choose a simulation accuracy of about N^−η,0 < η < ¹₂, to

Keywords and phrases. random field simulation and approximation, anisotropic fractional Brownian sheet.

1 Universit´e Paul Sabatier, 31062 Toulouse cedex 04;[email protected];

[email protected]

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/psor http://dx.doi.org/10.1051/ps:2007010

produce anIsize image, we need to generateIindependent random variables and a 2nGaussian vector where n=O(N^ηlogN) and the algorithm complexity isO(N^1+ηlogN).

This study is to be done in higher dimensions. For the 2−dimensional case and when the value of one of the Hurst parameters (α1, α2) is more than ¹₂,for computation reasons and not because of the model, Chan et Wood’s algorithms [19] failed. S. L´eger used them in [13] to test some estimators of parametersαi successfully whenαi<¹₂.But, whenα1∨α2> ¹₂, the extended 2–dimension circulant embedding of the covariance matrix should be theoretically non negative definite, but practically it is not; these matrices are not well-conditioned and so Choleski’s method can’t be used.

The generalization from 1−dimension to 2−dimension is not so easy: for instance, Taylor’s formula in 2−dimension involves the cross derivatives and to control the constants we need to detail the computations.

Moreover, as is pointed out in [10], in the one dimensional case, our method can be applied to any Gaussian sheet. It could also be used for any process written as a multiple integral of any function satisfying smooth assumptions. For instance look at

[0,+∞[²g(x, y)µ(dx,dy) where µ is a random measure and g is a Laplace transform. The aim here is to approximate this double integral.

Concerning our algorithm, if we choose a simulation accuracy of aboutI^−η,0 < η < ¹₂,to produce aI² size image, we need to generateI² independent random variables, a 2n×2nGaussian matrix,I(1 +I) 2n-Gaussian vectors wherenis about [logI]I^η, then the algorithm complexity isO([logI]²I^2(1+η)).

The paper is organized as follows: first the problem is set, the 2−dimensional Liouville Brownian sheet is defined, as is the 2−dimensional fractional Brownian sheet that we want to simulate. In Section 3, we first recall a set of deterministic tools built in [7] in order to obtain a discrete approximation of the 1−dimensional Brownian process. We follow the same scheme as the one in [7]: Section 4 extends all these results to our 2−dimensional Brownian sheet; first a theorem for deterministic integrals on the plane is proved, then operators on the set of H¨older 2−dimensional functions are defined and their properties are studied. All this is used to produce a discrete approximation of the 2−dimensional Brownian sheet and the errors are controlled. Finally, an iterative algorithm of the 2−dimensional Brownian sheet synthesis is given thanks to a kind of Markov property. This property relies on the fact that the fractional Brownian motion can be considered as an Ornstein-Uhlenbeck process superposition (cf. [5]). The rate of its convergence is given in Section 5 with a constant which is a random variable, the law of which is known (its extreme values have very low probability). This constant also belongs to anyL^p so, as a byproduct, the approximating sheet uniformly converges to the fractional Brownian sheet in anyL^p. The algorithm parameters are chosen with respect to a given accuracy of the approximation.

The limit of this algorithm is stressed when the parameters α are very near ¹₂,0,1. The largest proofs are provided in Section 7.

2. Problem setting

Let (Ω,A,

P

) be a probability space and dBa white noise sheet on it. The “rectangular” fractional Brownian sheet W^α1,α2 is defined in [13] or [14]:

W_s,t^α1,α2 =

R

^2[(s−u)

α₁−¹₂

+ −(−u)^α+¹⁻¹²][(t−v)^α₊²⁻¹² −(−v)^α+²⁻¹²]dBu,v, (1) (α1, α2)∈]0,1[² and (s, t)∈

R

²+. This random field is null on the axes. Similarly we introduce the Liouville Brownian sheet defined when (s, t)∈

R

²+ by

V_s,t^α1,α2 =

[0,s]×[0,t]

(s−u)^α1−12(t−v)^α2−12dBu,v, (2)

but this one doesn’t have stationary increments. Recall these two expressions in one-dimension:

W_t^α=

R

^[(t−u)

α−¹₂

+ −(−u)^α+⁻¹²]dBu, V_t^α= t

0

(t−u)^α−12dBu, t∈

R

^{. (3)}

We can’t here approximate them directly by a sum since a recursive computation is not feasible (cf. Taqqu [17]). To produce an iterative process of the trajectories of these two random fields, we generalize Carmona et al.’ method [7] to the 2−dimensional case. For the sake of clearness, we start with the Liouville Brownian sheet.

2.1. Liouville Brownian sheet as a superposition of Ornstein-Uhlenbeck processes

Recall the equality for 0< α < ¹₂:

(s−u)^α−12 = 1 Γ(¹₂−α)

_∞

0

x^−α−12e^−x(s−u)dx, s > u. (4)

The key is the following so called stochastic Fubini lemma (cf. Carmonaet al. [7] for instance). Let (n, p)∈

N

^2,

the mixed Lebesgue space and its norm [18] are:

fp₁,p₂ =

R

ⁿ

R

^{p|f(u, a)|}

p₁

du ^p_p²

1 da _p¹

2

(5) Lp₁,p₂(

R

^p×

R

^{n) =} {f :

R

^p×

R

ⁿ→

R

^{, Borelian,}fp₁,p₂<+∞}.

Let us remark that iff ∈ L1,2(

R

ⁿ×

R

^p) using the Cauchy-Schwartz’ inequality:

f²1,2=

R

^p

R

^{n|f(u, a)|da}

2

du

=

R

ⁿ

R

ⁿ

R

^{p|f(u, a)||f}(u, b)|dudadb≤ f²2,1, so yields the inclusionL2,1(

R

^p×

R

ⁿ⁾⊂ L1,2(

R

ⁿ×

R

^p).

Lemma 2.1. Let f ∈ L2,1(

R

^p×

R

^n)anddB be a white noise sheet on

R

^p,then almost surely :

R

ⁿ⁽

R

^{pf(u, a)dBu)da=}

R

^p(

R

^nf(u, a)da)dBu. (6)

Proof. The mapY1:f →

R

ⁿ

R

^{pf(u, v)dBudv}is a continuous linear map on the step functions inL2,1taking its values inL²(Ω).The set of these step functions is dense in L2,1 (cf. Lem. 6.2.11 p. 124 [18]) so this map admits a unique continuous linear extension onL2,1.

Let the mapY2(f) :f →

R

^p

R

^{nf(u, v)dvdBu.}It is a linear continuous map onL2,1(

R

^p×

R

ⁿ⁾⊂ L1,2(

R

ⁿ×

R

^p) with a norm 1 fromL1,2(

R

ⁿ×

R

^{p) toL2}(Ω) so ‘a fortiori’ onL2,1(

R

^p×

R

^n):

Y2(f)²2=

R

^p(

R

^{nf(u, a)da)}

2du=f²1,2≤ f²2,1.

Finally, the mapsYi, i= 1,2,are well defined and coincide on the step functions.

A similar lemma is given in [11] (Lem. 4.1, p. 116) but the assumptions are quite different and are not satisfied in our cases.

This lemma and (4) allow us to prove:

Proposition 2.2. Let (α1, α2) ∈]0,¹₂[²; the process V^α1,α2 admits the following representation ∀(s, t)∈

R

²+, almost surely:

V_s,t^α1,α2 = 1 Γ(¹₂−α1)

1 Γ(¹₂−α2)

R

²+

x^−α1−12y^−α2−12X(x, y, s, t)dxdy (7) where

X(x, y, s, t) =

[0,s[×[0,t[

e^−x(s−u)e^−y(t−v)dBu,v. (8)

Proof. The stochastic Fubini Lemma 2.1 applied to (s, t) ∈

R

²+, n = p = 2, u = (u, v), a = (x, y) and to f(u, v, x, y) =x^−α1−1/2y^−α2−1/2e^{−x(s−u)−y(t−v)}1_[0,t](v)1[0,s](u)1]0,+∞[²(x, y),is correct since

f2,1 =

R

²+

x^−α1−1/2y^−α2−1/2 t

0

s 0

e^{−x2(s−u)−y2(t−v)}dudvdxdy

=

R

^+x−

α₁−1/2

1−e^−2xs

2x dx

R

^+y−

α₂−1/2

1−e^−2yt

2y dy <+∞,

and−αi−1/2∈]−1,−¹₂[, i= 1,2,−αi−1<−1, i= 1,2.

Now forα∈]¹₂,1[ ands≥u,use the identity:

(s−u)^α−12 = s

u

α−1

2

(r−u)^α−32dr=− 1 Γ(¹₂−α)

R

^+x

12−α

s u

e^−x(r−u)drdx. (9) In the caseα1∨α2∈]¹₂,1[,let us introduce the notation

ai =αi+1 2.sign

1 2−αi

, i= 1,2. (10)

Proposition 2.3. Letα1∨α2∈]¹₂,1[. The processV^α1,α2 admits the following representation on(

R

^+)2,∀(s, t), almost surely:

V_s,t^α1,α2 = 1 Γ(¹₂ −α1)

1 Γ(¹₂−α2)

R

²+

x^−a1y^−a2U(x, y, s, t)dxdy (11) whereU =Y, T or Z andY, T, Z are given by:

Y(x, y, s, t) =

[0,s[×[0,t[

X(x, y, r1, r2)dr1dr2, ifαi>1

2, i= 1,2, T(x, y, s, t) =

t 0

X(x, y, s, v)dv,ifα1<1

2 < α2, (12)

Z(x, y, s, t) = s

0

X(x, y, u, t)duifα1>1

2 > α2. (13)

Proof. We only detail the proof when αi > ¹₂, i = 1,2. Once again, the stochastic Fubini Lemma 2.1 is used ((s, t) being fixed) withn=p= 2, u= (u, v)∈[0, s]×[0, t], a= (x, y)∈

R

²+and

f(u, v, x, y) =x^12−α1y^12−α2 s

u

t v

e^{−x(r−u)−y(z−v)}drdz

sinceV_s,t^α1,α2=Y2(f) andf2,1<∞.ThusV_s,t^α1,α2 =Y1(f), meaning that V_s,t^α1,α2=

R

²+

( s

0

t 0

f(u, v, x, y)dBu,v)dxdy.

But s

0

t 0

f(u, v, x, y)dBu,v=x^12−α1y^12−α2 s

0

t 0

s u

t v

e^−x(r−u)e^−y(z−v)drdz

dBu,v. Using once again Lemma 2.1 (here the integrand is continuous with compact support):

s 0

t 0

s u

t v

e^−x(r−u)e^−y(z−v)drdz

dBu,v= s

0

t 0

X(x, y, r, z)drdz,

and the proof is concluded. Whenα1∈]¹₂,1[ orα2∈]¹₂,1[,the proof is almost the same.

Remark 2.4. The process X is continuous with respect the four parameters. Moreover, T, Z, Y are also continuous with respect the four parameters as integrals ofX on the compact sets [0, t] or [0, s] or [0, s]×[0, t].

Proof. cf. Section 7.

These processes, made discrete with respect to (s, t), look like ARMA processes. These fields can be seen as extended 2–dimensions Ornstein-Uhlenbeck processes in the sense of Proposition 2.5. Thus, the fractional Brownian sheet can be seen as a Ornstein-Uhlenbeck processes superposition.

Proposition 2.5. Let (x, y)∈

R

²+, U=X, Y, Z or T thenU(x, y, ., .)is solution to the integral equation:

U(x, y, s, t) = U(0,0, s, t) +xy s

0

t 0

U(x, y, z, τ)dzdτ

−x s

0

U(x,0, z, t)dz−y t

0

U(0, y, s, τ)dτ, (s, t)∈

R

²+. (14) For instance, U(x,0, s, t) =U(0,0, s, t)−xs

0 U(x,0, z, t)dz and U(0, y, s, t) =U(0,0, s, t)−s

0 U(0, y, s, τ)dτ, where

U(0,0, s, t) =Bs,t1_α1∨α2<1 2 +

t 0

Bs,udu1_α1<1 2<α₂+

s 0

Bu,tdu1_α2<1 2<α₁+

s 0

t 0

Bu,vdudv1_α1∧α2>1 2. Proof. Above, we proved that any Gaussian field in this proposition admits a continuous modification. So it is enough to prove the result (s, t) being fixed.

Remark the identity when 0≤u≤s,0≤v≤t:

e^−x(s−u)e^−y(t−v)−1 =xy s

u

t v

e^−x(z−u)e^−y(τ−v)dzdτ−x s

u

e^−x(z−u)dz−y t

v

e^−y(τ−v)dτ.

This identity is integrated on [0, s]×[0, t] with respect to the white noise sheet dB.Using the stochastic Fubini Lemma 2.1 we invert the two integrals (the lemma assumptions are satisfied: the integrands have compact support and are continuous).

So the first term in the decomposition ofX(x, y, s, t)−Bs,t is:

xy s

0

t 0

s u

t v

e^−x(z−u)e^−y(τ−v)dzdτ

dBu,v=xy s

0

t 0

z 0

τ 0

e^−x(z−u)e^−y(τ−v)dBu,v

dzdτ

which is xys 0

t

0X(x, y, z, τ)dzdτ. The two following terms are identified with −xs

0 X(x,0, z, t)dz and

−yt

0X(0, y, s, τ)dτ. This yields (14) for U = X. We now integrate this identity with respect to the two last arguments on [0, s]×[0, t],(respectively with respect to the last argument on [0, t] or the third argument

on [0, s]), yields (14) for U =Y, Z, T.

2.2. Fractional Brownian sheet

Similar results can be obtained for the fractional Brownian sheet due to the equality,s > u,0< α < ¹₂: (s−u)^α₊⁻¹²−(−u)^α+⁻¹² = 1

Γ(¹₂ −α) _∞

0

x^−α−12[e^−x(s−u)1_{u<s}−e^xu1_{u<0}]dx. (15) This equality allows us to show:

Proposition 2.6. Let (α1, α2) ∈]0,¹₂[²; the process W^α1,α2 admits the following representation ∀(s, t) ∈

R

²+, almost surely:

W_s,t^α1,α2 = 1 Γ(¹₂−α1)

1 Γ(¹₂−α2)

R

²+

x^−α1−12y^−α2−12X˜(x, y, s, t)dxdy, (16) where

X˜(x, y, s, t) =

R

^{2fs(x, u)ft(y, v)dBu,v (17)}

andfs(x, u) =1[0,s](u)e^−x(s−u)+1

R

−(u)e^xu(e^−xs−1).

Remark 2.7. The product expansionfs(x, u)ft(y, v) yields that ˜X(x, y, s, t) could also be defined as following:

X(x, y, s, t) +X2(x, y, s, t) +X3(x, y, s, t) +X4(x, y, s, t) where

X2(x, y, s, t) = (e^−xs−1)

]−∞,0[×]0,t[

e^xue^−y(t−v)dBu,v, X3(x, y, s, t) = (e^−yt−1)

]0,s[×]−∞,0[

e^−x(s−u)e^yvdBu,v, (18) X4(x, y, s, t) = (e^−xs−1)(e^−yt−1)

]−∞,0[×]−∞,0[

e^xue^yvdBu,v.

Proof. The function fs,t : (u, v, x, y) → x^−α1−12y^−α2−12fs(x, u)ft(y, v) belongs to L2,1(

R

^2,

R

²+) (cf. (5)), (s, t) being fixed (indeed fs,t2,1 =

R

^{+x−α1−12 1−ex−xs dx.}

R

^{+y−α2−12 1−ey−yt dy <} ∞) and we apply

Lemma 2.1 to this function.

In the case ¹₂ < α <1,the relation (9) is applied tos≥uand s= 0, and solving the integrals with respect tor yields:

(∗) (s−u)^α₊⁻¹² −(−u)^α+⁻¹² =−

R

⁺

x^12−α Γ(¹₂−α)

1_[0,s](u)1−e^−x(s−u)

x +1

R

−(u)e^xu1−e^−xs x

dx.

This relation allows us to prove:

Proposition 2.8. Let α1∨α2∈]¹₂,1[; the processW^α1,α2 admits the following representation on

R

²+, denoting ai=αi+¹₂sign(¹₂−αi), almost surely:

Ws,t^α1,α2 = 1 Γ(¹₂ −α1)

1 Γ(¹₂−α2)

R

²+

x^−a1y^−a2U(x, y, s, t)dxdy, (19) where U(x, y, s, t) =

R

^{2h1s(x, u)h2t(y, v)dBu,v, hi = f if αi < 12, and hi = g if αi > 12, gs(x, u) =}

1_[0,s](u)^{1−e−x(s−u)}_x +1

R

^{−(u)exu1−ex−xs, f} is defined in Proposition 2.6.

Definition 2.9. Let us denote ˜Y(x, y, s, t) = B(gs(x, .)gt(y, .)), Z(x, y, s, t) =˜ B(gs(x, .)ft(y, .)), and T˜(x, y, s, t) =B(fs(x, .)gt(y, .)).

Using Remark 2.4, these fields ˜X,Y ,˜ Z,˜ T˜ are continuous with respect to the four parameters, as sum of continuous fields.

2.3. Algorithm

In Section 5, we will provide a recursive algorithm to approximateW_s,t^α1,α2 by the following Definition 2.10. Letn∈

N

^∗, r, h, k >0:

Wˆ_n,r,h,k^α1,α2(ih, jk) = n j₁,j₂=−n+1

c¹_j1c²_j2U^h,k(r^j1−1, r^j2−1, ih, jk)

wherec^l_j

l =_Γ(1¹ 2−α_l)

(r¹⁻al−1)

1−a_l r^{(1−al)(jl−1)},alis defined in (10) and U^h,k will be defined below depending on the position ofαi with respect to ¹₂.

U^h,k= ˜X^h,k1_]0,1

2[²(α) + ˜Y^h,k1_]1

2,1[²(α) + ˜T^h,k1_]0,1

2[×]¹₂,1[(α) + ˜Z^h,k1_]1

2,1[×]0,¹₂[(α). (20) The key of the recursive algorithm is Proposition 2.5. Let (B_ij^hk,(i, j)∈

N

²) be a Gaussian white noise with variancehk,(B₂^k(x), x∈

R

^{+) and (B}3^h(y), y∈

R

⁺) be Gaussian vectors with covariance function equal to _x+x^k respectively _y+y^h Concerning the first term in U^hk (cf. (18)), we need a double induction as following, given x=r^j1−1, y=r^j2−1, ji∈ {−n+ 1,· · ·, n}, i= 1,2:

∀j∈

N

^, X(x,0, jk) = 0,

X(x, ih, jk) = e^−xhX(x,(i−1)h, jk) +1−e^−xh xh B^h,k_ij , Xˆ(x, ih, jk) = 1

x[B^h,k_ij − X(x, ih, jk)].

In a second step, we set∀i∈

N

^{, X(x, y, ih,0) =}Z(x, y, ih,0) = 0:

X(x, y, ih,(j+ 1)k) = e^−ykX(x, y, ih, jk) +1−e^−yk

yk X(x, ih, jk),

α1∨α2<1 2

, (21)

Z(x, y, ih,(j+ 1)k) = e^−ykZ(x, y, ih, jk) +1−e^−yk

yk X(x, ih, jk),ˆ

α2<1 2 < α1

, (22)

Y(x, y, ih,(j+ 1)k) = 1 y

_j+1

l=1

Xˆ(x, ih, jk)−Z(x, y, ih,(j+ 1)k)

,

α1∧α2> 1 2

. (23)

We now deal with the three other terms in (18) but only in caseα1∨α2<¹₂.

Concerning the fourth term in (18), an exact simulation is possible since X4(x, y, ih, jk) = (1−e^−xih)(1− e^−yjk)B4(x, y) whereB4is a centered Gaussian matrix with covariance function Γ4(x, x, y, y) =_x+x¹ _y+y¹ .

The two other terms in (18), denoted asX2 andX3,are symmetrically obtained by induction:

∀i, X2(x, y, ih,0) = 0,

X2(x, y, ih,(j+ 1)k) = e^−ykX2(x, y, ih, jk)−1−e^−yk

yk (1−e^−xih)B₂^k(x),

∀j, X3(x, y,0, jk) = 0,

X3(x, y,(i+ 1)h, jk) = e^−xhX3(x, y, ih, jk)−1−e^−xh

xh (1−e^−yjk)B3^h(x).

Finally, ˜X^hkis defined as the sum (X+X2+X3+X4)(x, h, ih, jk) (there is similar definitions of ˜Y^hk, T˜^hk, Z˜^hk) and the following will be proved in Theorem 5.3.

Theorem. For all ε >0 there exist n, r, h, k so, ∀T >0, there exists a random variable Cn,r,h,k admitting exponential moments such that the error is uniformly bounded:

sup

s,t∈[0,T]2|W^α1,α2(s, t)−Wˆ_r,n,h,k^α1,α2(s, t)| ≤εCn,r,h,k. (24)

3. Approximation of a fractional Brownian motion

This section develops results of [7] where all the proofs are omitted. Moreover here we bound the temporal approximation error. Using Fubini’s Lemma, we get the representations of the Liouville Brownian motion.

Almost surely,∀α∈]0,¹₂[, ∀t∈

R

^+:

V_t^α= 1 Γ(¹₂−α)

R

^+x−

α−¹₂X(x, t)dxwhereX(x, t) =

[0,t[

e^−x(t−u)dBu; (25)

∀α∈]¹₂,1[,∀t∈

R

^+:

V_t^α= 1 Γ(¹₂ −α)

R

^+x

12−α

Y(x, t)dxwhereY(x, t) =

[0,t[

X(x, z)dz. (26)

Then, we have similar results concerning the fractional Brownian motion, which has stationary increments.

Almost surely,∀α∈]0,¹₂[,∀t∈

R

^+:

W_t^α= 1 Γ(¹₂ −α)

R

+

x^−α−12X(x, t)dx˜ (27)

where

X˜(x, t) =

[0,t[

e^−x(t−u)dBu+ (e^−xt−1)

[−∞,0[

e^xudBu; (28)

∀α∈]¹₂,1[∀t∈

R

^+:

W_t^α= 1 Γ(¹₂−α)

R

^+x

12−αY˜(x, t)dx (29)

where

Y˜(x, t) =Y(x, t) +1−e^−xt x

[−∞,0[

e^xudBu. (30)

The aim here is to approximate the integrals in the representations (27), (29) summarized as W_t^α= 1

Γ(¹₂−α)

R

+

x^−aU(x, t)dx, (31)

whereU = ˜X1_]0,1

2[(α) + ˜Y1_]1

2,1[(α) andaisα+¹₂.sign(¹₂−α).We can deduce from the equation (25) that the processX(x, .) is an Ornstein-Uhlenbeck process:

X(x, t) =Bt−x t

0

X(x, u)du, t≥0. (32)

3.1. Approximation of a deterministic integral

We recall Lemma 13 in [7] using the norm.∞,d,e,(d, e)∈[0,1]², d < e,onC(]0,+∞[,

R

) and the measure µd,eon

R

⁺ which defines the L¹-normfd,e:

g_∞,d,e= sup

x∈]0,+∞[

[x^d1_{x<1}+x^e1_{x≥1}]|g(x)|, µd,e(dx) = min(x^−d, x^−e).dx.

Proposition 3.1. Let(d, e)∈[0,1]², d < e, a functionf ∈L¹([0,+∞[,

R

^{+, µd,e),andg∈C2(]0,+∞),}

R

^)such

that the norms._∞,d,e of the maps g and hg :x → |x∇g(x)|+|D²g(x)|x² are finite. Letr∈]1,2[, n∈N^∗ to define a geometric subdivision of ]0,+∞[, π= (r⁻ⁿ,· · ·, rⁿ): Ij = [r^j−1, r^j] and:

cj=

I_j

f(x)dx, j=−n+ 1,· · ·, n.

Then:

[0,+∞[

g(x)f(x)dx− n j=−n+1

cjg(r^j) ≤ 1

4r^e(r−1)hg∞,d,efµ_d,e

+g∞,d,ef(1]0,r⁻ⁿ]+1_]rn,+∞[)µ_d,e. (33) This proposition will be applied to f : x → x^−a, a = α+ ¹₂.sign(₂¹−α), g(x) = ˜X(x, t) or ˜Y(x, t), to approximateW^α(t),respectively whenα < ¹₂, α > ¹₂.

The proof is omitted: it is quite similar to Theorem 4.1 proof (cf. Sect. 7).

The following corollary will be useful for the time discretization.

Corollary 3.2. Let (d, e, f)andg, hsatisfying the assumptions of Proposition 3.1, then

n i=−n+1

ci[g(rⁱ)−h(rⁱ)]

≤r^efµ_d,eg−h_∞,d,e, (34) wheren etrare defined in Proposition 3.1.

Proof. LetDi= [g(rⁱ)−h(rⁱ)]|ci|.Using the normg−h∞,d,edefinition, we boundDi by

|ci|(r^−id1_{ri<1}+r^−ie1_{ri≥1})g−h∞,d,e≤ |ci|(rⁱ⁻¹)^−d∧(rⁱ⁻¹)^−e.g−h∞,d,e. (35)

Butci=

[rⁱ⁻¹,rⁱ]f(x)dxandµd,e(dx) =x^−d∧x^−edxso

|Di| ≤r^e

[rⁱ⁻¹,rⁱ]

f(x)µd,e(dx)g−h_∞,d,e.

To sum these bounds with respect to iget the conclusion.

3.2. 1–dimensional operators on the H¨ older functions

In the aim to compute by approximation the fractional Brownian process W_t^α (31), t being fixed, we use Proposition 3.1 withg(x) =U(x, t), f(x) =x^−a,(d, e) such thata+d <1< a+e. First, we have to study the smoothness of the Gaussian processesU and the associated functionshU, and their norm.∞,d,e.That is to show the existence of (d, e) as above and moreover that:

(i) U = ˜X,Y˜ belongs toC²(

R

^∗+),tfixed;

(ii) sup_x∈

R

^∗+(x^d1x<1+x^e1x≥1)|xⁱ∂_xⁱiU(x, t)|<∞, i= 0,1,2 (∂_xⁱiU denotes _∂x^∂iiU).

The point (i) will be a consequence of the fact thatX is the image of the Brownian motion by an operatorψ defined below and some similar tricks to be shown for the other processes. The point (ii) (see Corollary 3.10 below) relies on deterministic properties of the operatorsψandθdefined below and on path-wise properties of the Brownian motionB. These operators are defined on the set of H¨older functions polynomially increasing at

−∞(for instance as are the Brownian motion paths).

Define the set of α-H¨older functions onI⊂

R

^,taking value 0 at 0:

H⁰α(I) =

f : f(0) = 0, sup

s,s∈I²,s=s

|f(s)−f(s)|

|s−s|^α <∞

.

Finally, the Banach spaceSαis defined as the subset of the functions inC(]− ∞, T],

R

) such that:

Sα=

f ∈ C(]− ∞, T],

R

^{), sup}

x≤−1

|f(x)|

|x|^1−α <∞

∩ H⁰α([−1, T]),

and iff ∈ Sα let us denote the norm:

fα= sup

s,s∈[−1,T]²,s=s

|f(s)−f(s)|

|s−s|^α + sup

x≤−1

|f(x)|

|x|^1−α· Remark that if f ∈ Sα,∀u≤ −1, |f(u)| ≤ fα|u|^1−αand∀u≥ −1, |f(u)| ≤ fα|u|^α.

For instance, the Brownian motion process Bt =t

0dBu belongs to Sα, ∀α < ¹₂. (cf. Revuz and Yor [16], Th. 2.2 in I.2 and Prop. 1.10(iv) in I.1.)

We introduce two linear operators on Sα, useful to manage the Liouville process since the Fubini Lemma shows thatX(x, s) =ψ(B)(x, s) and we will see below that Y(x, s) = ˆψ(B)(x, s) where:

ψ(f) : (x, s) → f(s)e^−xs+x s

0

e^−xr[f(s)−f(s−r)]dr, ψ(fˆ ) : (x, s) →

s 0

e^−x(s−u)f(u)du.

We now study the smoothness of these functions and we bound their partial derivatives with respect to x, uniformly on

R

^∗+×[0, T].