Ying Hu
email: ying.hu@univ-rennes1.fr

Jianhui Huang
email: jhuang@polyu.edu.hk

Xun Li

Equilibrium for Time-Inconsistent Stochastic Linear-Quadratic Control under Constraint

Keywords: Time-inconsistency, stochastic linear-quadratic control, uniqueness of equilibrium control, forward-backward stochastic differential equation, mean-variance portfolio selection

Introduction

In dynamic decision making, the presence of time inconsistency is often identified in socioeconomic systems and accordingly, its study has important values in various fields, such as engineering, management science, finance and economics (for example, see Kydland and Prescott [START_REF] Kydland | Rules rather than discretion: The inconsistency of optimal plans[END_REF]). More recently, considerable research attention has been paid in studying this family of stochastic time-inconsistent control problems as well as their financial applications. The study on time-inconsistency by economists can be traced back to Strotz [START_REF] Strotz | Myopia and inconsistency in dynamic utility maximization[END_REF] in the 1950s, who initiated the formulation of time-inconsistent decision making as a game between incarnations of the decision maker himself. For the sake of motivation and to make our discussion concrete, let us briefly lay out a simple but illustrating example of time inconsistency in dynamic setting.

                 min u J(u) := Var(X T ) -2λE[X T ],
s.t. dX s = θ ′ u s ds + u ′ s dW s , s ∈ [0, T ],

X 0 = x 0 ,
where X • ∈ R, u • ∈ R 2 , θ = (1, 1) ′ and W • is a two-dimensional standard Weiner process. Its dynamic counterpart yields the following optimization problem over [t, T ], for any t ∈ [0, T ],

(1.2)

                 min u J t (u) := Var t (X T ) -2λE t [X T ], s.t. dX s = θ ′ u s ds + u ′ s dW s , s ∈ [t, T ], X t = x t , where E t [•] = E [•|F t ]
is the conditional expectation and Var t (•) is the conditional variance

under E t [•].
An admissible control u * ,0,x 0

• is optimal for the problem (1.1) if J(u * ,0,x 0 • ) = min u J(u).

Also, we define the optimal control u * ,t,xt

• for the problem (1.2) similarly. We say the problem (1.2) is time-consistent if, for any t ∈ [0, T ], it holds that (1.3) u * ,t,X * ,0,x 0 t s = u * ,0,x 0 s for t ≤ s ≤ T.

However, applying results obtained in [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF][START_REF] Hu | Constrained stochastic LQ control with random coefficients, and application to portfolio selection[END_REF][START_REF] Li | Continuous-time Markowitz's model with constraints on wealth and portfolio[END_REF][START_REF] Li | Dynamic mean-variance portfolio selection with no-shorting constraints[END_REF], we have an optimal control u * ,0,x 0 s = (x 0 -X * ,0,x 0 s + λe 2T )θ, 0 ≤ s ≤ T for Problem (1.1), where X * ,0,x 0 s = x 0 + 2 s 0 (x 0 -X * ,0,x 0 v + λe 2T )dv + s 0 (x 0 -X * ,0,x 0 v + λe 2T )θ ′ dW s .

Again, as before we should have an optimal control on [t, T ], u * ,t,X * ,0,x 0 t s = (X * ,0,x 0 t -X * ,t,X * ,0,x 0 t s + λe 2(T -t) )θ, t ≤ s ≤ T for Problem (1.2), where X * ,t,X * ,0,x 0 t s = X * ,0,x 0 t + 2 s t (X * ,0,x 0 t -X * ,t,X * ,0,x 0 t v + λe 2(T -t) )dv + s t (X * ,0,x 0 t -X * ,t,X * ,0,x 0

t v + λe 2(T -t) )θ ′ dW v .
It is obvious that u * ,t,X * ,0,x 0 t s = u * ,0,x 0 s for t ≤ s ≤ T . The dynamic optimization problem (1.2) is called time-inconsistent since (1.3) fails to hold. Therefore, time inconsistency reflects that an optimal strategy at present may no longer be optimal in the future.

In response, Strotz suggested two possible fundamental schemes to circumvent time inconsistency: (i) "He may try to precommit his future activities either irrevocably or by contriving a penalty for his future self if he should misbehave", which is named the strategy of pre-commitment; and (ii) "He may resign himself to the fact of intertemporal conflict and decide that his 'optimal' plan at any date is a will-o'-the-wisp which cannot be attained, and learn to select the present action which will be best in the light of future disobedience", which is termed the strategy of consistent planning. The strategy of consistent planning is also called the time-consistent policy in the literature. For a dynamic mean-variance model, Basak and Chabakauri [START_REF] Basak | Dynamic mean-variance asset allocation[END_REF] reformulated it as an intrapersonal game model where the investor optimally elicits the policy at any time t, on the premise that he has already decided time-consistent (equilibrium) policies applied in the future.

The game formulation is tractable to capture time inconsistency when the underlying time setting is (finite or countable) discrete. Nevertheless, when the time setting is continuous, the formulation should be generalized or modified in different ways. Additionally, some tailor-made arguments, to be shown later, should also be introduced to handle the continuous-time setting. We remark that it is still unclear which is the best one among different definitions of a solution to time-inconsistent decision problem. Mathematically, both the existence and the uniqueness of a solution make a definition more acceptable. Although it is common that a game problem admits multiple solutions, the time-inconsistent decision problem is a decision problem for single player, and hence, an identical value process for all solutions is considered to be more reasonable even if the control may allow multiple solutions. Instead of seeking an "optimal control", some kind of equilibrium controls are worthy to be developed in both theoretical methodology and numerical computation algorithm. This is mainly motivated by practical applications in statistical economics and has recently attracted considerable interest and attempts.

Yong [START_REF] Yong | Time-inconsistent optimal control problems and the equilibrium HJB equation[END_REF] and Ekeland and Pirvu [START_REF] Ekeland | Investment and consumption without commitment[END_REF] established the existence of equilibrium solutions, with their own definitions for equilibrium solutions, for the time inconsistency caused by hyperbolic discounting. Grenadier and Wang [START_REF] Grenadier | Investment under uncertainty and timeinconsistent preferences[END_REF] also studied the hyperbolic discounting problem in an optimal stopping model. In a Markovian system, Björk and Murgoci [START_REF] Björk | A general theory of Markovian time inconsistent stochastic control problems[END_REF] proposed a definition of a general stochastic control problem with time-inconsistent terms, and presented some sufficient condition for a control to be a solution by a system of partial differential equations. They constructed some solutions for some examples including an LQ one, but it looks very hard to find not-too-harsh condition on parameters to ensure the existence of a solution. Björk, Murgoci and Zhou [START_REF] Björk | Mean-variance portfolio optimization with state dependent risk aversion[END_REF] also derived an equilibrium for a meanvariance portfolio selection with state-dependent risk aversion. Basak and Chabakauri [START_REF] Basak | Dynamic mean-variance asset allocation[END_REF] studied an equilibrium strategy for a mean-variance portfolio selection problem with constant risk aversion and got more details on the constructed solution. Hu, Jin and Zhou in [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF] generalized the discrete-time game formulation for an LQ control problem with time-inconsistent terms in a non-Markovian system, which is slightly different from the one in Björk and Murgoci [START_REF] Björk | A general theory of Markovian time inconsistent stochastic control problems[END_REF], and constructed an equilibrium strategy for quite general LQ control problem, including a non-Markovian system, and then in [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium[END_REF], they proved that the constructed equilibrium strategy is unique. Bensoussan, Frehse and Yam [START_REF] Bensoussan | Mean Field Games and Mean Field Type Control Theory[END_REF] introduced a class of time-inconsistent game problems of mean-field type and provided their equilibrium solutions; Karnam, Ma and Zhang [START_REF] Karnam | Dynamic approaches for some time inconsistent problems[END_REF] introduced the idea of "dynamic utility" under which the original time-inconsistent problem (under the fixed utility) is transferred to time-consistent one. In addition, Cui, Li, Wang and Zhu [START_REF] Cui | Better than dynamic mean-variance: Time inconsistency and free cash flow stream[END_REF] showed that the multi-period mean-variance problem does not satisfy time consistency in efficiency and developed a revised mean-variance strategy. By relaxing the self-financing restriction to allow the withdrawal of money from the market, the revised mean-variance strategy dominates the original dynamic mean-variance strategy in the mean-variance space. Furthermore, Cui, Li, Li and Shi [START_REF] Cui | Time consistent behavioral portfolio policy for dynamic mean-variance formulation[END_REF] further investigated the time-consistent strategy for a behavioral risk aversion model by solving a nested mean-variance game formulation.

Recently, Bensoussan, Wong, Yam and Yung [START_REF] Bensoussan | Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting[END_REF] studied the time-consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete-time and continuous-time settings and showed that the discrete-time equilibrium controls converge to that in the continuous-time setting. In their work, the cost functional just includes the terminal mean and variance terms without the running cost part. In this paper, we further consider a class of time-inconsistent stochastic LQ control problems under control constraint involving the integral part in the cost functional. Also, we investigate these problems within the framework of random coefficients. Hu, Jin and Zhou [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF][START_REF] Hu | Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium[END_REF] introduced the new methodology which distinguishes significantly from those in classic control (without constraints) theory in the literature, to tackle time-inconsistent stochastic LQ control problem without constraints. Our work aims to further develop the new methodology proposed in [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF] to cope with the mathematical difficulties rooted in the presence of control constraints. We first prove that the existence of an equilibrium solution is equivalent to the existence of a solution to some forward-backward stochastic differential equations (FBSDE) with constraints. Then we present an explicit solution to equilibrium for mean-variance portfolio selection under convex cone constraint and show that the constructed solution is unique. Finally, we illustrate the established results using examples. In particular, we compare our results with that in Bensoussan, Wong, Yam and Yung [START_REF] Bensoussan | Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting[END_REF] for the deterministic coefficients. Our current work is one further step toward understanding the role of input constraint in time-inconsistency decision making, and we expect to see more research progress along this direction.

The rest of this paper is organized as follows. In Section 2, we give the formulation of the LQ control problem without time consistency under constraint. Then we give an equivalent characterization of a solution by a system of forward-backward stochastic differential equations in Section 3. Finally in Section 4, we give an explicit solution to equilibrium for mean-variance portfolio selection under convex cone constraint and show that the thus constructed solution is unique.

Problem Formulation

Let (W t ) 0≤t≤T = (W 1 t , • • • , W d t )
0≤t≤T be a d-dimensional Brownian motion on a probability space (Ω, F , P). Denote by (F t ) the augmented filtration generated by (W t ).

We will use the following notation. Let p ≥ 1.

S l : the set of symmetric l × l real matrices.

L p G (Ω; R l ): the set of random variables ξ : (Ω, G) → (R l , B(R l )) with E [|ξ| p ] < +∞. L ∞ G (Ω; R l ): the set of essentially bounded random variables ξ : (Ω, G) → (R l , B(R l )). L p G (t, T ; R l ): the set of {G s } s∈[t,T ] -adapted processes f = {f s : t ≤ s ≤ T } with E T t |f s | p ds < ∞. L ∞ G (t, T ; R l ): the set of essentially bounded {G s } s∈[t,T ] -adapted processes. L p G (Ω; C(t, T ; R l )): the set of continuous {G s } s∈[t,T ] -adapted processes f = {f s : t ≤ s ≤ T } with E sup s∈[t,T ] |f s | p < ∞.
We will often use vectors and matrices in this paper, where all vectors are column vectors. For a matrix M, define M ′ as transpose and |M| = i,j m 2 ij as Frobenius norm of a matrix M, respectively. Now we introduce the model under consideration in this paper.

Let T > 0 be given and fixed. The controlled system is governed by the following stochastic differential equation (SDE) on [0, T ]:

(2.1)

dX s = [A s X s + B ′ s u s + b s ]ds + d j=1 [C j s X s + D j s u s + σ j s ]dW j s , X 0 = x 0 ,
where A is a bounded deterministic function on [0, T ] with values in R n×n , B, C j , D j are all essentially bounded adapted processes on [0, T ] with values in R l×n , R n×n , R n×l , respectively, and b and σ j are stochastic processes in

L ∞ F (0, T ; R n ). Let K be a given convex set in R l . The process u ∈ p>2 L p F (0, T ; K) is the control, and X ∈ L p F (Ω; C(0, T ; R n )) is
the corresponding state process with initial value x 0 ∈ R n and with u ∈ L p F (0, T ; K). When time evolves to t ∈ [0, T ], we need to consider the controlled system starting from t and state x t ∈ L p Ft (Ω; R n ):

(2.2)

dX s = [A s X s + B ′ s u s + b s ]ds + d j=1 [C j s X s + D j s u s + σ j s ]dW j s , X t = x t .
For any control u ∈ L p F (t, T ; K), there exists a unique solution

X t,xt,u ∈ L p F (Ω; C(t, T ; R n )).
At time t with the system state X t = x t , our aim is to minimize

J(t, x t ; u) △ = 1 2 E t T t [ Q s X s , X s + R s u s , u s ] ds + 1 2 E t [ GX T , X T ] - 1 2 hE t [X T ] , E t [X T ] -µ 1 x t + µ 2 , E t [X T ] (2.3) over u ∈ L p F (t, T ; K), where X = X t,xt,u
, and

E t [•] = E [•|F t ].
In the above Q and R are both positive semi-definite and essentially bounded adapted processes on [0, T ] with values in S n and S l respectively, G, h, µ 1 , µ 2 are constants in S n , S n , R n×n , R n respectively, and moreover G is positive semi-definite.

We define an equilibrium (control) in the following manner. Given a control u * , for any

t ∈ [0, T ), ε > 0 and v ∈ p>2 L p F (t, T ; K), define (2.4) u t,ε,v s = u * s + (v s -u * s )1 s∈[t,t+ε) , s ∈ [t, T ]. Definition 2.1 Let u * ∈ p>2
L p F (0, T ; K) be a given control and X * be the state process corresponding to u * . The control u * is called an equilibrium if

lim inf ε↓0 J(t, X * t ; u t,ε,v ) -J(t, X * t ; u * ) ε ≥ 0,
where u t,ε,v is defined by (2.4), for any t ∈ [0, T ) and v ∈ p>2 L p F (t, T ; K).

Remark 2.2

There is some difference between our definition and that of [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF], because there is a control constraint K in our situation. Note that the convexity of K is not needed in our definition.

Necessary and Sufficient Condition of Equilibrium Controls

In this section, we present a general necessary and sufficient condition for equilibria. This condition is made possible by a stochastic Lebesgue differentiation theorem involving conditional expectation.

To proceed, we start with some relevant known result from [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF]. Let u * be a fixed control and X * be the corresponding state process. For any t ∈ [0, T ), define in the time

interval [t, T ] the processes (p(•; t), (k j (•; t)) j=1,••• ,d ) ∈ L 2 F (t, T ; R n ) × (L 2 F (t, T ; R n )) d as the unique solution to (3.1)                      dp(s; t) = -A ′ s p(s; t) + d j=1 (C j s ) ′ k j (s; t) + Q s X * s ds + d j=1 k j (s; t)dW j s , s ∈ [t, T ], p(T ; t) = GX * T -hE t [X * T ] -µ 1 X * t -µ 2 . Notice that if u * ∈ L p F (0, T ; K), then p(•; t) ∈ L p F (t, T ; R n ) in fact. Furthermore, define (P (•; t), (K j (•; t)) j=1,••• ,d ) ∈ L ∞ F (t, T ; S n ) × (L 2 F (t, T ; S n )) d as the unique solution to (3.2)                              dP (s; t) = -A ′ s P (s; t) + P (s; t)A s + d j=1 [(C j s ) ′ P (s; t)C j s + (C j s ) ′ K j (s; t) + K j (s; t)C j s ] + Q s ds + d j=1 K j (s; t)dW j s , s ∈ [t, T ], P (T ; t) = G.
Notice that neither the terminal condition nor the coefficients of this equation depend on t; so it can be taken as a BSDE on the entire time interval [0, T ]. Denote its solution as (P (s), K(s)), s ∈ [0, T ]. It then follows from the uniqueness of the solution to BSDE that (P (s; t), K(s; t)) = (P (s),

K(s)) at s ∈ [t, T ] for any t ∈ [0, T ].
The following estimate under local spike variation is reproduced from [10, Proposition

3.1]. Proposition 3.1 For any t ∈ [0, T ), ε > 0 and v ∈ p>2 L p F (0, T ; K), define u t,ε,v by (2.4).
Then

(3.3) J(t, X * t ; u t,ε,v )-J(t, X * t ; u * ) = E t t+ε t Λ(s; t), v s -u * s + 1 2 H(s)(v s -u * s ), v s -u * s ds+o(ε),
where Λ(s; t)

△ = B s p(s; t) + d j=1 (D j s ) ′ k j (s; t) + R s u * s and H(s) △ = R s + d j=1 (D j s ) ′ P (s)D j s .
In view of Proposition 3.1 and the fact that H(s) 0, it is straightforward to get the following sufficient condition of an equilibrium.

Corollary 3.2 A control u * ∈ L p F (0, T ; K) is an equilibrium if (3.4) lim inf ε↓0 1 ε t+ε t E t [ Λ(s; t), v s -u * s ] ds ≥ 0, a.s., ∀t ∈ [0, T ).
The necessary condition is somewhat different.

Proposition 3.3 If a control u * ∈ L p F (0, T ; K) is an equilibrium, then for θ ∈ (0, 1], (3.5) lim inf ε↓0 1 ε E t t+ε t Λ(s; t), v s -u * s + θ 2 H(s)(v s -u * s ), (v s -u * s ) ds ≥ 0. Proof: We set, for v ∈ p>2 L p F (0, T ; K) and θ ∈ (0, 1], vs = u * s + θ(v s -u * s ) ∈ L p F (t, T ; K). Then (3.6) J(t, X * t ; u t,ε,v ) -J(t, X * t ; u * ) = E t t+ε t θ Λ(s; t), v s -u * s + θ 2 2 H(s)(v s -u * s ), (v s -u * s ) ds + o(ε).
Hence,

lim inf ε↓0 1 ε E t t+ε t Λ(s; t), v s -u * s + θ 2 H(s)(v s -u * s ), (v s -u * s ) ds ≥ 0. Q.E.D.
The next result provides a key property for the solution to BSDE (3.1), and represents the process Λ(s; t) in a special form.

Proposition 3.4 For any given pair of state and control processes (X * , u * ), the solution to BSDE (3.1) satisfies k(s; t 1 ) = k(s; t 2 ) for a.e. s ≥ max (t 1 , t 2 ). Moreover, there exist

λ 1 ∈ L p F (0, T ; R l ), λ 2 ∈ L ∞ F (0, T ; R l×n ) and ξ ∈ L p (Ω; C(0, T ; R n )), such that Λ(s; t) has the representation Λ(s; t) = λ 1 (s) + λ 2 (s)ξ t .
           dp(s; t) = - d j=1 ψ(s)(C j s ) ′ ψ(s) -1 kj (s; t) + ψ(s)Q s X *
; t) = ψ(s) -1 p(s) -ψ(s) -1 (hE t [X * T ] + µ 1 X * t + µ 2 ) = p(s) + ψ(s) -1 ξ t ,
where

ξ t := -hE t [X * T ] -µ 1 X * t -µ 2 defines the process ξ ∈ L p F (Ω; C(0, T ; R n )) and p(s) := ψ(s) -1 p(s) defines the process p ∈ L p F (Ω; C(0, T ; R n )). Hence, Λ(s; t) = B s p(s; t) + d j=1 (D j s ) ′ k j (s; t) + R s u * s = B s p(s) + d j=1 (D j s ) ′ k j (s) + R s u * s + B s ψ(s) -1 ξ t = λ 1 (s) + λ 2 (s)ξ t ,
where λ 1 (s) := B s p(s)

+ d j=1 (D j s ) ′ k j (s) + R s u * s and λ 2 (s) := B s ψ(s) -1 . Q.E.D.
We now set out to derive our general necessary and sufficient condition for equilibrium controls. Although (3.4) and (3.5) already provide characterizing conditions, they are nevertheless not very useful because they involve a limit. It is tempting to expect that the limit therein is Λ(t; t), in the spirit of the Lebesgue differentiation theorem. However, one needs to be very careful, since in both (3.4) and (3.5), the conditional expectation with respect to F t is involved. The following lemma generalizes Lemma 3.4 in [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium[END_REF] from q = 2 to any q > 1, and we provide a complete proof here for the sake of self-containedness.

Lemma 3.5 Let Y ∈ L q F (0, T ; R), q > 1 be a given process. If lim inf ε↓0 1 ε t+ε t E t [Y s ] ds ≥ 0, a.e. t ∈ [0, T ), a.s., then Y t ≥ 0, a.e. t ∈ [0, T ), a.s..
Proof: Since L q * F T (Ω; R + ) (q * is the conjugate of q) is a separable space, it follows from the (deterministic) Lebesgue differentiation theorem that there is a countable dense subset

D ⊂ L q * F T (Ω; R + ) ∩ L ∞ F T (Ω; R + )
, such that for almost all t, we have

(3.8) lim ε↓0 1 ε t+ε t E [Y s η] ds = E [Y t η] , ∀η ∈ D,
and lim

ε↓0 1 ε t+ε t E [Y q s ] ds = E [Y q t ] . For any η ∈ D, define η s = E s [η]. Then E [Y s η] = E [Y s η s ].
We have the following estimates:

lim ε↓0 1 ε t+ε t E [Y s (η s -η t )] ds ≤ lim ε↓0 1 ε t+ε t E [Y q s ] ds 1 q t+ε t E (η s -η t ) q * ds 1 q * = lim ε↓0 1 ε t+ε t E [Y q s ] ds 1 q 1 ε t+ε t E (η s -η t ) q * ds 1 q * ≤ lim ε↓0 1 ε t+ε t E [Y q s ] ds 1 q sup s∈[t,t+ε] E (η s -η t ) q * 1 q * ≤ q lim ε↓0 1 ε t+ε t E [Y q s ] ds 1 q E (η t+ε -η t ) q * 1 q * = 0,
where the last inequality is due to Doob's martingale inequality as η s is an L q * -integrable martingale. Hence for any η ∈ D,

E [Y t η t ] = E [Y t η] = lim ε↓0 1 ε t+ε t E [Y s η] ds = lim ε↓0 1 ε t+ε t E [Y s η s ] ds = lim ε↓0 1 ε t+ε t E [Y s η t ] ds = lim ε↓0 1 ε t+ε t E [(E t [Y s ])η t ] ds = lim ε↓0 E 1 ε t+ε t E t [Y s ] ds η t .
Since (recall that q * is the conjugate of q)

E 1 ε t+ε t E t [Y s ] ds q = 1 ε q E t+ε t E t [Y s ] ds q ≤ 1 ε q E t+ε t ds q/q * t+ε t E t [Y s ] q ds = 1 ε q E ε q/q * t+ε t E t [Y s ] q ds = 1 ε E t+ε t E t [Y s ] q ds ≤ 1 ε t+ε t E [Y q s ] ds,
and lim

ε↓0 1 ε t+ε t E [Y q s ] ds = E [Y q t ]
, there exists a constant δ t > 0, such that

E 1 ε t+ε t E t [Y s ] ds q ≤ 2E [Y q t ] , ∀ ε ∈ (0, δ t ).
This implies that Proof: Recall that we have the representation Λ(s; t) = λ 1 (s) + λ 2 (s)ξ t . Then

1 ε t+ε t E t [ Λ(s; t), v s -u * s ] ds - 1 ε t+ε t E t [ Λ(s; s), v s -u * s ] ds = 1 ε t+ε t E t [ λ 2 (s)(ξ t -ξ s ), v s -u * s ] ds. Hence (3.10) lim ε↓0 1 ε t+ε t E t [ Λ(s; t), v s -u * s ] ds - 1 ε t+ε t E t [ Λ(s; s), v s -u * s ] ds = 0.
If (3.9) holds, then from (3.10),

lim inf ε↓0 1 ε t+ε t E t [ Λ(s; t), v s -u * s ] ds = lim inf ε↓0 1 ε t+ε t E t [ Λ(s; s), v s -u * s ] ds ≥ 0,
i.e. (3.4) holds, and from Corollary 3.2, u * is an equilibrium. Now we suppose that u * is an equilibrium, then from (3.5) and (3.10), lim inf

ε↓0 1 ε E t t+ε t Λ(s; s), v s -u * s + θ 2 H(s)(v s -u * s ), (v s -u * s ) ds ≥ 0.
Then, from Lemma 3.5, for any θ ∈ (0, 1],

Λ(t; t), v t -u * t + θ 2 H(t)(v t -u * t ), (v t -u * t ) ≥ 0.
Sending θ → 0 + , we obtain (3.9).

Q.E.D.

When n = 1, the state process X is a scalar-valued process evolving by the dynamics

(3.11) dX s = [A s X s + B ′ s u s + b s ]ds + [C s X s + D s u s + σ s ] ′ dW s ; X 0 = x 0 ,
where A is a bounded deterministic scalar function on [0, T ]. The other parameters B, C, D are all essentially bounded and F t -adapted processes on [0, T ] with values in R l , R d , R d×l , respectively. Moreover, b ∈ L ∞ F (0, T ; R) and σ ∈ L ∞ F (0, T ; R d ). In this case, the two adjoint equations for the equilibrium become

         dp(s; t) = -[A s p(s; t) + C ′ s k(s; t) + Q s X * s ]ds + k(s; t) ′ dW s , s ∈ [t, T ], p(T ; t) = GX * T -hE t [X * T ] -µ 1 X * t -µ 2 ;
(3.12)

                 dP (s; t) = -[(2A s + |C s | 2 )P (s; t) + 2C ′ s K(s; t) + Q s ]ds + K(s; t) ′ dW s , s ∈ [t, T ], P (T ; t) = G.
(3.13) For reader's convenience, we state here the n = 1 version of Theorem 3.6.

Theorem 3.7 An admissible control u * ∈ L p F (0, T ; R l ) is an equilibrium control if and only if, for any time t ∈ [0, T ), (i) the system of SDEs

(3.14)                          dX * s = [A s X * s + B ′ s u * s + b s ]ds + [C s X * s + D s u * s + σ s ] ′ dW s , s ∈ [0, T ], X * 0 = x 0 , dp(s; t) = -[A s p(s; t) + C ′ s k(s; t) + Q s X * s ]ds + k(s; t) ′ dW s , s ∈ [t, T ], p(T ; t) = GX * T -hE t [X * T ] -µ 1 X * t -µ 2 , t ∈ [0, T ],
admits a solution (X * , p, k);

(ii) Λ(•; t) △ = p(•; t)B • + D ′ • k(•; t) + R • u *
• satisfies the condition (3.9).

Mean-Variance Equilibrium Strategies in a Market under Convex Cone Constraint

As an application of the time-inconsistent LQ theory, we consider the continuous-time Markowitz mean-variance portfolio selection model in a market under convex cone constraint with random model coefficients. We aim to establish the existence and uniqueness of the equilibrium strategy. The model is mathematically a special case of the general LQ problem formulated earlier in this paper, with n = 1 naturally.

We use the classical setup. The wealth equation is governed by the SDE (4.1)

         dX s = [r s X s + θ ′ s u s ]ds + u ′ s dW s , s ∈ [t, T ], X t = x t ,
where r is the (bounded) deterministic interest rate function, and θ is the essentially bounded stochastic risk premium process. In particular, x 0 > 0.

The objective at time t with state X t = x t is to minimize

J(t, x t ; u) △ = 1 2 Var t (X T ) -γ t (x t )E t [X T ] (4.2) = 1 2 E t [X 2 T ] -(E t [X T ]) 2 -µ 1 x t E t [X T ].
There are two sources of time-inconsistency in this model, one from the variance term and the other from the state-dependent tradeoff between the mean and the variance. We suppose that the portfolio constraint K is a convex cone here.

The FBSDE (2.1) and (3.1) specializes to

(4.3)                  dX * s = [r s X * s + θ ′ s u * s ]ds + (u * s ) ′ dW s , X * 0 = x 0 , dp(s; t) = -r s p(s; t)ds + k(s; t) ′ dW s , p(T ; t) = X * T -E t [X * T ] -µ 1 X * t ,
and the process Λ(s; t) in condition (3.9) is Λ(s; t) = p(s; t)θ s + k(s; t).

We require that (4.4) Λ(t; t), v tu * t ≥ 0.

Existence

In this subsection, we construct a solution to (

Let us first assume the following Ansatz:

p(s; t) = M s X * s -E t [M s X * s ] -ρ s µ 1 X * t ,
with

dM s = -F M,U (s)ds + U ′ s dW s , and
ρ s = e T s rvdv . Applying Itô's formula to M s X * s , we get d(M s X * s ) = (-F M,U (s) + r s M s )X * s + (θ s M s + U s ) ′ u s ds + (X * s U s + M s u s ) ′ dW s ,
and then

dE t [M s X * s ] = E t (-F M,U (s) + r s M s )X * s + (θ s M s + U s ) ′ u s ds. Hence k(s) = X * s U s + M s u s , and
p(t; t) = -ρ t µ 1 X * t .
Then (4.4) becomes

-ρ t µ 1 θ t X * t + X * t U t + M t u t , v t -u * t ≥ 0.
We will construct a solution with X * t ≥ 0, thus

u * t = Proj K M -1 t (ρ t µ 1 θ t -U t ) X * t .
Denoting

α t = Proj K M -1 t (ρ t µ 1 θ t -U t ) ,
and coming back to (4.3), we get:

F M,U (s) = 2r s M s + (θ s M s + U s ) ′ α s .
To proceed, let us recall some facts about bounded-mean-oscillation (BMO) martingales; see Kazamaki [START_REF] Kazamaki | Continuous Exponential Martingales and BMO[END_REF]. The process

Z • W △ = • 0 Z ′ s dW s is a BMO martingale if and only if there exists a constant C > 0 such that E T τ |Z s | 2 ds F τ ≤ C
= E T (V • W ), W Q t △ = W t - t 0 V s ds is a standard Brownian motion, and Z • W Q is a BMO martingale.
dM s = -2r s M s + (θ s M s + U s ) ′ Proj K M -1 s (ρ s µ 1 θ s -U s ) ds + U ′ s dW s , M T = 1, admits a solution (M, U) ∈ L ∞ F (0, T ; R) × L 2 F (0, T ; R d ) satisfying M ≥ c for some constant c > 0. Moreover, U • W is a BMO martingale.
Proof: We can prove the existence by a truncation argument together with a Girsanov transformation.

Let c > 0 be a given number to be chosen later. Consider the following quadratic BSDE:

(4.6)          dM s = -2r s M s + θ s (M s ∨ c) + U s ′ Proj K (M s ∨ c) -1 (ρ s µ 1 θ s -U s ) ds + U ′ s dW s , M T = 1.
This BSDE is a standard quadratic BSDE. Hence there exists a solution (M c , U c ) ∈

L ∞ F (0, T ; R) × L 2 F (0, T ; R d ),
and U c • W is a BMO martingale; see [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] and [START_REF] Morlais | Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem[END_REF]. We can rewrite the above BSDE as, by noticing that θ

′ s Proj K (θ s ) = |Proj K (θ s )| 2 because K is a convex cone, (4.7)          dM c s = -(2r s M c s + ρ s µ 1 |Proj K (θ s )| 2 )ds + (U c s ) ′ [dW s -β s ds], M c T = 1,
where

β s = Proj K (M c s ∨c) -1 (ρ s µ 1 θ s -U c s ) + θ ′ s Proj K (ρ s µ 1 θ s -U c s ) -Proj K (ρ s µ 1 θ s ) |U c s | 2 U c s 1 {U c s =0} .
It is easy to see that |β| ≤ C(1

+ |U c |), hence β • W is a BMO martingale.
As β • W is a BMO martingale, there exists a new probability measure Q such that

W Q t = W t - t 0 β s ds is a Brownian motion under Q.
Hence,

M c s = E Q s e 2 T s rtdt + T s ρ v µ 1 e 2 v s rtdt |Proj K (θ v )| 2 dv ,
from which we deduce that there exists a constant η > 0 independent of c such that M ≥ η.

Taking c = η, we obtain a solution.

Q.E.D.

Now we can state our main existence theorem:

Theorem 4.2
The following feedback

u * s = Proj K M -1 s (ρ s µ 1 θ s -U s ) X * s is an equilibrium strategy. Proof: Set dX * s = r s X * s + θ ′ s Proj K M -1 s (ρ s µ 1 θ s -U s ) X * s ds + Proj K M -1 s (ρ s µ 1 θ s -U s ) X * s ′ dW s , X * 0 = x 0 ; u * s = Proj K M -1 s (ρ s µ 1 θ s -U s ) X * s , p(s; t) = M s X * s -E t [M s X * s ] -ρ s µ 1 X * t , and
k(s) = X * s U s + M s u * s .
Then, X * s > 0, and

dX * s = [r s X * s + θ ′ s u * s ]ds + (u * s ) ′ dW s , X * 0 = x 0 .
Let us now prove that u * is in

p>2 L p F (0, T ; K). Applying Ito's formula to M s (X * s ) 2 , we obtain (recall that α s = Proj K M -1 s (ρ s µ 1 θ s -U s ) ), d M s (X * s ) 2 = -(X * s ) 2 2r s M s + (θ s M s + U s ) ′ α s ds + (X * s ) 2 U ′ s dW s + 2M s [r s + θ ′ s α s ] + M s |α s | 2 (X * s ) 2 ds + 2M s α ′ s (X * s ) 2 dW s + 2(X * s ) 2 U ′ s α s ds = θ s + U s M s ′ α s + |α s | 2 M s (X * s ) 2 ds + M s (X * s ) 2 2α s + U s M s ′ dW s .
As K is a convex cone,

a ′ Proj K (a) = |Proj K (a)| 2 ,
from which we deduce that

d M s (X * s ) 2 = 1 + M -1 s ρ s µ 1 θ ′ s α s M s (X * s ) 2 ds + M s (X * s ) 2 2α s + U s M s ′ dW s . Hence M t (X * t ) 2 = M 0 x 2 0 e t 0 (1+M -1 s ρsµ 1 )θ ′ s αsds E 2α + U M • W t .
From John-Nirenberg's inequality (see Kazamaki [14, Theorem 2.2, p.29]), we deduce that there exists ε > 0 such that E e ε T 0 |αs| 2 ds < +∞. Thus, e

T 0 (1+M -1 s ρsµ 1 )θ ′ s αsds ∈ p>1 L p . Moreover, as (2α + U M ) • W is a BMO martingale, sup t E 2α + U M • W t ∈ p>1 L p . As M ≥ c > 0, we deduce that sup t |X * t | is in p>2 L p , and then u * is in p>2 L p F (0, T ; K).
Now we calculate dp. Applying Ito's formula to M s X * s , we obtain

d(M s X * s ) = -X * s 2r s M s + (θ s M s + U s ) ′ α s ds + X * s U ′ s dW s +M s [r s + θ ′ s α s ]X * s ds + M s α ′ s X * s dW s + X * s U ′ s α s ds = -r s M s X * s ds + k(s) ′ dW s ,
and then

dp(s; t) = -r s M s X * s ds + k(s) ′ dW s + r s E t [M s X * s ]ds + r s ρ s µ 1 X * t ds = -r s p(s; t)ds + k(s) ′ dW s .
Hence (X * , u * , p, k) is a solution to (4.3), and (4.4) is easily checked.

Q.E.D.

Uniqueness

Theorem 4.3 The following feedback

u * s = Proj K M -1 s (ρ s µ 1 θ s -U s ) X * s
is the unique equilibrium strategy.

Proof: Suppose that (X, u, p, k) is a solution to (4.8)

                 dX s = [r s X s + θ ′ s u s ]ds + (u s ) ′ dW s , X 0 = x 0 , dp(s; t) = -r s p(s; t)ds + k(s; t) ′ dW s , p(T ; t) = X T -E t [X T ] -µ 1 X t ,
and the process Λ(s; t) in condition (3.9) is Λ(s; t) = p(s; t)θ s + k(s; t).

We require that (4.9) Λ(t; t), v tu t ≥ 0.

There exist two adapted processes α + and α + with 0 ≤ α + ≤ 1, -1 ≤ α -≤ 0 and

α + -α -= 1, such that X + t = α + (t)X t ,
′ M-1 s Proj K (ρ s µ 1 θ s -Ũs ) α + (s) +Proj K -(ρ s µ 1 θ s -Ũs ) α -(s) ds + Ũ′ s dW s .
We note that

(θ s Ms ) ′ M-1 s Proj K (ρ s µ 1 θ s ) α + (s) + Proj K (-ρ s µ 1 θ s ) α -(s) = θ ′ s Proj K (ρ s µ 1 θ s ) α + (s) + Proj K (-ρ s µ 1 θ s ) α -(s) = ρ s µ 1 [|Proj K (θ s )| 2 α + (s) -|Proj K (-θ s )| 2 α -(s)] ≥ 0.
Applying the same method as that of Lemma 4.1, the quadratic BSDE (4.10) admits a Applying the same method as in [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium[END_REF], we deduce that p(s; t) = 0 and k(s) = 0. Therefore

solution ( M , Ũ ) ∈ L ∞ F (0, T ; R) × L 2 F (0, T ; R d ) satisfying M ≥ c for some constant c > 0. Moreover, Ũ • W is a BMO martingale. Let us take any such solution ( M, Ũ ). It is important to note that M-1 s Proj K (ρ s µ 1 θ s -Ũs )X s = M-1 s Proj K (ρ s µ 1 θ s -Ũs ) α + (s) + Proj K -(ρ s µ 1 θ s -Ũs ) α -(s) X s = αs X s , where αs = M-1 s [Proj K (ρ s µ 1 θ s -Ũs ) α + (s) + Proj K -(ρ s µ 1 θ s -Ũs ) α -(s)]. Setting p(s; t) = p(s; t) -Ms X s -E t [ Ms X s ] -ρ s µ 1 X t , and
u t = αt X t ,
and then X t > 0, α + = 1, α -= 0.

As X t > 0, replacing ( M , Ũ) by (M, U), and using the above procedure again, we deduce then

u s = Proj K M -1 s (ρ s µ 1 θ s -U s ) X s ,
and we conclude the proof.

Q.E.D.

From the above theorem, we deduce immediately the uniqueness of solution to BSDE (4.5).

Corollary 4.4

The solution to BSDE (4.5) is unique.

Proof: Let ( M , Ū) be another such solution. Then from the above theorem, we deduce that Proj K M -1 s (ρ s µ 1 θ s -U s ) X * s = Proj K M-1 s (ρ s µ 1 θ s -Ūs ) X * s .

As X * s > 0, we deduce that, Proj K M -1 s (ρ s µ 1 θ s -U s ) = Proj K M-1 s (ρ s µ 1 θ s -Ūs ) , from which we deduce the uniqueness of solution.

Q.E.D.

Deterministic Risk Premium

Let us first consider the case when the risk premium is a deterministic function of time.

Then U = 0 and

M s = e 2 T s rvdv 1 + µ 1 T s e -T v rzdz |Proj K (θ v )| 2 dv .
The equilibrium strategy is given by

u * s = µ 1 e T s rvdv
M s Proj K (θ s )X * s .

In the appendix, we obtain that the precommitted optimal control for the problem starting at t = 0 is also in an affine feedback form u * pre (s, x) = -Proj K (θ s )x + e s 0 rvdv x 0 + µ 1 x 0 e T 0 (|Proj K (θv)| 2 -rv)dv Proj K (θ s ).

In [START_REF] Bensoussan | Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting[END_REF], the equilibrium is defined for the class of feedback controls as in [START_REF] Björk | Mean-variance portfolio optimization with state dependent risk aversion[END_REF]. Therein the equilibrium strategy is derived in a linear feedback form u * f be t = c f be t X * t with c f be t uniquely determined by an integral equation and iterated by numerical method. However, we can show that the linear coefficient of our equilibrium above is not a solution of the integral equation in [START_REF] Bensoussan | Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting[END_REF]. This implies the difference between the two definitions of equilibrium (open-loop and feedback).

To compare the performance of these two different equilibrium controls, together with the precommitted optimal control at time t = 0, we calculate J(0, x 0 ; u) for u = u * , u * f be , and u * pre , respectively. Denote c s = µ 1 e T s rv dv

Ms

Proj K (θ s ), then it is an easy exercise to get

Example 1 . 1

 11 Let λ > 0 be a constant. Then we consider the following dynamic meanvariance portfolio problem(1.1)

Proof:

  Define the function ψ(•) as the unique continuous solution to the following matrixvalued ordinary differential equation (ODE) dψ(t) = ψ(t)A(t) ′ dt, ψ(T ) = I n , where I n denotes the n × n identity matrix. It is clear that ψ(•) is invertible, and both ψ(•) and ψ(•) -1 are bounded.

  for any stopping time τ ≤ T . For every such Z, the stochastic exponential of Z •W denoted by E(Z • W ) is a positive martingale, and for any p > 1, there exists a constant C p > 0 such that E T τ |Z s | 2 ds p F τ ≤ C p for any stopping time τ ≤ T . Moreover, if Z • W and V • W are both BMO martingales, then under the probability measure Q defined by dQ dP

Lemma 4 . 1

 41 The following quadratic BSDE(4.5)

X * T = x 0 e T 0

 0 (rs+c ′ s θs-|cs| 2 2 )ds+ T 0 c ′ s dWs . Hence E [X * T ] = x 0 e T 0 (rs+c ′ s Proj K (θs))ds , Var(X * T ) = x 2 0 e 2 T 0 (rs+c ′ s Proj K (θs))ds (e T 0 |cs| 2 ds -1), leading to J(0, x 0 ; u * ) = x 2

  Let p(s; t) = ψ(s)p(s; t) + hE t [X * T ] + µ 1 X * t + µ 2 and kj (s; t) = ψ(s)k j (s; t) for j = 1, • • • , d. Then by Itô's formula, on the time interval [t, T ], (p(•; t), k(•; t)) satisfies

	(3.7)

  It is clear that neither the terminal condition nor the coefficients of this equation depend on t; so it can be taken as a BSDE on the entire time interval [0, T ]. We denote its solution as (p(s), k(s)), s ∈ [0, T ]. It follows from the uniqueness of the solution to BSDE

	d
	s ds +	kj (s; t)dW j s ,
	j=1
	p(T ; t) = GX * T .

that (p(s; t), k(s; t)) = (p(s), k(s)) at s ∈ [t, T ] for any t ∈ [0, T ]. As a result, k(s; t) = ψ(s) -1 k(s) := k(s), proving the first claim of the proposition.

Next, from the definition of p(s; t), p(s

  Ms = -2r s Ms + (θ s Ms + Ũs )

	quadratic BSDE:
	d
	(4.10)

and X - t = α -(t)X t . Consider the following

  k(s) = k(s) -(X s Ũs + Ms u s ). from which we deduce that there exists one bounded adapted process A, such thatu t = M-1 t Proj K -p(t; t)θ t -k(t) + (ρ t µ 1 θ t -Ũt )X t ) = M-1 t A t -p(t; t)θ t -k(t) + M-1 t Proj K (ρ t µ 1 θ t -Ũt )X t = M-1 t A t -p(t; t)θ t -k(t) + αt X t . =r s p(s; t) + (θ s + Ũs M-1 s ) ′ A s [-θ s p(s; s) -k(s)] -E t (θ s + Ũs M-1 s ) ′ A s [-θ s p(s; s) -k(s)] ds + k(s) ′ dW s , s ∈ [t, T ],

	After some calculus, we arrive at:
	
	   	dp(s; t)
	
	
	
	
	
	
	
	
	(4.11)
	
	
	
	
	
	
	
	
	
	
	
	

Then Λ(s; t) = p(s; t) + Ms X s -E t [ Ms X s ]ρ s µ 1 X t θ s + k(s) + X s Ũs + Ms u s ,

and then condition (4.9) becomes: for any v t ∈ K,

[p(t; t)ρ t µ 1 X t ]θ t + k(t) + X t Ũt + Mt u t , v tu t ≥ 0, p(T ; t) = 0.

partially supported by Lebesgue center of mathematics "Investissements d'avenir" program -ANR-11-LABX-0020-01, by ANR CAESARS -ANR-15-CE05-0024 and by ANR MFG -ANR-16-CE40-0015-01. G-UA7U, Hong Kong RGC grants 15224215 and 15255416.

Concluding Remarks

In this paper, we consider some time-inconsistent LQ control problem under constraint.

We define the equilibrium strategy via spike perturbation of open control and deduce the necessary and sufficient condition by applying the stochastic maximum principle, following the ideas of [START_REF] Hu | Time-inconsistent stochastic linear-quadratic control[END_REF][START_REF] Hu | Time-inconsistent stochastic linear-quadratic control: characterization and uniqueness of equilibrium[END_REF]. LQ control problem with control constraint is useful because of its wide applications in finance and economics. Our necessary and sufficient conditions are general enough to cover many interesting time-inconsistent LQ control problem under various constraint. We also shed light on important application in mean-variance portfolio under convex cone constraint and present its explicit equilibrium. In particular, we can treat the random coefficient case, while the HJB method used by Bensoussan, Wong, Yam and Yung [START_REF] Bensoussan | Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting[END_REF] seems not applicable in random coefficient case.

A Appendix.

Precommitted Mean-Variance Portfolio with Cone Constraint

We consider the precommitted optimal control problem at time t = 0, (A.1) min J(t, x 0 ; u)

From the existing study on precommitted mean-variance problems such as [START_REF] Hu | Constrained stochastic LQ control with random coefficients, and application to portfolio selection[END_REF][START_REF] Li | Continuous-time Markowitz's model with constraints on wealth and portfolio[END_REF][START_REF] Li | Dynamic mean-variance portfolio selection with no-shorting constraints[END_REF], it follows that, when the parameters r • and θ • are deterministic, we can get the explicit optimal value for the precommitted problem (A.1):

Furthermore, the corresponding optimal control can be written as the affine feedback con-


Nguyet Trang

Thanh Chau

Stéphane Menuel

Sophie Colombel-Rouen

Miguel Guerrero

Eric Monflier

Karine Philippot

Audrey Denicourt-Nowicki

Alain Roucoux

Active hydrogenation Rh nanocatalysts protected by new selfassembled supramolecular complexes of cyclodextrins and surfactants in water

The stability of inclusion complexes between randomly methylated β-cyclodextrin (RaMeCD) or its leucine-grafted analogue (RaMeCDLeu) with two hydroxylated ammonium surfactants was investigated. The binding isotherms and complexation constants were measured by Isothermal Titration Calorimetry (ITC) technique. These host-guest inclusion complexes were used as protective agents during the formation of rhodium (0) nanoparticles by chemical reduction of rhodium trichloride in water. The amount of protective agents was adjusted in order to ensure both stability and reactivity of the rhodium nanocatalysts under the catalytic conditions. The size and dispersion of air-stable and water-soluble rhodium suspensions were determined by Transmission Electron Microscopy (TEM) analyses. These spherical nanoparticles, with sizes between 1.20 to 1.50 nm according to the nature of inclusion complexes, were evaluated in the biphasic hydrogenation of various reducible compounds (olefins, linear or aromatic ketones), showing promising results in terms of activity and selectivity.

Introduction

The development of highly selective and readily recyclable catalysts, 1 as well as the substitution of volatil organic solvents with novel reaction media (e.g. ionic liquids, supercritical fluids or water), [2][3] constitute crucial parameters in the quest towards sustainable and more environmentally-friendly chemical processes. [4][5] Recently, nanometer-sized metallic particles have been developed as relevant catalysts for performing green organic transformation processes because of their outstanding intrinsic properties, combining the advantages of homogeneous and heterogeneous catalysts. 6-9 Besides their pertinent activities owing to a high number of surface-exposed metal atoms and their original selectivities regarding to their unique surface structures and combined a Ecole Nationale Supérieure de Chimie de Rennes, CNRS UMR 6226, 11 Allée de Beaulieu, CS 50837, 35708 Rennes Cedex 7, France. Fax: (+33) 02 2323 8199; Tel: (+33) 02 2323 8037; E-mail: alain.roucoux@ensc-rennes.fr; audrey.denicourt@enscrennes.fr. b Université d'Artois, CNRS UMR 8181, Faculté des Sciences Jean Perrin, Rue Jean Souvraz, SP 18, F-62307 Lens Cedex, France. c CNRS; LCC (Laboratoire de Chimie de Coordination), 205 Route de Narbonne, F-31077 Toulouse, France. d Université de Toulouse; UPS, INPT, LCC, F-31077 Toulouse, France. † Electronic Supplementary Information (ESI) available: [T-ROESY NMR spectra of inclusion complexes have been described]. See DOI: 10.1039/b000000x/ supramolecular assemblies, 10 they could also provide improved recovery potentialities, which constitute sine qua non conditions for eco-responsible applications.

11 Thus, the use of colloidal metallic particles finely dispersed in water constitutes one of the most effective and sustainable approaches to achieve the targeted recycling objective, through a biphasic water-substrate catalytic process. 12 For this purpose, highly water-soluble protective agents, such as surfactants, [START_REF] Denicourt-Nowicki | Metal Nanoparticles for Catalysis. Advances and Applications[END_REF] dendrimers [START_REF] Ji | Dendrimers[END_REF] or novel ligands such as Nheterocyclic carbenes [START_REF] Ferry | [END_REF][16] are necessary to maintain the nanospecies within the aqueous phase. More recently, as a class of nontoxic oligosaccharides, cyclodextrins have been extensively investigated in host-guest chemistry for construction of versatile supramolecular complexes owing to their special hydrophobic inner cavities. 17 For instance, the inclusion complexation of randomly methylated cyclodextrins (RaMeCD) with an ammonium surfactant 18 (HEA16Cl: N,N-

dimethyl-N-hexadecyl-N-(2-hydroxyethyl)-ammonium chloride)
or a sulfonated phosphine 19 as guest molecules has been used to efficiently stabilize nanosized ruthenium(0) hydrogenation catalysts in aqueous media and thus tune the catalytic performances of the metallic centers. Although ruthenium(0) nanospecies have been successfully protected by cyclodextrins alone, [20][21] the less oxophilic rhodium metal proved to be more difficult to stabilize with the formation of visual aggregates. Based on these results, we investigated the R e v i s e d m a n u s c r i p t use of original inclusion complexes formed between various cyclodextrins and ammonium surfactants as pertinent supramolecular assemblies to protect rhodium(0) species. These nanocatalysts were evaluated and compared in terms of included inside the CD cavity whatever the inclusion complex (ESI-Figure S2-S4). According to the literature, dipolar contacts between the alkyl chain protons of the hexadecyltrimethylammonium bromide and protons inside the CD cavity provide a binding - 1 24 stability, catalytic activity, and selectivity in the hydrogenation constant about 67700 M , as initially reported by Holzwarth -1 and of various substrates: a functionalized olefin (methyl-2-more recently revised to 60733 ± 11484 M 25 by B. Tutaj and acetamidoacrylate), an activated ketone (ethyl pyruvate) and coworkers. Here, the slightly higher values achieved with the acetophenone, a pertinent substrate in terms of chemoselectivity (aromatic ring vs. carbonyl group). Among surfactants bearing a hydroxylated polar heads, could be attributed to the formation of a hydrogen bond between the flexible hydroxy 18 the classical RaMeCD cyclodextrin or HEA16Cl ammonium salt, groups of the surfactant's polar head and the CD. The a methylated cyclodextrin grafted with a L-Leucine moiety (RaMeCDLeu) 22 and an optically active ammonium salt (QCD16Br=(1S,2R,4S,5R)-(+)-N-hexadecyl-5-vinyl-2quinuclidinium-methanol bromide)

23

were also used as potential guest/host molecules to tune the selectivity, and potentially the asymmetric induction as a more challenging objective (Figure 1). thermodynamic parameters determined from ITC experiments also confirm that the surfactants have the same structural motif included into the -CD cavity. Indeed, the inclusion processes are all mainly entropy driven, showing similar positive ΔS (Table 1). These results confirmed the formation of stable inclusion complexes between methylated CDs and hydroxylated ammonium surfactants, which could potentially be pertinent protective assemblies of rhodium(0) nanoparticles (Scheme 1).

Results and discussion

Design and characterization of rhodium(0) nanoparticles

The Isothermal Titration Calorimetry (ITC) measurements were Scheme 1. Synthesis of rhodium(0) nanoparticles stabilized by methylated βcyclodextrin/ammonium surfactant inclusion complexes. Rhodium(0) nanoparticles protected by these inclusion complexes were synthesized, using a two-step methodology as previously reported for the preparation of Ru 0 @RaMeCD-HEA16Cl NPs. 18 The rhodium trichloride hydrate (RhCl .3H O) 3 2 achieved to check the stability and to determine equilibrium constants of the novel inclusion complexes obtained from the combination of methylated CDs with ammonium surfactants, including RaMeCD-QCD16Br, RaMeCDLeu-HEA16Cl and RaMeCDLeu-QCD16Br supramolecular assemblies (ESI-Figure S1). The association constant values (Table 1) were compared to the one of RaMeCD-HEA16Cl inclusion complex. was chemically reduced, using sodium borohydride as reducing agent, in the presence of RaMeCD and HEA16Cl. As previously described in the literature 18,26 the metallic surface is surrounded by a double layer of stabilizers (Scheme 2). The first layer is composed of surfactant molecules, whose polar head is directed towards the metallic surface, according to the adsorption of counter-ion to electron-deficient metal surfaces. [27][28][29] The second layer is constituted of cyclodextrins interacting with the lipophilic tail of the surfactant. The association constants were found to be similar to that of RaMeCD-HEA16Cl inclusion complex. In fact, T-ROESY experiments indicated that the alkyl chain of ammonium surfactants was inclusion complexes near the nanoparticle surface.

The Zeta potential value of the well-dispersed Rh@RaMeCD-HEA16Cl suspension, measured by electrophoretic light scattering (DLS spectroscopy), was used to determine the apparent charge of the nanoparticle in solution. A significant positive apparent charge of + 50 mV for the rhodium species R e v i s e d m a n u s c r i p t coated with RaMeCD-HEA16Cl assemblies was obtained as usually observed with ammonium stabilizers. 30 This value, which explains the good stability of the aqueous colloidal suspensions, characterizes the difference of potential between the solution far from the metal/solution interface (Stern layer based on adsorbed ionic species) and the mobile part of the disperse layer where salt concentration decreases when the distance increases.

The RaMeCD/ HEA16Cl ratio has been optimized since the use of a stoechiometric ratio leads to the destabilization of the rhodium(0) particles in the 1-tetradecene hydrogenation under standard conditions (1 bar H2, room temperature, 3h). Two equivalents of each protective agents proved to be efficient for the stabilization of Rh 0 NPs, under atmospheric hydrogenation, as well as high pressure conditions (10 bar H2).

The particle size, morphology and dispersion were determined by Transmission Electron Microscopy (TEM) analyses (ESI-Figure S5-S7). Whatever the stabilizer association, the rhodium particles were well-dispersed on the grid, with spherical morphology. The Rh 0 NPs displayed a mean size in the range 1.2-1.5 nm, depending on the cyclodextrin/surfactant association. Based on the same surfactant (QCD16Br), the inclusion complex containing the more sterically hindered L-leucine-grafted cyclodextrin (Figure 2a vs. Figure 2c) led to the formation of larger NPs than the RaMeCD (1.5 nm vs. 1.2 nm, respectively). Similarly, compared to Rh 0 @RaMeCDLeu/QCD16Br system, the colloids stabilized by the RaMeCD combined with the less sterically hindered surfactant (HEA16Cl) exhibited slightly smaller diameters (1.3 nm vs. 1.5 nm) (Figure 2b vs. Figure 2c). In all cases, the metal core size of these original Rh 0 NPs were smaller than the host-guest inclusion complex-capped Ru 0 NPs with an average diameter of about 4 nm. 18 This better control of the NPs growth could be easily explained by the presence of higher quantity of protective supramolecular assemblies (inclusion complex/Rh ratio = 2/1) which provide a better control during the growth step of the nanoparticles.

Catalytic hydrogenation using rhodium(0) nanoparticles

The catalytic performances of Rh 0 @RaMeCD-QCD16Br, Rh 0 @RaMeCDLeu-HEA16Cl and Rh 0 @RaMeCDLeu-QCD16Br NPs were evaluated in the hydrogenation of various model substrates, such as an activated olefin (methyl 2acetamidoacrylate), an activated -ketoester (ethylpyruvate) and an aromatic ketone (acetophenone). The reduction of these prochiral compounds was carried out under pure biphasic conditions (water/substrate), at room temperature and under 10 bar H2 with 1 mol% of the nanocatalysts (cyclodextrins/surfactant/rhodium = 2/ 2/ 1) as optimized in 1-tetradecene hydrogenation (3h, 10 bar of H2 for 100% conv. vs 20h under 1 bar H2). These conditions were chosen in order to provide good yields in reduced products. Moreover, the stability of the triazole ring under the catalytic reduction conditions was also checked. The conversion was monitored by gas chromatography analysis. The specific activity (SA) was calculated, considering an optimized reaction time for a complete conversion of the substrate and based on the total introduced amount of metal and not on the exposed surface metal. These specific activity values were clearly underestimated but were suitable from an economic point of view. 2). The transformation of ethylpyruvate into the corresponding alcohol was totally achieved, with various specific activities values according to the supramolecular assemblies in the following order: RaMeCDLeu-HEA16Cl > RaMeCDLeu-QCD16Br > RaMeCD-QCD16Br (SA of 67, 17 and 6 h -1 , respectively)., These results could be justified by a more favorable displacement during the catalysis of free RaMeCDLeu vs RaMeCD in the aqueous phase starting from inclusion complexes, as mentioned in Scheme 2, thus providing an efficient solubilisation of the organic substrate inside the RaMeCDLeu's cavity and a better diffusion towards the metal surface. Moreover, based on the observed conversions at 1.5h (Table 3) and the comparison of the head group inside QCD16Br and HEA16Cl, it seems that kinetics is slower when the QCD16Br surfactant is located at the vicinity of the particle's surface, due to an increased steric hindrance, which probably limits the access of the substrate. Finally, the chemoselectivity of the catalytic system was investigated towards the hydrogenation of acetophenone (reduction of the ketone vs. the aromatic ring (Table 4). Table 4. Hydrogenation of acetophenone using rhodium(0) nanoparticles a Whatever the inclusion complexes (RaMeCD-QCD16Br, RaMeCDLeu-HEA16Cl or RaMeCDLeu-QCD16Br), the reduction of methyl 2-acetamidoacrylate occurred exclusively on the activated C=C double bond with complete conversion in an optimized 1.5 h time. It is noteworthy to report that the reduction of this substrate by Rh 0 suspensions, solely stabilized by cyclodextrins (RaMeCD or RaMeCDLeu), could not be achieved in these experimental conditions due to the destabilization of the nanocatalysts (Table 2, Entries 4-5).

Finally, the use of the sole ammonium surfactant (HEA16Cl or QCD16Br) as protective agents leads to well stable colloidal suspensions with pertinent catalytic results in the reduction of methyl 2-acetamidoacrylate ( previous results based on similar investigations performed in suspension, 10 bar H2, rt. b Selectivity determined by GC analysis. c Catalytic our group, 23,30 these efficient activities could be explained by the dynamic behaviour of the double layer which facilitates the access of the substrate at the particle's surface. The reduction of the carbonyl function in ethylpyruvate as prochiral model substrate was also performed in similar reaction conditions (Table 3). From kinetics investigations, two reaction pathways could be involved for the formation of the totally hydrogenated product (1-cyclohexylethanol 3), either via 1-cyclohexylethanone 1 from the reduction of aromatic ring, or either via 1phenylethanol 2 from the reduction of carbonyl group. The higher ratio of 2 over 1 suggests that the carbonyl group is more easily reduced than the aromatic ring. However, the significant quantity of 1 (up to 24%) proves that the RaMeCD or RaMeCDLeu could not efficiently wrap the aromatic ring and avoid its hydrogenation. In addition, after 1.5 h of reaction, complete conversions were observed in the case of Rh 0 NPs stabilized by the less sterically hindered inclusion complexes (RaMeCD-QCD16Br and RaMeCDLeu-HEA16Cl), compared to 60% conversion with Rh 0 @RaMeCDLeu-QCD16Br catalytic system. Increasing reaction time up to 3 h improved significantly the conversion by a factor of 2.5, however with a slightly destabilization of the Rh 0 suspension with formation of aggregates. These results could be attributed to the competitive affinity of the substrate (acetophenone) and the surfactants inside the RaMeCDLeu's cavity. During the catalytic R e v i s e d m a n u s c r i p t process, a partial release of the CDs in the bulk aqueous media allows the solubilisation of organic substrate by some free CDs (Scheme 2) and, as a consequence, enables a better diffusion of the substrates towards the metallic surface. Nevertheless, a strong affinity between organic substrate and CDs could disturb the dynamic organization of the protective agents around the NPs. Unfortunately, no selectivities could be achieved with the use of chiral molecules as QCD16Br, RaMeCDLeu alone or associated, whatever the investigated prochiral substrate is (Tables 234). Based on previous investigations in our group on chiral ammonium-capped rhodium(0) nanocatalysts, 23 the lack of enantiodiscrimination during these asymmetric catalytic reactions could be attributed to a significant mobility of the inclusion complex at the metal surface in water. Moreover, in contrast with suitable coordinating ligands, the interaction of the HEA16Cl and QCD16Br surfactants within the particle's core seems to be too weak to provide asymmetric induction.

Conclusions

To conclude, the formation of supramolecular inclusion complexes between ammonium surfactant and cyclodextrin-based chiral molecules was investigated. The existence of a stable interaction between the surfactant and cyclodextrin was clearly evidenced by NMR spectroscopy and isothermal titration calorimetry measurements showed a strong affinity between the lipophilic chain of the surfactant and the hydrophobic cavity of cyclodextrin compounds, which was corroborated with their significant association constant values. These host-guest inclusion complexes were successfully used as pertinent supramolecular assemblies to protect rhodium(0) nanospecies. TEM analyses showed very small (1.2-1.5 nm) and well-dispersed spherical nanoparticles without aggregates. According to optically active skeletons grafted on randomly methylated -cyclodextrin (RaMeCDLeu) or surfactant (QCD16Br), these assemblies alone or associated produced a chiral environment onto the nanoparticle surfaces. Finally Rh 0 @RaMeCD-QCD16Br, Rh 0 @RaMeCDLeu-HEA16Cl and Rh 0 @RaMeCDLeu-QCD16Br were evaluated in the biphasic liquid-liquid hydrogenation of various functional groups (C=O, C=C) and demonstrated efficiencies in terms of stability and catalytic performances. During acetophenon hydrogenation, various differences in terms of conversions and selectivities relative to the competitive hydrogenation of the C=O bond versus the aromatic ring were observed, evidencing a direct impact of the steric hindrance of the supramolecular association. Nevertheless, in all cases, no significant induction was observed as expected. Finally, taking advantage of the supramolecular properties of assemblies to modulate the surface reactivity of nanoparticles, this original work may open new opportunities in the field of nanocatalysis and desirably in asymmetric reaction.

Experimental

Reagents and chemicals N,N-Dimethyl-N-hexadecyl-N-(2-hydroxyethyl)ammonium chloride (HEA16Cl) and (1S,2R,4S,5R)-(+)-N-hexadecyl-5-vinyl-2-quinuclidinium-methanol bromide (QCD16Br) were synthesized from an already described procedure. 23,30 Randomly methylated -cyclodextrin (RaMeCD (1.8)) was purchased from Sigma-Aldrich. This cyclodextrin was partially methylated at positions 2, 3 or 6 and 1.8 hydroxyl groups per glucopyranose unit were statistically modified. L-leucine-N-[(1randomlymethylated--cyclodextrinyl-1H-1,2,3-triazol-4-yl) methylamide] (RaMeCDLeu) was prepared according to the procedure described in the literature. 31 Rhodium(III) chloride hydrate was obtained from Strem Chemicals. Sodium borohydride and all substrates were purchased from Sigma-Aldrich, Acros Organics or Alfa Aesar and were used without further purification. Water was purified using Millipore Elix 5 (type MSP 100) system.

Analytical procedures

ITC analysis. Formation constants of methylated CDammonium surfactant complexes were determined by an isothermal calorimeter (ITC200, MicroCal Inc., USA), in a phosphate buffer (pH 6.5) at 298 K. In the case of RaMeCD, titrations of a 0.1 mM surfactant solution (QCD16Br or HEA16Cl) were realized with a 5 mM solution of RaMeCD. The inclusion ability of RaMeCDLeu was then investigated by competitive experiments with RaMeCD (injection of a 5 mM RaMeCD solution into a 0.1 mM surfactant-0.1 mM RaMeCDLeu solution). Each experiment was performed three times and consisted in the addition of an initial aliquot of 0.4 μL, followed by ten injections of 3.7 μL, delivered over 7.4 s, within a 204.5 μL cell. Time interval between two consecutive injections was 60 s and agitation speed was 1000 rpm. The corresponding heat flow was recorded as a function of time, and the heat effects due to dilution were taken into account by performing blank titrations.

Binding constants were determined by nonlinear regression analysis of binding isotherms (obtained by integration of the raw signal) using a dedicated homemade program.

32

TEM analysis. Transmission electron microscopy (TEM) images were performed at the "Service Commun de Microscopie Electronique de l'Université Paul Sabatier" (UPS-TEMSCAN, Toulouse) and recorded with a JEOL 1011 electron microscope operating at 100 kV with resolution point of 4.5 Å. A drop of rhodium nanoparticles in water was deposited on a carboncoated copper grid and dried in air. The size distributions were determined through a manual analysis of enlarged micrographs with Image J software using Microsoft Excel to generate histograms of the statistical size distribution and a mean diameter.

Dynamic Light Scattering Spectroscopy. The zeta potential of the nanoparticle aggregates were measured by dynamic light scattering (DLS) using a DelsaNano C instrument (Beckman Coulter). The aqueous suspensions of rhodium nanoparticles R e v i s e d m a n u s c r i p t were analysed at 25

• C and measurements were started 10 min after the cell was placed in the DLS apparatus to allow the temperature to equilibrate. Gas chromatography. For hydrogenation reactions, the conversion and the selectivity were determined by gas chromatography using Fisons Instruments GC 9000 series with FID detector equipped with a chiral Varian Chiralsil-Dex CB capillary column (30 m, 0.25 mm i.d.). Parameters were as follows: isotherm program with oven temperature, 90°C (ethylpyruvate) or 130°C (acetophenone and 2-acetamido acrylate); carrier gas pressure, 50 kPa.

General procedure for the preparation of surfactant/modified βcyclodextrin stabilized rhodium(0) nanoparticles RhCl . 3H O (10 mg, 3.8 х 10 -2 mmol, 1 eq) was dissolved into 10 K. An, G. A. Somorjai, ChemCatChem 2012, 4, 1512-1524.

distilled water (6 mL) and maintained under vigorous stirring. At the same time, cyclodextrin (RaMeCD or RaMeCDLeu) (7.6 х 10 -2 mmol, 2 eq) was added to a warm aqueous solution (14 mL) of appropriate surfactant (HEA16Cl or QCD16Br) (7.6 х 10 -2 mmol, 2 eq). The inclusion complex was stirred for 30 min before addition of sodium borohydride (3.6 mg, 9.5 х 10 -2 mmol, 2.5 eq). This mixture was quickly added to rhodium solution previously prepared above. The obtained rhodium nanoparticles were stirred for 24 h before used in catalytic test.

General procedure for atmospheric hydrogenation reactions

A 25 mL round bottom flask, charged with the aqueous colloidal rhodium(0) suspension (10 mL, 1.9 x 10 -2 mmol) and appropriate substrate ([Substrate]/[Rh 0 ] ratio = 100/1), was connected to a gas burette and a flask to balance the pressure. Then the system was filled with hydrogen (PH2 = 1 bar) and the mixture was stirred vigorously at room temperature. Samples were collected from time to time to monitor the reaction by gas chromatography in previously mentioned conditions.

General procedure for high pressure hydrogenation reactions

The stainless steel autoclave was charged with the aqueous colloidal rhodium(0) suspension (10 mL, 1.9 x 10 -2 mmol) and appropriate substrate ([Substrate]/[Rh 0 ] ratio = 100/1). The autoclave was degassed three times and hydrogen gas was admitted to the system at a constant pressure (10 bar H2). The mixture was stirred vigorously at room temperature. Samples were removed from time to time to monitor the reaction by gas chromatography in previously mentioned conditions. R e v i s e d m a n u s c r i p t

Figure 1 .

 1 Figure 1. Structure of CDs and surfactants

Figure 2 .

 2 Figure 2. TEM pictures and size distributions of (a) Rh 0 @RaMeCD-QCD16Br, (b) Rh 0 @RaMeCDLeu-HEA16Cl and (c) Rh 0 @RaMeCDLeu-QCD16Br NPs In a first set of experiments, RaMeCD-QCD16Br, RaMeCDLeu-HEA16Cl and RaMeCDLeu-QCD16Br rhodium nanocatalysts

  Substrate]/[Rh 0 ] molar ratio = 100/1, 10mL of aqueous

a

  Conditions: [Substrate]/[Rh 0 ] molar ratio = 100/1, 10mL of aqueous suspension, 10 bar H2, rt.b Time for a complete conversion determined by GC analysis. c Specific Activity (SA) defined as number of mol of transformed substrate per mol of Rh per hour. d Checked twice system started to be destabilized.

Table 1 . Thermodynamic data obtained by ITC of various inclusion complexes a

 1

Table 2 .

 2 Hydrogenation of methyl-2-acetamidoacrylate using rhodium(0) nanoparticles a

	Entry	Stabilizing agent	Conv.	b (%)	SA c (h -1 )
	1	RaMeCD-QCD16Br	100	66.7
	2	RaMeCDLeu-HEA16Cl	100	66.7
	3 4 5 6 7	RaMeCDLeu-QCD16Br RaMeCD d RaMeCDLeu d HEA16Cl QCD16Br	100 n.d e n.d e 100 f 100 f	66.7 e n.d e n.d 66.7 66.7

a Conditions: [Substrate]/[Rh 0 ] molar ratio = 100/1, 10mL of aqueous suspension, 10 bar H2, rt, 1.5 h. b Determined by GC analysis. c Specific Activity (SA) defined as number of mol of transformed substrate per mol of Rh per hour. d Formation of metal aggregates. e Non determined. f not optimized time

Table 2

 2

	b

, Entries 6-7). According to Entry Stabilizing agent Time (h) 1:2:3 (%)

Table 3 .

 3 Hydrogenation of ethylpyruvate using rhodium(0) nanoparticles a

	Entry	Stabilizing agent	Conv. at 1.5h (%) Time	b (h) SA c (h -1 )
	1	RaMeCD-QCD16Br	30	17 d	6
	2	RaMeCDLeu-HEA16Cl	100	1.5	66.7
	3	RaMeCDLeu-QCD16Br	45		6	17

Acknowledgements

The authors are grateful to CNRS and the Agence Nationale de la Recherche for the financial support of the SUPRANANO program (ANR-09-BLAN-0194). We also thank David Landy and Eléonore Bertaut (Université du Littoral Côte Opale) for Isothermal Titration Calorimetry measurements and Gregory Crowyn (Université d'Artois ) for NMR measurements. We thank Buchler GmbH for Quincoridine precursor gift.


Tien Duc Cao
email: tien-duc.cao@inria.fr

Ioana Manolescu
email: ioana.manolescu@inria.fr

Xavier Tannier
email: xavier.tannier@limsi.fr

Extracting Linked Data from statistic spreadsheets

Statistic data is an important sub-category of open data; it is interesting for many applications, including but not limited to data journalism, as such data is typically of high quality, and reflects (under an aggregated form) important aspects of a society's life such as births, immigration, economic output etc. However, such open data is often not published as Linked Open Data (LOD) limiting its usability.

We provide a conceptual model for the open data comprised in statistic files published by INSEE, the leading French economic and societal statistics institute. Then, we describe a novel method for extracting RDF LOD populating an instance of this conceptual model. Our method was used to produce RDF data out of 20k+ Excel spreadsheets, and our validation indicates a 91% rate of successful extraction.

Introduction

The tremendous value of Big Data has been noticed of late also by the media, and the term "data journalism" has been coined to refer to journalistic work inspired by data sources [START_REF] Gray | The Data Journalism Handbook: How Journalists can Use Data to Improve the News[END_REF]. While data of some form is a natural ingredient of all reporting, the increasing volumes of available digital data as well as its increasing complexity lead to a qualitative jump, where technical skills for working with data are stringently needed in journalism teams.

A class of data sources of particularly high value is that comprised into government statistics. Such data has been painstakingly collected by state administrations which sometimes have very significant human and technology means at their disposal, especially in countries like France with an old history of statistics gathering1 . This makes the data oftentimes of high quality. Another important quality of such government statistic data is that it logically falls into the open data category, since it is paid for with taxpayer money.

Such high-quality statistic data is a very good background information to be used when trying to assess the truthfulness of a claim. A general description of a factchecking scenario can be constructed as follows: first, a claim is made. Second, journalists or other fact-checkers look up reference sources providing relevant statistics for the assessment of the claim; Third, some post-processing may need to be applied to compute from the reference source values, the value corresponding to the claim. This post-processing may involve elaborate steps.

In this work, we focus on the extraction of reference sources. In the best-case scenario, reference data sources are available to fact-checkers in a format and organized according to a model that they understand well. For instance, in [WAL + 14], unemployment data is assumed available in a temporal database relation. This assumption is very reasonable for statisticians and other scientists working within the statistics institutes. However, unfortunately, most fact-checking actors are outside these institutes, and only have access to the data through publication by the institute. This raises the question of the format and organization of the data. While the W3C's best practices argue for the usage of RDF in publishing open data [START_REF]The World Wide Web Consortium (W3C). Best practices for publishing linked data[END_REF], in practice open data is often published in other formats, such as Excel, HTML or PDF; in the latter formats, statistic data is typically shown as tables or charts. In the particular case of the INSEE 2 , the leading French social and economic statistics institute, while searching for some hot topics in the current French political debate (youth unemployment, immigrant population etc.) we found many very useful datasets, available as Excel and/or HTML and sometimes accompanied by a PDF report explaining and commenting the data, but we did not find the data in a structured, machine-readable format.

Motivated by this limitation, we propose an approach for extracting Linked Open Data from Below, we describe the data sources we work with and the model for the extracted data (Section 2) and our extraction method (Section 3), outline the format of the extracted RDF data (Section 4), and present an evaluation of our extraction process (Section 5), before discussing related work and concluding.

Reference statistic data

In Section 2.1, we present the input INSEE statistics, while Section 2.2 describes our conceptual model for this data. could be quite easily adapted to HTML tables in the future. Also, while we targeted INSEE in this work, we found similar-structure files published in other Open Data government servers in the UK3 and the US4 . Thus, we believe our tool can be used to handle more than the data sources we considered so far.

INSEE data sources

We view each spreadsheet file as a collection of data sheets D 1 , D 2 , . . . Each data sheet D has a title D.t and optionally an outline D.o. The title and outline are given by the statisticians producing the spreadsheets in order to facilitate their understanding by human users, such as the media and general public. The title is a short nominal phrase stating what D contains, e.g., "Average mothers' age at the birth of their child per department in 2015". The outline, when present, is a short paragraph which may be found on a Web page from where the file enclosing D can be downloaded. The outline extends the title, for instance to state that the mothers' ages are rounded to the closest smaller integers, that only the births of live (not stillborn) children are accounted for etc.

In a majority of cases (about two thirds in our estimation), each data sheet D comprises one statistics table, which is typically a two-dimensional aggregate table. Data is aggregated since statistics institutes such as IN-SEE are concerned with building such global measures of society or economy phenomena, and the aggregate tends to be bidimensional since they are meant to be laid out in a format easy for humans to read. For instance, Figure 1 illustrates a data sheet for the birth statistic dataset mentioned above. In this example, the two dimensions are the mother's age interval, e.g., "16-20", "21-25", "26-30" etc. and her geographic area, specified as a department, e.g., "Essonne", "Val-de-Marne" etc. For a given age interval and region, the table includes the number of women which were first-time mothers in France in 2015, in that age interval and department. In Figure 1, we have included a top row and column on a gray background, in order to refer to the row and column numbers of the cells in the sheet. We will use the shorthand D r,c to denote the cell at row r and column c in D. We see that the table contains data cells, such as D 6,3 , D 6,4 , which hold data values, etc. and header cells, e.g., "Region", "Age below 30" etc., which (i) characterize (provide context for) the data values, and (ii) may occupy several cells in the spreadsheet, as is the case of the latter.

The basic kind of two-dimensional agregate in a data sheet is (close to) the result of a group-by query, as illustrated in Figure 1. However, we also found cases where the data sheet contains a data cube [GCB + 07], that is: the result of a group-by, enhanced with partial sums along some of the group-by dimensions. For instance, in the above example, the table may contain a row for every region labeled "Region total" which sums up the numbers for all departments of the region, for each age interval 5Dimension hierarchies are often present in the data. For instance, the dimension values "16-20", "21-25", "26-30" are shown in Figure 1 as subdivisions of "Age below 30", while the next four intervals may be presented as part of "Age above 31". The age dimension hierarchy is shown in magenta in Figure 1, while the geographical dimension hierarchy is shown in blue. In those cases, the lowergranularity dimension values appear in the table close close to the finer-granularity dimension values which they aggregate, as exemplified in Figure 1. Cell fusion is used in this case to give a visual cue of the finer-granularity dimension values corresponding to the lower-granularity dimension value which aggregates them.

A few more rules govern data organization in the spreadsheet:

• D 1,1 (and in some cases, more cells on row 1, or the first few rows of D) contain the outline D.o, as illustrated in Figure 1; here, the outline is a long text, thus a spreadsheet editor would typically show it overflowing over several cells. One or several fully empty rows separate the outline from the topmost cells of the data table; in our example, this is the case of row 2. Observe that due to the way data is spatially laid out in the sheet, these topmost cells are related to the lowest granularity (top of the hierarchy) of one dimension of the aggregate. For instance, in Figure 1, this concerns the cell at line 3 and columns 3 to 9, containing Mother's age at the time of the birth.

• Sometimes, the content of a header cell (that is, a value of an aggregation dimension) is not immediately understandable by humans, but it is the name of a variable or measure, consistently used in all IN-SEE publications to refer to a given concept. Exactly in these cases, the file containing D has a sheet immediately after D, where the variable (or measure) name is mapped to the natural-language description explaining its meaning. For example, we translate SEXE1 AGEPYR10186 into Male from 18

to 24 years old. Finally, note that our current data model does not account for data cubes, that is: we do not model the cases when some data cells are aggregations over the values of other data cells. Thus, our current extraction algorithm (described below) would extract data from such cubes as if they were result of a plain aggregation query. This clearly introduces some loss of information; we are working to improve it in a future version.

Extraction

We developed a tool for extracting data from spreadsheet files so as to populate an instance of our conceptual data model; it uses the Python library xlrd7 for accessing spreadsheet content. Below, we present the extraction procedure. Unlike related work (see Section 6), it is not based on a sequence classification with a supervised learning approach, but on a more global rule-based system aiming at framing the data cell range and extracting the headers around these data cell.

For each sheet, the first task is to identify the data cells. In our corpus, these cells contain numeric information, that is: integers or real numbers, but also interval specifications such as "20+" (standing for "at least 20"), special null tokens such as "n.s." for "not specified". Our approach is to identify the rectangular area containing the main table (of data cells), then build data cells out of the (row, column) locations found in this area. Note that the area may also contain empty locations, corresponding to the real-life situation when an information is not available or for some reason not reported, as is the case for D 9,9 in Figure 1.

Data cell identification proceeds in several steps.

Identify the leftmost data location (ldl) on each row:

this is the first location encountered from left to right which (i) follows at least one location containing text (such locations will be interpreted as being part of header cells), and (ii) has a non-empty content that is compatible with a data cell (as specified above). We store the column numbers of all such leftmost data locations in an array called ldl; for instance, ldl holds for rows 2 to 5, while ldl[i] = 3 for i = 6, 7 etc.

For each row i where ldl[i]

= -1, we compute a row signature sig[i] which is counts the non-empty locations of various types found on the row. For instance, sig[6] = {string : 2, num : 7} because the row has 2 string-valued cells (" Île-de-France" and "Essonne") and 7 numerical cells. For each signature thus obtained, we count the number of rows (with data cells) having exactly that signature; the largest row set is termed the data table seed. Note that the rows in the data table seed are not necessarily contiguous. This reflects the observation that slightly different-structured rows are sometimes found within the data table; for instance, D 9,9 is empty, therefore row 9 is not part of the seed, which in this example, consists of the rows {6,7,8,10,11} and possibly other rows (not shown) below.

3. We "grow" the data table seed it by adding every row at distance 1 or at most 2 above and below the data table seed, and which contains at least a data cell. We do this repeatedly until no such rows can be found. In our example, this process adds row 9, despite the fact that it does not have the same signature as the others. We repeat this step in an inflationary fashion until no row adjacent to the current data table qualifies. The goal of this stage is to make sure we do not miss interesting data cells by being overly strict w.r.t. the structure of the rows containing data cells.

At this point, all the data cells are known, and their values and coordinates can be extracted.

Identification and extraction of header cells

We first look for those organized in rows, e.g., rows 3, 4 and 5 in our example.

1. We attempt to exploit the visual layout of the file, by looking for a horizontal border which separates the first header row of the (usually empty) row above it.

(a) When such a border exists (which represents a majority but not all the cases), it is a good indicator that header cells are to be found in the rows between the border and going down until (and before) the first data row. (Note that the data rows are already known by now.) In our example, this corresponds to the rectangular area whose corners are D 3,1 and D 4,9 .

(b) When the border does not we look at the row immediately above the first data row, and then move up row by row to the top of the sheet; in each row i, we examine the locations starting at the column ldl[i] (if this is greater than 0). If the row doesn't contain empty locations, we consider that it is part of the header area. Intuitively, this is because no data cell can lack a value along one of the dimensions, and header cells carry exactly dimension values. An empty header cell would correspond precisely to such a missing dimension value.

2. Once the header area is known, we look for conceptual header cells. The task is complicated by the fact that such a cell may be spread over several locations: typically, a short text is spread from the top to the bottom over locations situated one on top of the other. This can also happen between locations in the same row (not in the same column), which are fused: this is the case of D 3,3 to D 3,9 in our example.

The locations holding a single dimension value are sometimes (but not always) fused in the spreadsheet; sometimes, empty locations are enclosed within the same border as some locations holding text, to obtain a visually pleasing aspect where the dimension value "stands out well". We detect such groups by tracking visible cell borders, and create one conceptual header cell out of each area border-enclosed area 8 .

Next, we identify and extract header cells organized in columns, e.g., columns 1 and 2 in our example. This is done by traversing all the data rows in the data area, and collecting from each row i, the cells that are at the left of the leftmost data cell ldl[i].

Finally, we populate the instance of the data model with the remaining relationships we need: we connect each data and header cell to its respective locations; we connect each data cell with its closest header cell on the same column, and with its closest header cell on the same row; and we connect each row (respectively, column) header 8 The reader may wonder at this point: if a horizontal border on top of the header row is absent, how can one detect multi-location header cells by following borders? This case has never turned out in the files we examined, which is reasonable if we consider the hard-to-interpret visual aspect it would lead to. Thus, in our experience, when the top border is absent, each row header cell occupies only one location.

cell to its closest header cell above (respectively, at left) through the relation aggregated by.

Linked Data vocabulary

To represent the extracted data in RDF, we created an RDF class for each entity, and assigned an URI to each entity instance; the properties and their typing follow directly from the entity attributes and relationships shown in Figure 2. Some sample triples resulting from the extraction are shown in Figure 3. We use the namespace inseeXtr.excel: for classes and properties we introduce in our extraction from INSEE Excel tables, and the namespace rdf: to denote the standard namespace associated to the W3C's type property. The snippet in Figure 3 corresponds to one Excel file, one sheet in the file, a row header cell, a column header cell, a data cell associated to them, and the location of the data cell.

Evaluation

From 20,743 Excel files collected (see 2.1), we selected at random 100 unseen files to evaluate the reliability of the extraction process; these files contained a total of 2432 tables. To avoid very similar files in our evaluation, e.g., tables showing the same metric for distinct years, we required the first three letters in the name of each newly selected file, to be different from the first three letters in the names of all the files previously selected to be included in the test batch.

We did the evaluation manually by looking at Excel files and identifying their header cells, data cells and header's hierarchy; then comparing them with the results from our extractor. We only assign the label "extracted correctly" for one table when there is an exact match of all cells and header's hierarchy. All of the remaining cases were marked as "extracted incorrect". The following table shows a summary of evaluation result:

Category

Number % Tables that were extracted correctly

91%

Tables that were extracted incorrectly

9%

Figure 3: Sample extracted RDF triples.

We have inspected the tables which were not handled correctly, and we saw that most of them had sheets with several tables (not just one). Our system is currently not able to correctly recognize these; we plan to address this limitation in the near future.

Overall, the extraction produced 2.68 • 10 6 row header cells, 122 • 10 6 column header cells (table height is often higher than the table width), and 2.24•10 9 data cells (three orders of magnitude more data cells than header cells). It took 5 hours to run our extraction algorithm.

Related work

Closest to our work, Chen et al [START_REF] Chen | Automatic web spreadsheet data extraction[END_REF] built a system that could automatically extract relational data from spreadsheets. Their system used conditional random field (CRF) [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting and labeling sequence data[END_REF] to identify data frames: such a frame is a value region which contains numeric cells, top attributes and left attributes corresponding to header cells and their hierarchies. From a collection of 410,554 Excel files collected from the Web, they selected randomly 100 files, labeled 27,531 non-empty rows (as title, header, data or footnote) manually by human experts, then used this dataset for training and evaluation. They obtained significant better precision and recall metrics when taking into account both textual and layout features for CRF. SVM was applied on header cells to learn about their hierachies. The model achieved a precision of 0.921 for top headers, and 0.852 for left headers. However, these numbers should not be compared with our results, because the datasets are different (tables encountered on the Web vs. government statistics), as well as the evaluation metrics (cell-by-cell assessment in [START_REF] Chen | Automatic web spreadsheet data extraction[END_REF] vs. binary assessment over the entire sheet extraction in our case). Overall, it can be said that their method is more involved whereas our method has the advantage of not requiring manual labeling at all. As we continue in our work (see below), we may experiment with learning-based approaches also.

Apart from spreadsheet data extraction, the high number of public available spreadsheets on the Web also attracted interest for other data management tasks. For example, Ahsan, Neamt ¸u et al. [START_REF] Ahsan | Towards spreadsheet integration using entity identification driven by a spatial-temporal model[END_REF] built an extractor to automatically identify in spreadsheets entities like location and time.

When one considers HTML and not Excel data, data extraction from tables encoded as HTML lists has been addressed e.g., in lists [START_REF] Elmeleegy | Harvesting relational tables lists on the web[END_REF]. Structured fact extraction from text Web pages is an extremely hot topic, and has lead to the construction of high-quality knowledge bases such as Yago [MBS15] and Google's Knowledge Graph [DGH + 14] built out of Freebase and enriched through Web extraction. Our work has a related but different focus as we focus on high-value, highconfidence, fine-granularity, somehow-structured data such as government-issue statistics; this calls different technical methods.

In prior work [GKK + 13], we devised a tool for exploiting background information (in particular, we used DBPedia RDF data) to add context to a claim to be fact-checked; in this work, we tackle the complementary problem of producing high-value, high-confidence Linked Open Data out of databases of statistics.

Conclusion and perspectives

We have described an ongoing effort to extract Linked Open Data from data tables produced by the INSEE statistics institute, and shared by them under the form of Excel tables. We do this as part of our work in the ContentCheck R&D project, which aims at investigating content management techniques (from databases, knowledge management, natural language processing, information extraction and data mining) to provide tools toward automating and supporting fact-checker journalists' efforts. The goal is to use the RDF data thus extracted as reference information, when trying to determine the degree of thruthfulness of a claim. We foresee several directions for continuing this work:

Improving the extractor As previously stated, we currently do not correctly detect several data tables in a single sheet; we believe this can be achieved with some weighted extraction rules (the current extraction method does not use weights). We also plan to extend the extraction to correctly interpret partial sum rows and columns in Excel files containing cubes, e.g., the example of footnote 5. Finally, we will evaluate (and possibly improve) our extraction process on other data sources such as government data from other countries.

Entity recognition in cells and data linking

The usefulness of the extracted data will be enhanced when we will be able to match data cells with entities known from existing data and knowledge bases, e.g., one can recognize "Essonne" in D 6,2 is the same as the DBPedia entity http://fr.dbpedia.org/resource/Essonne (d épartement) and use this URI instead of the "Essonne" string, to interconnect the data with other linked data sources.

Data set and fact search The ultimate goal is to enable users to find information in our extracted data, thus we have started to consider methods which, given a query of the form "youth unemployment in 2015", identify (i) the data files most relevant for the query, and (ii) finegranularity answers directly extracted from the file. Observe that the latter task is strongly facilitated by the finegranularity extraction presented here; we still need to develop and experiment with scoring techniques in order to solve (i).

Last but not least, we plan to approach INSEE concerning the authorization to share the RDF data thus produced.

We plan to make our code available in open-source by the time of the workshop (refactoring and cleanup is ongoing).

  INSEE publishes a variety of datasets in many different formats. In particular, we performed a crawl of the INSEE web site under the categories Data/Key figures, Data/Detailed figures, Data/Databases, and Publications/Wideaudience publications, which has lead to about 11,000 web pages including 20,743 Excel files (with total size of 36GB) and 20,116 HTML tables. In this work, we focused on the set of Excel files; we believe our techniques 2 INSEE: French National Institute of Statistics and Economic Studies, http://www.insee.fr.

Figure 2 :

 2 Figure 2: Conceptual data model.

  INSEE Excel tables. The two main contributions of our work are: (i) a conceptual data model (and concrete RDF implementation) for statistic data such as published by INSEE and other similar institutions, and (ii) an extraction algorithm which populates this model based on about 11,000 pages including 20,743 Excel files published by INSEE.

  For instance, the closest column header cell for the data cell at D 7,4 is D 5,4 , while the closest column header cell is D 7,2 . Further, the header cells are organized in an aggregation hierarchy materialized by the respective relation "aggregated by". For instance, D 5,4 is aggregated by D 4,3 , which is agregated by D 3,3 . In each sheet, we identify the top dimension values as those which are not aggregated by any other values: in our example, the top dimension values appear in D 3,3 , D 6,1 and D 9,1 . Note that such top dimension values are sometimes dimensions names, e.g., Region, and other times dimensions values, e.g., Île-de-France. Indeed, statistics in a sheet are not guaranteed to be organized exactly as predicted by the formal models of relational aggregation queries, even if they typically come very close.

	Variable	Meaning
		Sex
	1	Male
	2	Female
	AGEPYR10	Age group
	00	Less than 3 years old
	03	3 to 5 years old
	06	6 to 10 years old
	11	11 to 17 years old
	18	18 to 24 years old
	25	25 to 39 years old
	. . .	. . .
	80	80 years old
	2.2 Conceptual data model

From the above description of the INSEE statistic datasets, we extract a conceptual data model shown in Figure

2

. Entities are shown as boxes while attributes apper in oval shapes. We simplified the notation to depict relationships as directed arrows, labeled with the relationship name. Data cells and header cells each have an attribute indicating their value (content); by definition, any data or header cell has a non-empty content. Moreover, a data cell has one associated location, i.e., (row number, column number), while a header cell may occupy one or several (adjacent) locations. Each data cell has a closest header cell on its column, and another one on its line; these capture the dimension values corresponding to a data cell.

In France, this started with Jean-Baptiste Colbert (1619-1683), the acclaimed finance minister of king Louis XIV).

E.g., http://www.ic.nhs.uk/catalogue/PUB08694/ifs-uk-2010-chap1-tab.xls.

E.g., http://www.eia.gov/totalenergy/data/monthly/query/mer data excel.asp? table=T02.01.

A sample real-life file from INSEE illustrating such cube tables is: https://www.insee.fr/fr/statistiques/fichier/2383936/ARA CD 2016 action sociale 1-Population selon l age.xls.

https://www.insee.fr/fr/statistiques/fichier/2045005/BTX TD POP1A 2013.zip

https://pypi.python.org/pypi/xlrd

Acknowlegements

This work was supported by the Agence Nationale pour la Recherche (French National Research Agency) under grant number ANR-15-CE23-0025-0.

rdf:type <http://inseeXtr.excel:Dataset>; inseeXtr:source "http://insee.fr/.../Age Mère ....xls"; inseeXtr:name "Age Mère ....xls"; inseeXtr:description "The data reflects children..." . <http://inseeXtr.excel:Sheet0> rdf:type <http://inseeXtr.excel:Sheet>; inseeXtr:title "sheet title"; inseeXtr:belongsTo <http://inseeXtr.excel:File0> . <http://inseeXtr.excel:HeaderCellY0> inseeXtr:value "x1"; rdf:type <http://inseeXtr.excel:HeaderCellY>; inseeXtr:YHierarchy <http://inseeXtr.excel:Sheet0> . <http://inseeXtr.excel:HeaderCellX2> inseeXtr:value "a1"; rdf:type <http://inseeXtr.excel:HeaderCellX>; inseeXtr:aggregatedBy <http://inseeXtr.excel:HeaderCellX3>; inseeXtr:XHierarchy <http://inseeXtr.excel:Sheet0> . <http://inseeXtr.excel:DataCell0> inseeXtr:value "1"; rdf:type <http://inseeXtr.excel:DataCell>; inseeXtr:location <http://inseeXtr.excel:Location0>; inseeXtr:closestXCell <http://inseeXtr.excel:HeaderCellX2>; inseeXtr:closestYCell <http://inseeXtr.excel:HeaderCellY0>. <http://inseeXtr.excel:Location0> rdf:type <http://inseeXtr.excel:DataCell>; inseeXtr:Row "2"; inseeXtr:Col "3" .