L Huang
email: lhuang@hse.ru

DENSITY STABILITY FOR SOME L ÉVY-DRIVEN STOCHASTIC DIFFERENTIAL EQUATIONS

Keywords: 1991 Mathematics Subject Classification. 60H10, 60H30, 60F05. March 15, 2016 Stochastic Approximation, parametrix, Density Bounds

We consider a Stochastic Differential Equation driven by a Lévy process whose Lévy measure satisfy a tempered stable domination. We study how a perturbation of the coefficients reflects on the density of the solution. We quantify the distance between the densities in term of the proximity of the coefficients. This extend to the stable case the works of [KKM15], where the noise is Gaussian.

Introduction and Main Result

This work is motivated by the recent article of Konakov et al. [START_REF] Konakov | Stability of densities for perturbed diffusions and markov chains[END_REF], where the authors studied the sensitivity of the density of a diffusion process with respect to a perturbation on the coefficients. In this paper, we aim to extend their result, formulated for the Brownian motion to other types of noises. Specifically to the class of Lévy processes whose Lévy measure satisfy the domination [H-1] below. For such Lévy process Z, we consider the following SDE: dX t = b(t, X t )dt + σ(t, X t -)dZ t .

(1.1) We are interested in a perturbation of this equation:

dX n t = b n (t, X n t )dt + σ n (t, X n t -)dZ t . (1.2)
We assume that the coefficients b and σ can be obtained as a limit: b n → b and σ n → σ, and we aim to control the distance between the densities of X t and X n t . Quantifying such a distance can be useful for applicative purposes. See [START_REF] Konakov | Stability of densities for perturbed diffusions and markov chains[END_REF] for a review of the literature.

The idea is to exploit the parametrix series to obtain explicit representations for the densities. The expansions can in turn be linked to the coefficients of the SDE. This allows us to control the distance between the densities using the distance between the coefficients. Let us mention that the parametrix representation proves to be very useful when tracking the dependency of the density in some parameter of the coefficients. For instance, we mention the works of Bally and Kohatsu-Higa [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF] who discuss regularity of the density with respect to the coefficients of the SDE.

The parametrix technique for SDEs has been studied in many occurrences. Let us mention the works of Kolokoltsov [START_REF] Kolokoltsov | Symmetric stable laws and stable-like diffusion[END_REF] who first studied the stable case, more recent works of Knopova and Kulik [START_REF] Knopova | Parametrix method and the weak solution to an sde driven by an α-stable noise[END_REF][START_REF] Knopova | Parametrix construction for certain Lévy-type processes[END_REF], and Huang [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF]. In this paper, we aim to extend the result of Konakov et al. formulated for a Brownian noise to the setting of [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF].

L. HUANG

We denote by [H] the following set of assumptions.

[H-1] (Z t ) t≥0 is a symmetric Lévy process without Gaussian part. We denote by ν its Lévy measure. There is a non increasing function q : R + → R + , µ a bounded measure on S d-1 , and α ∈ (0, 2) such that:

ν(A) ≤ S d-1
+∞ 0 1 A (sθ) q(s) s 1+α dsµ(dθ) = m(A).

(1.3)

We assume one of the following:

[H-1a] µ has a density with respect to the Lebesgue measure on the sphere.

[H-1b] there exists γ ∈ [1, d] such that µ B(θ, r) ∩ S d-1 ≤ Cr γ-1 , with γ + α > d
, and for all s > 0, there exists C > 0 such that q(s) ≤ C q(2s). [H-2] Denoting by ϕ Z the Lévy-Kintchine exponent of (Z t ) t≥0 , there is K > 0 such that :

E e i p,Zt = e tϕZ (p) ≤ e -Kt|p| α , |p| > 1.
(1.4)

[H-3] b, b n : R + × R d → R d are
measurable and bounded and σ, σ n : R + × R d → R d ⊗ R d are bounded and η-Hölder continuous η ∈ (0, 1). Also, when α ≤ 1, we impose b, b n = 0. [H-4] σ, σ n are uniformly elliptic. For all x, ξ ∈ R d , there exists κ > 1 such that:

κ -1 |ξ| 2 ≤ ξ, σ(t, x)ξ ≤ κ|ξ| 2 , κ -1 |ξ| 2 ≤ ξ, σ n (t, x)ξ ≤ κ|ξ| 2 .
(1.5)

[H-5] For all A ∈ B, Borelian, we define the measure:

ν t (x, A) = ν {z ∈ R d ; σ(t, x)z ∈ A} . (1.6)
We assume these measures to be Hölder continuous with respect to the first parameter, that is, for all A ∈ B,

|ν t (x, A) -ν t (x ′ , A)| + |ν n t (x, A) -ν n t (x ′ , A)| ≤ Cδ ∧ |x -x ′ | η(α∧1) m(A).
[H-6] We assume the following stability conditions. There is a sequence ∆ n -→ n→+∞ 0 such that

• Stability for the Lévy measures for all A ∈ B, t ≤ T , sup

x∈R d |ν t (x, A) -ν n t (x, A)| + sup x =x ′ |ν t (x, A) -ν t (x ′ , A) -ν n t (x, A) + ν n t (x ′ , A)| δ ∧ |x -x ′ | η(α∧1)
≤ ∆ n m(A).

• Stability for the drift coefficient:

sup (t,x)∈R+×R d |b(t, x) -b n (t, x)| ≤ C∆ n .
Remark 1.1. The assumptions [H-5] and [H-6] can seem restrictive. However, in the case where σ is real and a general Lévy measurethis assumption follows from the control in Hölder norm formulated in [START_REF] Konakov | Stability of densities for perturbed diffusions and markov chains[END_REF].

It has been shown in [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF] that under assumptions [H-1] to [H-5], the parametrix representation for the densities of (X t ) t≥0 and (X n t ) t≥0 holds. We will use the following notations throughout this paper:

p(t, s, x, z)dz = P(X s ∈ dz|X t = x), p n (t, s, x, z)dz = P(X n s ∈ dz|X n t = x).
Moreover, the following density bound holds:

p(t, T, x, y) + p n (t, T, x, y) ≤ C p(t, T, x, y) := (T -t) -d/α 1 + |y-x| (T -t) 1/α γ+α Q(|y -x|),
where when [H-1a] holds, we take Q = q and set γ = d and when [H-1b] holds, we take Q(ρ) = min(1, ρ γ-1 )q(ρ). The aim of this paper is to quantify the distance between p and p n using p and ∆ n . Our main result is the following:

Theorem 1.1. Fix a finite time horizon T > 0. There exists C > 0, for all t ≤ T and all x, y ∈ R d , such that

|(p -p n )(t, T, x, y)| ≤ C∆ n p(t, T, x, y).
The rest of this paper is organised as follows. In Section 2, we recall the background on the parametrix technique. In Section 3, we prove the main result, using technical results that we prove in Section 4.

Remark 1.2. The bounded drift assumption comes from the fact that in the Lipschitz case, the density estimate for the solution of the SDE involves the flow of the ODE associated to the considered SDE. In our case, we would have to compare the flows θ t,T (y) and θ n t,T (y), respectively solutions of the ODEs:

dx t dt = b(t, x t ), dx n t dt = b n (t, x n t ), x T = x n T = y.
Thus, a comparison of the two flows has to be done. Let us mention a similar procedure developed in [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF],

where the initial flow is compared to its linearization.

The parametrix Setting

In this Section, we recall some facts about the parametrix representation. The reader may consult Kolokoltsov [START_REF] Kolokoltsov | Symmetric stable laws and stable-like diffusion[END_REF] or Huang [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF] for a more extensive presentation. The idea is to set up a parametrix for both the original SDE (X t ) t≥0 and the perturbed one (X n t ) t≥0 , and compare the two representations. To that end, we define two frozen processes. Fix a terminal position y ∈ R d and consider:

Xs = x + s t σ(u, y)dZu, Xn s = x + s t σ n (u, y)dZu.
We omit the dependencies Xs = Xs (t, x, y) and Xn s = Xn s (t, x, y) in order to simplify the notations. Formally, Xs (resp. Xn s ) is the candidate to approximate X s (resp. X n s ). From assumption [H-2], it is clear that the frozen processes have densities. Those densities depend on the frozen parameter y, and we write p(t, T, x, y) = py (t, T, x, y) (resp. pn (t, T, x, y) = py n (t, T, x, y)). The following estimate is proved in [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF]. Lemma 2.1. Fix a terminal time T > 0. There exist C > 0 such that for all x, y ∈ R d : p(t, T, x, y) + pn (t, T, x, y) ≤ C p(t, T, x, y). This estimate is established by comparing the frozen process to the driving noise. We know from Sztonyk [START_REF] Sztonyk | Estimates of tempered stable densities[END_REF] that under [H], density estimates holds for the driving process (Z t ) ≥0 . Then, we can transfer the estimates on the frozen process thanks to the uniform ellipticity of σ, σ n . See [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF] for details.

The second step in the parametrix technique is to link the frozen process to the initial SDE. To that end, we define the integro-differential operator:

L t (z, ∇ x )ϕ(x) = b(t, z), ∇ x ϕ(x) + R d ϕ(x + σ(t, z)ξ) -ϕ(x) -∇ x ϕ(x), σ(t, z)ξ 1 {|ξ|≤1} ν(dξ).
Observe that when x ∈ R d is the initial condition of (1.1), then L t (x, ∇ x ) is the generator of (X t ) t≥0 and for y ∈ R d a fixed terminal point, L t (y, ∇ x ) is the generator of the frozen process. With obvious notations, we also denote L n t (z, ∇ x ) the generators of the perturbed equation, with b n and σ n instead of b and σ. We have the following result.

L. HUANG

Theorem 2.2. Under assumptions [H], X T (resp. X n T ) has a density with respect to the Lebesgue measure, and the following representation holds:

p(t, T, x, y) = k≥0 p ⊗ H (k) (t, T, x, y), p n (t, T, x, y) = k≥0 pn ⊗ H (k) n (t, T, x, y),
where H(t, T, x, y) = L t (x, ∇ x )-L t (y, ∇ x ) p(t, T, x, y) and H n (t, T, x, y) = L n t (x, ∇ x )-L n t (y, ∇ x ) pn (t, T, x, y). Besides, ⊗ denote the space time convolution:

ϕ ⊗ ψ(t, T, x, y) = T t du R d ϕ(t, u, x, z)ψ(u, T, z, y)dz, ϕ (k) (t, T, x, y) = ϕ (k-1) ⊗ ϕ(t, T, x, y),
and ϕ (0) = Id. Finally, the following density bound holds:

p(t, T, x, y) + p n (t, T, x, y) ≤ C p(t, T, x, y).
We refer to Huang [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF] for the proof of this statement. The main idea of the proof is to prove a regularisation property for H that allows to compensate the singularities induced by the generators. We define:

ρ(t, T, x, y) = δ ∧ |y -x| η(α∧1) p(t, T, x, y), H(t, T, x, y) = δ ∧ |y -x| η(α∧1) T -t p(t, T, x, y).
It has been shown in [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF] that:

|H(t, T, x, y)| + |H n (t, T, x, y)| ≤ C H(t, T, x, y)
. Thus, to control the series, it suffices to control the iterated convolution of p and H. We define

ρ 2k (t, T, x, y) = (T -t) kω k!ω 2k (T -t) kω p + (p + ρ) (t, T, x, y) ρ 2k+1 (t, T, x, y) = (T -t) kω (k + 1)!ω 2k+1 (T -t) (k+1)ω p + (T -t) ω (p + ρ) + ρ (t, T, x, y).
The following estimate has been proved in [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF]. For all t ≤ T , there exists C, C > 1 such that for all x, y ∈ R d :

|p ⊗ H (m) |(t, T, x, y) + |p n ⊗ H (m) n |(t, T, x, y) ≤ C p ⊗ H(m) (t, T, x, y) ≤ C m ρ m (t, T, x, y). (2.1)
Using the last estimate, we can prove that the parametrix series is absolutely convergent. We can now use this representation to control the distance between p and p n , in the lines of Konakov et al. [START_REF] Konakov | Stability of densities for perturbed diffusions and markov chains[END_REF].

Proof of the Main Result

In this section, we prove our main result. We rely on technical results whose proofs are postponed to Section 4. The idea is to compare the two series representations and get an estimate for the difference of each term involving ∆ n p(t, T, x, y). The proof is very similar to the proof of the convergence of the parametrix series. Exploiting the parametrix representation for p and p n , we write:

|p(t, T, x, y) -p n (t, T, x, y)| = +∞ k=0 p ⊗ H (k) -pn ⊗ H (k) n (t, T, x, y)
We now control the series term by term. We proceed by induction. The first term above is the difference between the frozen densities. Taking β = 0 in Lemma 4.1 gives: |p(t, T, x, y) -pn (t, T, x, y)| ≤ C∆ n p(t, T, x, y).

(3.1)

To control the next terms, observe that we can split for a general m ∈ N:

p ⊗ H (m+1) -pn ⊗ H (m+1) n = p ⊗ H (m+1) -pn ⊗ H (m) n ⊗ H + pn ⊗ H (m) n ⊗ H -pn ⊗ H (m+1) n = p ⊗ H (m) -pn ⊗ H (m) n ⊗ H + pn ⊗ H (m) n ⊗ (H -H n ). (3.2)
We treat the two terms separately. First, we look of an estimate for p ⊗ H (m) -pn ⊗ H (m)

n . We have the following result:

Lemma 3.1. Fix T > 0. There exists C 3.1 > 1 such that for all x, y ∈ R d , ∀0 ≤ t ≤ T , p ⊗ H (m) -pn ⊗ H (m) n ≤ ∆ n (C 3.1 ) m ρ m (t, T, x, y).
Proof. We set C 3.1 = C 4.3 × C 0 , where C 4.3 is the constant appearing in Lemma 4.3, and we proceed by induction.

(H 0 ): The case k = 0 follows from equation (3.1) since ρ(t, T, x, y) > 0.

(H m ⇒ H m+1 ): We split: p ⊗ H (m+1) -pn ⊗ H (m+1) n = p ⊗ H (m) -pn ⊗ H (m) n ⊗ H + pn ⊗ H (m) n ⊗ (H -H n ).
Now we treat the two terms separately. It has been shown in [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF] that there exists C 0 > 1 such that:

ρ m ⊗ H(t, T, x, y) ≤ C 0 ρ m+1 (t, T, x, y), (3.3)
thus we have:

p ⊗ H (m) -pn ⊗ H (m) n ⊗ H (t, T, x, y) ≤ ∆ n (C 3.1 ) m ρ m ⊗ H(t, T, x, y) ≤ ∆ n (C 3.1 ) m C 0 ρ m+1 (t, T, x, y),
the estimate follows from the fact that

C 0 ≤ C 4.3 × C 0 = C 3.1 .
For the second term, we use estimate (2.1) and Lemma 4.

3 to get pn ⊗ H (m) n ⊗ (H -H n ) (t, T, x, y) ≤ ∆ n (C 3.1 ) m C 4.3 C 0 ρ m ⊗ H(t, T, x, y) ≤ ∆ n (C 3.1 ) m+1 ρ m+1 (t, T, x, y).
For the second part of the right hand side of (3.2), we can use the estimate (2.1) to control

|p n ⊗ H (m)
n | by ρ m and Lemma 4.3 to control |H -H n | by ∆ n H. We then estimate ρ m ⊗ H ≤ Cρ m+1 , thus we obtain a summable bound with ∆ n in factor. To complete the proof, it remains us point out that k≥0 (C 3.1 ) k ρ k < +∞, and that the sum of the series yields the right hand side in Theorem 1.1.

Preliminary results

In this Section, we prove the estimates used un the last section. The proof our main result relies on an estimation of the difference between the two parametrix representations. Therefore, we have to control the difference of the frozen densities, and the kernels H, H n .

Lemma 4.1. Let Φ t (x, ∇ x ) be a pseudo-differential operator, and denote by φ t (x, p) its symbol. Assume that |φ t (x, p)| ≤ C|p| β . There exists C > 0 for all t < T , for all x, y ∈ R d , such that:

Φ t (x, ∇ x ) p -pn (t, T, x, y) ≤ C ∆ n (T -t)
|β| α p(t, T, x, y).

L. HUANG

Note that taking β = 0 yields estimate (3.1).

To prove this Lemma, we rely on arguments developed in [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF]. Keeping the notations, we exhibited how for fixed times, the frozen process can be linked to the marginals of a Lévy process. Fix 0 < t < T , and

x, y ∈ R d . Then XT = x + T t σ(u, y)dZ u (d) = x + S T -t ,
where (S u ) u≥0 is a Lévy process satisfying assumptions [H]. Precisely, let us denote by ν S the Lévy measure of S. There exists µ S such that setting σ v = σ((T -t)v + t, y), we have:

ν S (A) = 1 0 R d 1 {σv ξ∈A} ν(dξ)dv ≤ +∞ 0 S d-1 1 {ρθ∈A} q(ρ) ρ 1+α dρµ S (dθ).
Exploiting the Lévy-Itô decomposition of S u = M u + N u , with M martingale and N compound Poisson process, we can write:

p(t, T, x, y) = R d p M (T -t, y -x -ξ)P NT -t (dξ),
where p M designates the density of the martingale and P NT -t (dξ) the law of the compound Poisson process. The same decomposition holds for Xn

T . Now, we point out that the operator Φ t (x, ∇ x ) acts on the variable x, which is only present in the density of the martingale. Consequently, we have:

Φ t (x, ∇ x ) p -pn (t, T, x, y) = R d Φ t (x, ∇ x ) p M -p M n (T -t, y -x -ξ)P NT -t (dξ) + R d Φ t (x, ∇ x )p M n (T -t, y -x -ξ) P NT -t (dξ) -P N n T -t (dξ) .
To prove Lemma 4.1, we establish the following estimates. We refer to the procedure developed in Sztonyk [START_REF] Sztonyk | Estimates of tempered stable densities[END_REF] to see how these estimates can be used to derive the final density estimate. Lemma 4.2. Fix m ≥ 1. There exists C m , C > 0, such that for all t < T , ∀y, x, ξ ∈ R d ,

Φ t (x, ∇ x ) p M -p M n (T -t, y -x -ξ) ≤ C m ∆ n (T -t) β/α (T -t) -d/α 1 + |y -x -ξ| (T -t) 1/α -m , (4.1)
P NT -t (dξ) -P N n T -t (dξ) ≤ C∆ n +∞ k=0 (T -t) k k! m * k (dξ) + +∞ k=0 (T -t) k ν * k S n (dz) k! , (4.2)
where m(dξ) = 1 {|ξ|≥(T -t) 1/α } m(dξ), and m * k denotes the k th fold convolution of m with itself.

Proof of Lemma 4.2. We prove the first estimate by invariance of the Schwartz's space for the Fourier transform. The Fourier transform of M writes:

ϕ M (p) = R d e i p,ξ -1 -i p, ξ 1 {|ξ|≤(T -t) 1/α } ν S (dξ).
From [H-2], this Fourier transform is integrable, thus we can express the density of M as a Fourier inverse:

p M (T -t, y -x -ξ) = 1 (2π) d R d e -i p,y-x-ξ exp (T -t)ϕ M (p) dp.
Consequently, the action of the pseudo-differential operator Φ t (x∇ x ) writes:

Φ t (x∇ x )p M (T -t, y -x -ξ) = 1 (2π) d R d e -i p,y-x-ξ φ t (x, p) exp (T -t)ϕ M (p) dp.
Since we have the same results for the density of Xn t , when taking the difference, we can write using the mean value theorem:

Φ t (x∇ x ) p M -p M n (T -t, y -x -ξ) = 1 0 dλ 1 (2π) d R d dpe -i p,y-x-ξ φ t (x, p)e (T -t) λϕM (p)+(1-λ)ϕ M n (p) (T -t) ϕ M (p) -ϕ M n (p) = ∆ n (T -t) β α 1 0 dλ 1 (2π) d R d dpe -i p,y-x-ξ × φ t (x, p)e (T -t) λϕM (p)+(1-λ)ϕ M n (p) (T -t) 1+ β α ϕ M (p) -ϕ M n (p) ∆ n = ∆ n (T -t) β α 1 0 dλ (T -t) -d/α (2π) d R d dpe -i q, y-x-ξ (T -t) 1/α (4.3) × φ t x, q (T -t) 1/α e (T -t) λϕM ( q (T -t) 1/α )+(1-λ)ϕ M n ( q (T -t) 1/α ) (T -t) 1+ γ α (ϕ M -ϕ M n )( q (T -t) 1/α ) ∆ n ,
where to get to the last equality, we changed variables to q = (T -t) 1/α p. The announced estimate will now hold when we prove that uniformly in the parameters λ, t, T, n, the expression between brackets is in Schwartz's space.

In the q variable, the above expression is smooth, thanks to the cut-off (see [START_REF] Sztonyk | Estimates of tempered stable densities[END_REF] and the references therein).

For the integrability, observe that from assumptions on φ t (x, p), we can bound φ t x,

q (T -t) 1/α ≤ |q| β (T -t) β/α .
On the other hand, we have for all p ∈ R d :

λϕ M (p) + (1 -λ)ϕ M n (p) = R d e i p,ξ -1 -i p, ξ 1 {|ξ|≤(T -t) 1/α } λν S (dξ) + (1 -λ)ν S n (dξ) ≤ R d e i p,ρς -1 -i p, ρς 1 {ρ≤(T -t) 1/α } q(ρ) ρ 1+α dρ(λµ S + (1 -λ)µ S n )(dς).
We take p = q(T -t) -1/α and change variables to r = ρ(T -t) -1/α . Recall that from the doubling condition, q((T -t) 1/α r) ≤ q(r), we get:

λϕ M (p) + (1 -λ)ϕ M n (p) ≤ C T -t R d e i q,rς -1 -i q, rς 1 {r≤1} q(r) r 1+α dρ(λµ S + (1 -λ)µ S n )(dς) ≤ C|q| α T -t .
The last inequality comes from assumption [H-2]. We now turn to the difference of the exponent ϕ M -ϕ M n . Similarly to the last computation, we deduce from

[H-6] that measures |ν S (dz) -ν S n (dz)| ≤ ∆ n m(dz), so that (ϕ M -ϕ M n ) q (T -t) 1/α ≤ R d e i q,z -1 -i q, z 1 {|z|≤(T -t) 1/α } (ν S -ν S n )(dz) ≤ 1 T -t R+×S d-1
e i q,rς -1 -i q, rς 1 {r≤1} q(r) r 1+α dρµ(dς)

where to get the last inequality, we changed variables to r = (T -t) -1/α |z|. Consequently, we can write:

φ t x, q (T -t) 1/α e (T -t) λϕM +(1-λ)ϕ M n ( q (T -t) 1/α ) (T -t) 1+ β α (ϕ M -ϕ M n )( q (T -t) 1/α ) ∆ n ≤ |q| β+α e -C|q| α .
Thus, the expression between brackets in (4.3) is in Schwartz's space uniformly in the parameters λ, t, T, n, thus so is its Fourier inverse, and estimate (4.1) follows.

We prove the next estimate by induction. We denote νS (dz) = 1 {|z|≤(T -t) 1/α } ν S (dz), the characteristic measure of the compound Poisson process (N u ) u≥0 . We know (see [START_REF] Sztonyk | Estimates of tempered stable densities[END_REF]) that the law of (N u ) u≥0 actually writes

P Nu = e -uνS (R d ) +∞ k=0 u k ν * k S (dz) k!
. The same decomposition holds for N n u . Consequently, when taking the difference:

P Nu (dz) -P N n u (dz) = e -uνS (R d ) +∞ k=0 u k k! ν * k S -ν * k S n (dz) + e -uνS (R d ) -e -uν S n (R d ) +∞ k=0 u k ν * k S n (dz) k! .
From Taylor's formula, we deduce that the second term is controlled by ∆ n

+∞ k=0 u k ν * k S n (dz) k! . It remains us to prove that |ν * k S (dz) -ν * k S n (dz)| ≤ ∆ n C k m * k (dz)
, which can be done by a direct induction since from [H-6] and the definition of the measures ν S , ν S n , we have |ν S n (dz) -ν S (dz)| ≤ ∆ n m(dz).

We now turn to the estimate on the difference of the parametrix kernels H -H n . We have the following estimate:

Lemma 4.3. There exists C 4.3 > 0 for all t < T , for all x, y ∈ R d , such that:

|H -H n |(t, T, x, y) ≤ C 4.3 ∆ n δ ∧ |y -x| η(α∧1) T -t p(t, T, x, y).
By definition of the parametrix kernel, we can split:

(H -H n )(t, T, x, y) = L t (x, ∇ x ) -L t (y, ∇ x ) p -pn (t, T, x, y)
+ L t (x, ∇ x ) -L t (y, ∇ x ) -L n t (x, ∇ x ) + L n t (y, ∇ x ) pn (t, T, x, y).
The first contribution is controlled by Lemma 4.1, since denoting by l t (z, p), l n t (z, p) the symbols of L t (z, ∇ x ), L n t (z, ∇ x ) respectively, we have |l t (z, p)| + |l n t (z, p)| ≤ C|p| α . Thus, we focus on the second contribution. Lemma 4.4. There exists C > 0 for all t < T , for all x, y ∈ R d , such that:

L t (x, ∇ x ) -L t (y, ∇ x ) -L n t (x, ∇ x ) + L n t (y, ∇ x ) pn (t, T, x, y) ≤ C∆ n δ ∧ |y -x| η(α∧1) T -t p(t, T, x, y).
Proof. We focus on the integro-differential part of the generator. For the gradient part, the estimate stems from the stability assumption on b -b n (see [H-6]). We recall the definition of:

ν t (x, A) = ν({z ∈ R d ; σ(t, x)z ∈ A}), ν n t (x, A) = ν({z ∈ R d ; σ n (t, x)z ∈ A}).
With this definition, observe that from the assumption on the Lévy measure, we can write:

L t (x, ∇ x ) -L t (y, ∇ x ) -L n t (x, ∇ x ) + L n t (y, ∇ x ) pn (t, T, x, y) = R d pn (t, T, x + z, y) -pn (t, T, x, y) -∇ x pn (t, T, x, y), z 1 {|z|≤C(T -t) 1/α } × ν t (x, dz) -ν t (y, dz) -ν n t (x, dz) + ν n t (y, dz) ≤ ∆ n δ ∧ |x -y| η(α∧1) Lp n (t, T, x, y),
where we define:

Lp n (t, T, x, y) = R d pn (t, T, x + ρθ, y) -pn (t, T, x, y) -∇ x pn (t, T, x, y), ρθ 1 {ρ≤C(T -t) 1/α } q(ρ) ρ 1+α µ(dθ).
The proof follows from the upper bound Lp n (t, T, x, y) ≤ C 1 T -t p(t, T, x, y). From now, we follow the proof in [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF]. The main idea is to split the operator in small and big jump contributions. Let us define the following operators:

L M ϕ(x) = (T -t) 1/α 0 S d-1 ϕ(x + ρθ) -ϕ(x) -∇ x ϕ(x), ρθ q(ρ) dρ ρ 1+α µ(dθ) L N ϕ(x) = +∞ (T -t) 1/α S d-1 ϕ(x + ρθ) -ϕ(x) q(ρ) dρ ρ 1+α µ(dθ).
Observe that due to the symmetry of µ, we can change the cut-off function so that the identity holds:

L t (x, ∇ x ) -L t (θ t,T (y), ∇ x ) ϕ(x) ≤ L M ϕ(x) + L N ϕ(x).
We now handle separately the small and big jumps. Let us start with the small jumps. We claim that:

L M p(t, T, x, y) ≤ C δ ∧ |y -x| η(α∧1)
T -t p(t, T, x, y). Plugging this identity in the definition of the operator L M yields:

L M p(t, T, x, y) = R d P NT -t (dξ) (T -t) 1/α 0 S d-1 p M (T -t, y -x -ρθ -ξ) -p M (T -t, y -x -ξ) -∇ x p M (T -t, y -x -ξ), ρθ q(ρ) dρ ρ 1+α µ(dθ).
Observe that since the jumps are truncated, the density of the martingale p M is smooth. Moreover, the successive derivatives of p M are be controlled as follows:

∇ β x p M (u, x) ≤ C m u -|β|/α u -d/α 1 + |x| u 1/α -m
, ∀m ≥ 1.

See the proof of Lemma 2 in [START_REF] Sztonyk | Estimates of tempered stable densities[END_REF] or [START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF]. We now use Taylor's formula to expand under the integral:

p M (T -t, y -x -ρθ -ξ) -p M (T -t, y -x -ξ) -∇ x p M (T -t, y -x -ξ), ρθ = 1 2 ∇ 2 x p M (T -t, y -x -ξ)ρ 2 θ (2) + 1 0 ∇ 3 x p M (T -t, y -x -(1 -λ)ρθ -ξ)ρ 3 θ (3) λ 2 2 dλ.

L. HUANG

Now, from the control on the derivative, we obtain for the main part:

R d P NT -t (dξ) (T -t) 1/α 0 S d-1 1 2 ∇ 2 x p M (T -t, y -x -ξ)ρ 2 θ (2) q(ρ) dρ ρ 1+α µ(dθ) ≤ C R d P NT -t (dξ) (T -t) 1/α 0 S d-1 ρ (T -t) 1/α 2 (T -t) -d/α 1 + |y-x-ξ| (T -t) 1/α -m q(ρ) dρ ρ 1+α µ(dθ) = R d P NT -t (dξ) (T -t) -d/α 1 + |y-x-ξ| (T -t) 1/α -m × (T -t) 1/α 0 ρ (T -t) 1/α 2 q(ρ) dρ ρ 1+α ≤ 1 T -t ×µ(S d-1 ).
Finally, we recall that Sztonyk [START_REF] Sztonyk | Estimates of tempered stable densities[END_REF] proved that: R d P NT -t (dξ) T,x,y). Consequently, the bound (4.4) hold for the main term. We now turn to the remainder. Putting the absolute value inside the integrals allows us to interchange the order of integration. We thus have:

(T -t) -d/α 1+ |θ t,T (y)-x-ξ| (T -t) 1/α m ≤ p(t,
S d-1 (T -t) 1/α 0 1 0 R d P NT -t (dξ) ∇ 3 x p M (T -t, y -x -(1 -λ)ρθ -ξ) ρ 3 |θ (3) | λ 2 2 dλq(ρ) dρ ρ 1+α µ(dθ).
We now use the control on the derivative and write:

R d P NT -t (dξ) ∇ 3 x p M (T -t, y -x -(1 -λ)ρθ -ξ) ≤ C (T -t) 3/α R d P NT -t (dξ) (T -t) -d/α 1 + |y-x-(1-λ)ρθ-ξ| (T -t) 1/α -m ≤ C (T -t) 3/α (T -t) -d/α 1 + |x+(1-λ)ρθ-y| (T -t) 1/α α+γ q(|x + (1 -λ)ρθ -y|).
Plugging the last inequality gives the upper bound for the remainder:

S d-1 (T -t) 1/α 0 1 0 (T -t) -d/α 1 + |x+(1-λ)ρθ-y| (T -t) 1/α α+γ q(|x + (1 -λ)ρθ -y|) ρ (T -t) 1/α 3 λ 2 2 dλq(ρ) dρ ρ 1+α µ(dθ).

Now, we have to discuss according to the global regime of the density. If

|x -y| ≤ 1 2 (1 -λ)ρ, or if 1 2 (1 -λ)ρ ≤ |x -y|, then (T -t) -d/α 1 + |x+(1-λ)ρθ-y| (T -t) 1/α α+γ q(|x + (1 -λ)ρθ -y|) ≤ (T -t) -d/α 1 + |x-y| (T -t) 1/α α+γ q(|x -y|) ≍ p(t, T, x, y).
On the other hand, when 1 2 |x -y| ≤ (1 -λ)ρ ≤ 3/2|x -y|, since ρ ≤ (T -t) 1/α , the diagonal regime holds and

(T -t) -d/α 1 + |x+(1-λ)ρθ-y| (T -t) 1/α α+γ q(|x + (1 -λ)ρθ -y|) ≤ (T -t) -d/α ≍ p(t, T, x, y).
Consequently, in both cases, we have:

(T -t) -d/α 1 + |x+(1-λ)ρθ-y| (T -t) 1/α α+γ q(|x + (1 -λ)ρθ -y|) ≤ p(t, T, x, y),
and we thus have:

S d-1 (T -t) 1/α 0 1 0 (T -t) -d/α 1 + |x+(1-λ)ρθ-y| (T -t) 1/α α+γ q(|x + (1 -λ)ρθ -y|) ρ (T -t) 1/α 3 λ 2 2 dλq(ρ) dρ ρ 1+α µ(dθ) ≤ p(t, T, x, y) S d-1 (T -t) 1/α 0 1 0 ρ (T -t) 1/α 3 λ 2 2 dλq(ρ) dρ ρ 1+α µ(dθ) ≤ C 1 T -t p(t, T, x, y).
We now turn to the large jumps. Observe that for the large jumps, the Lévy measure is not singular and we can write:

|L N p(t, T, x, y)| ≤ +∞ (T -t) 1/α S d-1 p(t, T x + ρθ, y)q(ρ) dρ ρ 1+α µ(dθ) + p(t, T, x, y) +∞ (T -t) 1/α S d-1 q(ρ) dρ ρ 1+α µ(dθ) ≤C 1 T -t .
We can thus focus on the remaining integral. Assume first that the diagonal regime holds, that is |x -y| ≤ (Tt) 1/α . In that case, note that p(t, T, x, y) ≍ (T -t) 1/α . Also, since we have the global estimate p(t, T, x+ρθ, y) ≤ C(T -t) -d/α , we can take out the density and the integral yields the estimate L N p(t, T, x, y) ≤ C T -t p(t, T, x, y). Assume now that the off diagonal regime holds, that is |y -x| ≥ (T -t) 1/α . The regime of p(t, T, x + ρθ, y) is given by |y -x -ρθ|. Thus, thanks to the triangle inequality, when |y -x| ≤ 1/2ρ, or when ρ ≤ 1/2|y -x|, the density p(t, T, x + ρθ, y) is off-diagonal with p(t, T, x + ρθ, y) ≤ C p(t, T, x, y), and we can conclude as in the diagonal case. Consequently, the problematic case is when ρ ≍ |y -x|. Indeed, in this case, p(t, T, x + ρθ, y) can be in diagonal regime, when p(t, T, x, y) is still in the off-diagonal regime.

Assume first that [H-1-a] holds, that is that µ has a density with respect to the Lebesgue measure in the sphere: µ(dθ) = f µ (θ)dθ. The remaining integral becomes: p(t, T, x + ρθ, y)1 {ρ≥(T -t) 1/α } q(ρ) ρ d+α f µ (θ)ρ d-1 dρdθ. Now, since ρ ≍ |y -x|, we can take q(ρ) ρ α+d out of the integral to get: Bounding f µ by some constant, we recover the integral of the density p which is lower that one. Consequently, we have: Now, assume that [H-1-b] holds. In this case, we can take out q(r) r 1+α . In the last inequality, we recall that γ ≤ d, so that (T -t) In other words, we can correct the wrong decay by deteriorating the temperation. Consequently, the global upper bound for the kernel is the one announced.

  of[START_REF] Huang | Density estimates for sdes driven by tempered stable processes[END_REF], the frozen density decomposes asp(t, T, x, y) = p S (T -t, y -x) = R d p M (T -t, y -x -ξ)P NT -t (dξ).(4.5)

  , T, x + ρθ, y)1 {ρ≥(T -t) 1/α } q(ρ) ρ d+α f µ (θ)ρ d-1 dρdθ ≤ C q(|y -x|) |y -x| α+d +∞ 0 S d-1 p(t, T, x + ρθ, y)1 {ρ≥(T -t) 1/α } f µ (θ)ρ d-1 dρdθ.

  3/2|y-x| 1/2|y-x| S d-1 p(t, T, x + ρθ, y)1 {ρ≥(T -t) 1/α } q(ρ) ρ 1+α dρf µ (θ)dθ ≤ C T -t T -t |y -x| α+d q(|y -x|) ≍ C T -tp(tT, x, y).

  , T, x + ρθ, y)1 {ρ≥(T -t) , T, x + ρθ, y)1 {ρ≥(T -t)Rewriting the right hand side to make the time dependencies appear :q(|y -x|) |y -x| 1+α = 1 T -t (T -t) 1+ γ-d α |y -x| 1+α q(|y -x|) × (T -t) t) 1+ γ-d α |y -x| 1+α q(|y -x|).

  Now, we write:(T -t) 1+ γ-d α |y -x| 1+α q(|y -x|) = (T -t) 1+ γ-d α |y -x| α+γ × |y -x| γ-1 q(|y -x|).Define now Q(|y -x|) = min(1, |y -x| γ-1 )q(|y -x|), we finally obtain:C T -t (T -t) 1+ γ-d α |y -x| 1+α q(|y -x|) ≤ C T -t (T -t) 1+ γ-d α |y -x| α+γ Q(|y -x|) ≤ C T -tp(t, T, x, y).

Acknowledgments

This article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation if the Global Competitiveness Program.


Luc Bouganim
email: luc.bouganim@inria.fr

Athanasia Katsouraki
email: athanasia.katsouraki@inria.fr

Benjamin Nguyen
email: benjamin.nguyen@insa-cvl.fr

DatShA :A Data Sharing Algebra for access control plans

INTRODUCTION

Online social networks (OSN) are one of the most successful applications that have been created this last decade. Central to these applications is the problem of sharing data, such as texts, photos, geolocation, etc. In most cases, this data is private, and thus is only shared with "friends", a loose concept. Some OSN, such as Google+ let you define circles in order to categorize your friends: friends, close friends, acquaintances, etc. Data can then be shared on finer grain using these circles. However, there is no automatic way to control the simultaneous sharing of data to several circles, with different data precision granularities, such as in the following scenario: Alice wants to share a set of photos with her family, photos with no metadata with her close friends, photos without faces (and without metadata) in a reduced definition with her acquaintances, and does not want to share anything with anyone else.

In this article, we will show how the use of a data sharing algebra to write a variety of access control plans (ACP) can overcome these current limitations of OSN access control. Moreover, by using an algebra, it becomes simple to modify, compose, and share these ACPs. Thus less advanced users can easily reuse ACPs shared on a marketplace by more experienced users. A prototype of the DatShA system has been implemented using XQuery 3.0 and is briefly described.

OVERVIEW OF DatShA

In current OSNs, users have on one side vast quantities of personal data, and on the other side numerous "friends" with whom they wish to share (or sometimes hide) this data. In the current systems, it is not simple to share a specific piece of data while modifying it (e.g. changing its precision or removing some information) depending on the target with whom it is to be shared.

Consider the examples mentioned in the introduction. The ACP related to Alice's close friends should transform a set of photos to another set where metadata is removed. This could be done by simply specifying a regular expression to identify images files to be shared (FileSearch operator -see Figure 1.e), "type" this file to images (PathToImage operatorsee Figure 1.e), then remove metadata (RemoveMeta operator). For Alice's acquaintances, other operators could be invoked: ExtractFaces, ExtractMeta, Select and ReduceDefinition operators (not detailed here).

Thus the objective of DatShA is to provide the infrastructure and an extensible set of generic operators to describe how users want to process their data before sharing it. The operators must be able to be combined on any sort of (semi-structured) data to form an algebra. Finally, ACP may include user-dependent data (e.g., contact files) such that it can also compute the set of users with whom the data is shared, thus linking a plan with its grantee.

BACKGROUND AND RELATED WORKS

Access Control. Many different access control models exist, such as DAC, MAC, or RBAC. Many works exist on enforcing such models in OSN [START_REF] Carminati | Enforcing access control in Web-based social networks[END_REF]. We adopt a complementary approach: the goal of DatShA can be seen as helping the user to write complex views of her data, on which she can then apply any existing AC model (most often, DAC or RBAC).

Data Sharing on OSN. Current works on secure data sharing in OSNs consider various problems such as securing communications, i.e. how to securely share data, once access control has been checked [START_REF] Qinlong | Improving security and efficiency for encrypted data sharing in online social networks[END_REF], or how to write access control policies over data concerning several users [START_REF] Hu | Multiparty Access Control for Online Social Networks: Model and Mechanisms[END_REF]. XQuery 3.0. XQuery 3.0. is not only a declarative query language, it is also Turing complete. Rather than using a traditional language such as Java or C, we have chosen to use XQuery and XQuery Update Facility 3.0. Indeed, evaluating an ACP is done through modifications to a structured document (that we chose to code in XML). Generic operators can be completed by snippets of XPath or XQuery code referring to this data structure, which are directly evaluated by the DatShA system.

THE DATA SHARING ALGEBRA 4.1 General principle

An ACP is seen as a set of sequences of (polymorphic) operators, serialized as an XML file (see Figure 1.a). It takes as input an XML file containing or referencing private sensitive data and produces an XML file containing or referencing data that can be shared or published (See Figure 1.c). Users or sets of users (such as G+ circles) can be given access rights both on atomic data, and on ACPs. As with traditional access control through views, when access rights are given on an ACP, the data accessed during the process is done with the rights of the grantor. For example, if Alice grants Bob the right to view the country she is in, which is computed using her precise GPS coordinates, the execution of the ACP will use Alice's rights, but only return to Bob the final result.

Sharing ACPs through a marketplace

Operators and ACPs can be published on a "marketplace", and described by a short text explaining their goal. They can be downloaded by users in order to fine tune their data sharing policies. Thus, it is possible, even for non-expert users to apply complex access control policies, by combining existing operators or using existing policies. Search, recommendation, or ranking of ACP or operators based on their level of intrusiveness or their usability is possible within the marketplace. The only complexity is to link groups of users to their ACPs, but as the data shared is defined intentionally rather than extensionally, we believe this is much easier to do than with current privacy settings in OSN. most photographed place on Earth, which can be done by computing a "fuzzy" location of all her photos, where the "fuzzy" location is defined by GPS coordinate and an error bar e.g. X=45.23+/-0.01 Y=27.67+/-0.01. Note that this error bar could also be function of the density of photos in a given area.

ACP Example

We present, in Figure 1, the corresponding ACP. This ACP can be written as a sequence of operators (Due to space limitations, the actual XML ACP file is available online at: http://www.benjamin-nguyen.fr/data/ACP.xml). Each operator takes as input an XML file, and produces as output an XML file. It is possible to type-check the ACP at compile time, given that the operators are typed, but this discussion is beyond our scope here. The sequence is the following: for every image in the file path given in the input file, meta-data of the image is extracted, and an operator to reduce the precision of the GPS coordinates is executed. All meta-data apart from the blurred GPS coordinates is removed, pictures are grouped together by fuzzy GPS location, then counted.

Operators can in most cases be implemented in XQuery 3.0 but they can also be ad-hoc operators. Note that DatShA also defines many other operators, including binary (or even n-ary) operators such as the join operator which can be used to join two different sequences, thus needing two input files, and producing a single output file. All these operators have been implemented using XQuery. The framework executing a DatShA ACP has been written in Java, using eXistDB to execute the XQuery fragments.

We believe that the use of operators to build ACPs drastically simplifies the creation, reuse, combination and correctness checking of ACPs.

CONCLUSION AND FUTURE WORK

DatShA can be used to create ACPs to manage access control to one's data. We believe that the full power of DatShA appears when users start sharing ACPs between each other, either by simply reusing an ACP written by another user, or by integrating such an ACP into a more complex one. Indeed, any ACP can be encapsulated as a DatShA operator. Creating an online marketplace, and testing its usability and adoptability by real users is the next step of our work.

  We propose the following example which illustrates well DatShA potential : Alice wants to participate in a survey to determine the Published in Proc. 19th International Conference on Extending Database Technology (EDBT), March 15-18, 2016 -Bordeaux, France: ISBN 978-3-89318-070-7, on OpenProceedings.org. Distribution of this paper is permitted under the terms of the Creative Commons license CCby-nc-nd 4.0

FileSearch: replaces a <fileSpec/> with jokers in a set of file paths looking recursively or not (input "mode") in the directory indicated by input "target" every <fileSpec/> should be replaced by a set of path.

PathToImage: is an operator that replaces every occurrence of a path by an image (an xsd type). An image is at least a <imgPath/> to an "image" file, i.e., a jpg, png, gif, etc.... Initially the image type only includes the <imgPath/> but metadata can be added using the ExtractMeta operator.

ExtractMeta: replaces every occurrence of an image by the same image (every field is copied), and adds metadata that can be extracted from the actual file (e.g. location information embedded in the image).

Fuzzy: is an operator that can be applied to many types. The global behavior is to replace any occurrence of the target by fuzzy values, the precision being informed by the "precision" input, which can be an XPath.

Project: this operator is used like the relational algebra P operator. It replaces the target subtree by the same subtree in which it keeps only the elements (or subtrees) that are mentioned in the "keep" parameters.

Group: replaces the "target" subtree by a restructured one which must be a set. It constructs a <set> of <tuple>s, each containing n+1 elements (where n is the number of "groupBy" elements in the operator specification, in this example, n = 1). The last element of the tuple is a set of elements that share the same value of groupBy (here a set of image having the same GPS value). This operator is implemented by XQuery 3.0. Group By.

Aggregate :

The aggregate operator replaces a set of elements ("target" input) by an aggregate value having the "AggName" name and applying the "AggOperation", which in this case is the XQuery function fn:count().

XQuery