Fida Karam
email: karam.f@gust.edu.kw

Wassim Daher
email: daher.w@gust.edu.kw

Insider Trading In A Two-Tier Real Market Structure Model

Keywords: JEL classification: G14, D82 Insider Trading, Two-Tier Market, Stackelberg, Correlated signals, Kyle model

), this two-tier real market structure does not alter the amount of information disseminated in the stock price or the level of insider trading. Next, we add a second insider to the model. We show that competition in the financial sector decreases the level of output produced by firm 1 and the price of the intermediate good with respect to initial model. Moreover, it affects the insiders' trades and increases the amount of information revealed in the stock price.

Introduction

It is not a coincidence that corporate executives seem to always buy and sell their company's securities at the right times. After all, they have access to every bit of company information, due to their participation to real activities underlying the financial assets that are the subject of insider trading. Because real and financial decisions are inevitably intertwined, the theoretical research on insider trading has started to extend the pure financial models in the spirit of [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] to include the real aspects of the firm (Jain and[START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF][START_REF] Jain | Effects of Insider Trading under Different Market Structures[END_REF]Daher and[START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF]2007). The insider is modeled as the manager of the firm who chooses how much stock to buy and how much output to produce in the real market, thus affecting the profitability of the firm. Sometimes, the publicly-owned firm is a quantity-setting monopolist in the real market [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF][START_REF] Daher | Market Structure and Insider Trading[END_REF], and sometimes it is competing with another privately-owned firm in a Cournot way to determine the quantity produced of the homogeneous good [START_REF] Jain | Effects of Insider Trading under Different Market Structures[END_REF]; [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF]. In both cases and to simplify the analysis, the real good was produced at a no cost.

However, the production process involves the use of primary production factors such as labor, capital, and land, as well as intermediate goods. It is a characteristic feature of industrial economies that commodities are produced by means of commodities. Data from the OECD input-output tables (OECD, 2004) show that the share of intermediate goods in production ranges from 19% to 82% across different sectors. Any change in the cost of these intermediate goods will ripple throughout a market economy, affecting the market of the final good. In other words, buying and selling relationships link firms vertically, and through these links firms engage in market interactions while performing different functions in the value chain.

In this paper, we address this issue by assuming that the production of the final good by the publicly-owned firm (firm 1) involves constant labor costs as well as costs of intermediate goods. Those are locally produced by a privately-owned firm (firm 2). Specifically, we analyze insider trading in a static model in the spirit of [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF], where the insider of firm 1 is also the manager of that firm and thus makes both real and financial decisions.

However, in our model, the choice of the real variables will affect the market of the intermediate good, and vice versa. A sketchy analysis suggests that an increase in the price of the intermediate good is able to decrease the profit of firm 1, ceteris paribus. A lower profit induces the market maker to set a lower stock price. Inversely, any negative demand shock in the market of the final good is able to affect the demand for the intermediate good and thus its price.

In our model, firm 1, managed by the insider, is assumed to be a monopolist. We follow [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] in modeling the financial market environment and thus study a linear-normal equilibrium. Further, we assume that the stock orders are submitted by an insider as well as noise traders. A market maker sets stock prices competitively. The insider knows the true value of a random shock to the value of the firm whereas the market maker knows only the distribution of this shock. However, following Jain and Mirman (1999), the market maker observes the total stock order as well as the noisy market price of the real good before setting the price of the stock. The insider chooses the real output of the final good and the stock to be traded simultaneously. Firm 1 buys the necessary quantity of the intermediate good from firm 2, which also holds a monopoly power on the market of the intermediate good.

We show that insider trading affects the markets of both the final and the intermediate goods, and that the financial market variables change due to the insider's real decisions. For instance, the real output chosen by the insider or manager of firm 1 as well as the price of the intermediate good set by firm 2 are both greater than it would be in the absence of insider trading. Furthermore, the parameters of both real markets affect the stock price and the stock pricing rule. Finally, we show that, when compared to [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] and ( 2002), the two-tier real market structure does not alter the amount of information disseminated in the stock price or the level of insider trading.

Next, following [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] and ( 2007), we add a second insider, the owner of firm 1, and assume that he has no managerial responsibilities. Then, we carry out a comparative static analysis between model II (duopoly in the financial market) and model I (monopoly in the financial market). We show that competition in the financial sector affects the stock price coefficients as well as real variables. In particular, the level of output produced by firm 1 and the price of the intermediate good both decrease with respect to model I. Hence, the addition of another informed trader influences the production decision of firm 1, as well as the decision of firm 2. Moreover, competition in the stock sector affects the insiders' trades. Finally, the results show that Cournot competition in the stock market increases the amount of information revealed in the stock price.

Model I: The Monopoly Case

Let (Ω, F, P) be a probability space, all the random variables are defined with respect to this probability space. Consider an economy characterized by a financial sector and a two-tier real market structure in which the production of a final good requires an intermediate good (or service). More specifically, the real market is characterized by two firms, firm 1 holding a monopoly in producing the final good and firm 2 also holds a monopoly in the production of the intermediate good 2 .

The structure of the real market draws on existing models that explore the effect of foreign direct investment on backward linkages3 . More specifically, our model is a variant of the Cournot model developed in [START_REF] Lin | Multinational firms, exclisivity, and backward linkages[END_REF], which in turn is based on [START_REF] Salinger | Vertical Mergers and Market Foreclosure[END_REF]. For producing one unit of the final good, firm 1 requires l units of labor and c units of the intermediate good. The marginal cost of producing one unit of the final good is then equal to the sum of the labor cost and the cost of intermediate goods. For simplicity, the wage rate is normalized to 1. The price of the intermediate good w is set by firm 2. The inverse demand function for the final good is assumed to be linear and stochastic, i.e, q = (a -y)z and the unitary profit of firm 1 is also stochastic and given by,

= (a -y -l -cw)z
where a is positive constant and z is a random variable, normally distributed with mean z (assumed positive) and variance σ 2 z .

Firm 2 produces the intermediate good at constant marginal cost τ . Firm 1 and firm 2 interact in a two-stages process. First, firm 2 chooses the quantity of the intermediate good to supply, and accordingly, its unitary price w. Then, the manager of firm 1 decides the amount of the final good y to produce given the price of the intermediate good w.

On the other hand, we consider a financial market where firm 2 is privately owned and firm 1 is publicly owned. The stock of firm 1 is publicly traded in a competitive financial market. The value of the stock is the net profits of the firm per share. We assume, like in [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF], that there are three kinds of agents in the market. First, there is a risk-neutral rational trader, the manager of firm 1, who knows the realization z of z, the value of the stock. Second, there are non-rational noise traders, representing small investors with no information on z. The aggregate noise trade is assumed to be a random variable ũ, which is normally distributed with mean zero and variance σ 2 u . Finally, there are K(K ≥ 2) risk-neutral market makers who act like Bertrand competitors.

We follow [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] by allowing the market makers to observe two signals. The first one, from the real market, is denoted by q = (a -yl -cw)(z + ε) where ε is normally distributed with mean 0 and variance σ 2 ε . The second one is the total order flow, i.e., r = x+ ũ where x is the manager's trading order. We assume that z, ũ and ε are pairwise independent.

Following [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF], the financial trading mechanism is organized in two steps. In step one, a linear pricing rule and optimal order rule are determined by the market makers and the insider, respectively, as a Bayesian equilibrium. The market makers determine a (linear) pricing rule p, based on their a priori beliefs, where p is a measurable function p : R × R -→ R. The insider chooses a stock trade function x = x(z), where x : R -→ R is a measurable function. In the second step, the insiders observe the realization z of z, 4 and submit their stock order to the market makers based on the equilibrium stock trade functions. The market makers also receive orders from the noise traders, all these orders arrive as a total order flow signal r = x(z) + ũ. The two signals are used by the market makers to set the price p = p(q, r), based on the equilibrium price function, to clear the market.

The insider only knows the realization z of z and does not know the values of ũ, ε, r, z + ε before his order flow decisions are made. Moreover, neither the market maker nor firm 2 know the realization z of z but both know its distribution. Finally, the market makers cannot observe either x, u or ε.

The value per share of the firm 1 is the net profit of the firm per share, i.e.,

v = (a -y -l -cw)yz a, b > 0 (1)
and the profit of the manager is given by,

ψ 1 = (v -A -p) x1 + A x1 ( 2
)
where A is the compensation scheme received by the manager for his managerial tasks 5 . When the manager receives his compensation, the value of the firm becomes:

v = v -A.
Firm 2 chooses the quantity of the intermediate good y I that maximizes its profits, given the quantity of the final good y produced by firm 1, and sets accordingly the price w of the intermediate product. We assume that the intermediate product is produced at a constant marginal cost τ . Hence, firm 2's profits are given by,

ψ 2 = (w -τ )y I
such that w = f (y I ) is the inverse demand function for the intermediate good and cy = y I .

The Equilibrium Concept

This is a game of incomplete information because the market makers, unlike the insider, do not know the realization of z. Hence, we seek a Bayesian-Stackelberg equilibrium. A Bayesian-Stackelberg equilibrium is a vector of four functions [x(.), y(.), y I (.), p(., .)] such that:

(a) Profit maximization of the manager of firm 1,

E((a -y -l -cw)yz -p((a -y -l -cw)(z + ε), x + ũ))x ≥ E((a -y -l -cw)y z -p((a -y -l -cw)(z + ε), x + ũ))x (3)
for any level of trading order x and level of output y decided by the manager of firm 1 .

(b) Profit maximization of firm 2, (w(y I (y)) -τ )y I (y) ≥ (w(y

I (y))) -τ )y I (y) (4)
for any level of quantity y I of the intermediate good decided by firm 2 such that cy = y I and w is the inverse demand function for the intermediate good.

(c) Semi-Strong Market Efficiency: The pricing rule p(., .) satisfies,

p(q, r) = E[ṽ|q, r]. ( 5
)
An equilibrium is linear if there exists constants µ 0 , µ 1 , µ 2 such that, ∀q, r, p(q, r)

= µ 0 + µ 1 q + µ 2 r. (6)
Note that condition (3) defines the optimal strategy of the insider while (4) reflects the Stackelberg structure of the real market. Since firm 2 is the leader (firm 1 is the follower) so it takes the firm's 1 output decision into its maximization problem. Condition (5) guarantees the zero expected profits for the market makers. The stock price, set by the market makers, is equal to the conditional expectation of the asset given the available information. We restrict our study to linear equilibrium. The normal distributions of the exogenous random variables, together with the particular expression of the demand, enable us to derive and to prove the existence of a unique linear equilibrium.

Optimality Results

We have a game of incomplete information because the market makers, unlike the insider, do not know the realization of z. We solve backward the maximization problem of the insider in stage 2, given the assumption of linear equilibrium. The manager chooses (y, x) to maximize his expected profit, given z. So, Max

(y,x) G where G = E [ v -p]x + A.x = E [ v -p]x. This is equivalent to, Max (y,x) [(a -y -l -cw)yz -µ 0 -µ 1 (a -y -l -cw)z -µ 2 x]x.
The first order necessary conditions are,

x(z) = (a -y -l -cw)(y -µ 1 )z -µ 0 2µ 2 , (7)
and

y = (a -l -cw + µ 1 ) 2 . ( 8
)
From equation 8, the inverse demand for the intermediate goods equals:

w = (a -l -2y + µ 1 ) c . (9)
In stage 1, firm 2 maximizes its profits, given the quantity of the final good produced by firm 1. Plugging the expression of w in equation 9, the maximization problem of firm 2 becomes, Max

y I (( a -l -2y I /c + µ 1 c -τ )y I )
The first order condition implies

y I = (a -l + µ 1 )c -c 2 τ 4 (10)
w = (a -l + µ 1 + cτ ) 2c (11)
Now, we determine the price function coefficients. First, we have,

Lemma 1

The coefficients of the price function are, (i)

µ 1 = (a -l -cτ )σ 2 z 3σ 2 z + 8σ 2 ε = F k, (ii)
µ 2 = F 2 σ ε (1 -3k) (1 -3k)k 8 √ 2σ u ,
where

k = σ 2 z 3σ 2 z +8σ 2 ε and F = (a -l -cτ ) with F > 0 Proof: See the Appendix.
Second, as in Jain-Mirman (2000), the compensation scheme of the insider is needed to ensure the second order condition. Indeed, combining the second order condition 2x(z)z > 0 and µ 2 > 0, [START_REF] Julien | A note on Stackelberg Competition[END_REF] with expression of the µ 0 given by

µ 0 = v -µ 1 q -µ 2 r = (a -y -l -cw)(y -µ 1 )z -A
In proposition 1, we provide a sufficient and necessary condition to ensure the existence of linear equilibrium.

Proposition 1 A linear equilibrium exists if and only if,

µ 0 = 0 ⇔ A = (a -y -l -cw)(y -µ 1 ) 2 z.
Proof: See Jain and Mirman (2000).

We turn now to the characterization of the equilibrium. Proposition 2 whose proof is relegated in appendix B, characterizes the amount of information revealed in the equilibrium stock price as well as the insider's conditional profits.

Proposition 2

The information revelation and the insider's conditional profits are:

V ar(z|q, r) = σ 2 ε σ 2 z + 2σ 2 ε σ 2 z (13) π(z) = 2(a -l -cτ ) 2 σ u σ 4 ε z2 (3σ 2 z + 8σ 2 ε ) 2 σ z (14)

Discussion of the equilibrium

There are several properties of this equilibrium that are worth emphasizing. First, note that the total output produced by firm 1 and the price of the intermediate good charged by firm 2 are both more than what is expected in a two-tier real market structure without insider trading. Indeed, equations ( 8) and [START_REF] Jain | Insider Trading with Correlated Signals[END_REF] given by

y = (a -l -cw + µ 1 ) 2 w = (a -l + µ 1 + cτ ) 2c (15)
show that both the total output y and the intermediate good price w are increasing functions of µ 1 , the market maker response to the real signal. Thus insider trading increases the production of the final good, thereby lowering the price. This happens since an increased output has the effect of lowering the real output price. A lower real output price conveys, ceteris paribus, a lower realization (on average) of z to the market maker and thus induces him to set a lower stock price.

Moreover, insider trading increases the price of the intermediate good as well. Indeed, a greater production of the final output increases the demand of the intermediate good addressed to firm 2, which in turn, profits to charge a higher price. Inversely, any increase in the price of the intermediary good is supposed to decrease the quantity demanded of that good by firm 1, and thus decrease the production of the final good. The market price of the final good will then rise to balance the market. If the real output price increases more than the price of the intermediary good, the market maker observes a higher realization (on average) of z and thus sets a higher stock price.

Recall that our model adds a two-tier real market structure to the financial market of Jain and Mirman (1999) with one insider and two signals observed by the market maker. The effect of real market on Jain and Mirman (1999) has also been explored in Jain and Mirman (2000) and (2002). Indeed, Jain and Mirman (2000) investigated the effect of the monopoly structure in the real market on the model of Jain and Mirman (1999). Jain and Mirman (2002) added Cournot competition in the real market to [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF]. Both papers also showed that insider trading increases the production of the real good. One of the characteristics of the two-tier real market structure presented in this paper is that firm 1, that acts as a follower to firm 2 in the two-tier real market structure, is influenced by the decision of firm 2 which charges the price of the intermediate good. Thus, the insider is thus less able to influence the market price of the final good (and the signal available to the market maker) by increasing his firm's output. Indeed, the signal observed by the market maker now depends on both the price of final good and the price of the intermediate good.

Second, note that the two-tier real market structure does not alter the amount of information disseminated in the stock price. The conditional variance in this model is the same found in Jain and Mirman (2000) and ( 2002) (see equation [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF], implying that the real market structure has no effect on information revelation. This is due to the fact that to the deterministic nature of the real market variables.

Third, the level of insider trading remains the same as in the previous works.

Intuitively, this is due to the fact that the effect of any change in output on the two signals is predictable for the market maker (since the outputs are deterministic) and thus is correctly incorporated in the stock price function (through lower weights on the two signals). Thus the insider's optimal response remains unchanged.

In the next section, we extend our model to two insiders. In other words, we introduce Cournot competition among the insiders in the financial sector. Formally,

Model II: The Duopoly Case

In this model, we extend model I to incorporate Cournot competition between two insiders in the financial market: the manager of the firm and the owner who has no managerial responsibilities 6 . We assume that the two insiders know the realization z of z and trade based on their private information. Introducing Cournot competition in the financial market allows us to study the effect of the strategic competition in the financial market on the equilibrium outcomes.

Under Cournot competition in the financial market, a Bayesian-Stackelberg equilibrium is a vector of five functions [x 1 (.), x 2 (.), y(.), y I (.), p(., .)] such that:

(a) Profit maximization of the manager of firm 1,

E((a -y -l -cw)yz -p((a -y -l -cw)(z + ε), x1 + x2 + ũ))x 1 ≥ E((a -y -l -cw)y z -p((a -y -l -cw)(z + ε), x 1 + x2 + ũ))x 1 (16)
for any level of trading order x 1 and level of output y decided by the manager of firm 1 .

(b) Profit maximization of the owner,

E((a -y -l -cw)yz -B -p((a -y -l -cw)(z + ε), x1 + x2 + ũ))x 2 ≥ E((a -y -l -cw)yz -p((a -y -l -cw)(z + ε), x1 + x 2 + ũ))x 2 (17)
for any level of trading order x 2 decided by the owner of firm 1 .

(c) Profit maximization of firm 2, (w(y I (y)) -τ )y I (y) ≥ (w(y I (y))) -τ )y I (y) [START_REF] O'hara | Market Microstructure Theory[END_REF] for any level of quantity y I of the intermediate good decided by firm 2 such that cy = y I and w is the inverse demand function for the intermediate good.

(d) Semi-Strong Market Efficiency: The pricing rule p(., .) satisfies,

p(q, r) = E[ṽ|q, r]. ( 19
)
Proposition 3, we characterize the unique linear equilibrium.

Proposition 3 Under Cournot competition in the financial market, a linear equilibrium exists and is unique. It is characterized by

x1 = (a -y -l -cw)(y -µ 1 )z 3µ 2 x2 = (a -y -l -cw)(y -µ 1 )(z -z) 3µ 2 (20)
y = (a -l -cw + µ 1 ) 2 w = (a -l + µ 1 + cτ ) 2c (21)
µ 0 = B = (a -y -l -cw)(y -µ 1 )z 3 , µ 1 = (a -l -cτ )σ 2 z 3σ 2 z + 12σ 2 ε = F k 1 , (22)
µ 2 = F 2 σ ε (1 -3k 1 ) (1 -3k 1 )k 1 4 √ 6σ u , V ar(z|q, r) = σ 2 ε σ 2 z + 3σ 2 ε σ 2 z ( 23
)
and

π 1 = 9(a -l -cτ ) 2 σ u σ 4 ε z2 2 √ 2(3σ 2 z + 12σ 2 ε ) 2 σ z π 2 = 9(a -l -cτ ) 2 σ u σ 4 ε (z -z) 2 2 √ 2(3σ 2 z + 12σ 2 ε ) 2 σ z (24)
where

k 1 = σ 2 z 3σ 2 z +12σ 2 ε and F = (a -l -cτ )

Comparative statics

In this section, we present several comparisons between the results of models I and II. Note that these comparisons highlight the effects of competition in the financial sector on both the real and financial sectors as well as the decisions of the insiders. The effect of competition in the financial sector on financial and real variables has been studied in several works before: For instance, Daher and Mirman (2007) investigated the effect of Cournot competition in the financial sector with a monopoly structure in the real market, on the model of [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], characterized by a monopoly structure in both the real and financial markets. Moreover, [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] introduced Cournot competition in the financial sector to the model of [START_REF] Jain | Effects of Insider Trading under Different Market Structures[END_REF] characterized by a monopoly structure in the financial market and Cournot competition in the real market.

Introducing Cournot competition to model I has two effects. On the one hand, it enables us to study the effect of Cournot competition between the insiders as compared to the monopoly case studied in the first model of this paper. On the other hand, it allows us to capture the effect of the two-tier real market structure on the equilibrium variables, as compared with the monopoly and duopoly real market structure existing in previous works.

Equilibrium outcomes

Lemma 2 provides a comparative static analysis of equilibrium outcomes between the Cournot competition model (model II) and the previous model (model I). It shows that the stock price coefficients are affected by Cournot competition in the financial sector. Moreover, the production of firm 1 and the price charged by firm 2 decrease with respect to model I. Finally, we can notice that the manager trades less and earns less than in model 1. Formally

Lemma 2

The effects of Cournot competition on equilibrium outcomes relative to the monopoly case are given by7 ,

µ 0 > µ M 1 0 µ 1 < µ M 1 1 µ 2 < µ M 1 2 (25) w < w M 1 y < y M 1 x1 < xM1 (26)
First, competition in the financial sector affects the stock price set by the market makers. Lemma 2, shows that the intercept coefficient of the stock price, µ 0 , is greater than in model I. Indeed, to satisfy the second order condition of the maximization problem of the manager of firm 1, and to guarantee the existence of a linear equilibrium, competition in the financial sector requires that the compensation scheme of the manager (which is positive) has the value µ 0 . In the absence of competition in the financial sector (model I), µ 0 must be equal to zero. However, in the presence of competition in the financial sector, the market makers set µ 0 different from zero in order to satisfy the zero expected profits.

The value of µ 1 , the response of the market makers to the real signal, is also less than in model I. The same results hold in [START_REF] Daher | Market Structure and Insider Trading[END_REF] who model Cournot competition in the financial market with a monopoly structure in the real market, when compared with [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] where a monopoly structure characterizes both markets. They also hold in [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] where Cournot duopoly exists in the financial and real markets, in comparison with [START_REF] Jain | Effects of Insider Trading under Different Market Structures[END_REF] where Cournot competition characterizes only the real market. In other words, the two-tier real market structure does not alter the previous findings concerning µ 0 and µ 1 . In order to understand this result recall the expression of µ 1 in model I and in our model, i.e.

µ M 1 1 = (a -l -cτ )σ 2 z 3σ 2 z + 8σ 2 ε and µ 1 = (a -l -cτ )σ 2 z 3σ 2 z + 12σ 2 ε
which depends on the value of the firm, the noisy signal and the total order flow (through the coefficient of σ 2 ε ). Hence, as the number of insiders increases, the total order flow increases. Thus the coefficient of the noise in the denominator increases which lowers the value of µ 1 . This result reflects the fact that with two insiders there is more information in the order flow signal which gets more weights, and thus makes the value of µ 1 lower than in model I.

However, and contrary to what is found in the previous works, the response of the market makers to the total order flow signal, µ 2 , is always lower than in model I. This is an interesting result which shows that, in the presence of the two-tier real market structure, the value of µ 2 the relationship which exists between exogenous variances of the model, mainly σ 2 ε and σ 2 z does not affect the value of µ 2 like in [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] and [START_REF] Daher | Market Structure and Insider Trading[END_REF].

Second, competition in the financial sector affects the real market variables. In fact, the level of output produced by firm 1 and the price of the intermediate good both decrease with respect to model I. To see this, recall the expressions of y and w in both models,

y = (a -l -cw + µ 1 ) 2 w = (a -l + µ 1 + cτ ) 2c (27)
Hence, y and w are both a function of µ 1 , the coefficient of the real sector signal of the stock price, which is set by the market makers. Plugging the equation of w in the equation of y gives a positive relation between y and µ 1 . Lemma 2 shows that competition in the stock sector reduces the value of µ 1 and thus, the output of firm 1 as well as the price of the intermediate good with respect to model I. Hence, the addition of another informed trader influences the production decision of firm 1, as well as the decision of firm 2.

Moreover, it worth mentioning that regardless of the type of competition between the insiders, the introduction of the financial sector increases the quantity produced of the final good as well as the price of the intermediate good. Indeed, equation ( 27) reveals the positive correlation between the quantity produced y and µ 1 as well as between the price of the intermediate good w and µ 1 . Thus, the financial market has a positive effect on the twotier real market variables.

Third, competition in the stock sector affects the insiders' trades. This result is consistent with the financial [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] model with one signal (the total order flow), going from one to two insiders [START_REF] Tighe | Three Essays on Insider Trading[END_REF]. Indeed, Lemma 2(iv) shows that the manager's trading level is less than in model I. The same result holds in [START_REF] Daher | Market Structure and Insider Trading[END_REF] where Cournot competition in the financial market is introduced to the real market monopoly structure of [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF]. This result is also true in [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] who extend the real market duopoly structure of [START_REF] Jain | Effects of Insider Trading under Different Market Structures[END_REF] to incorporate Cournot competition between the insiders.

Informativeness of stock price

In this section we examine the extent to which the stock price reveals information. This issue was originally studied by [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] as well as [START_REF] Rochet | Insider Trading without Normality[END_REF]. They found that the stock price reveals exactly half of the inside information, regardless of the parameter values of the model. Moreover, in [START_REF] Kyle | Continuous Auctions and Insider Trading[END_REF] with more than one informed trader, the amount of information revealed increases in the number of traders, but is still constant and independent of the parameter values of the model. This independence result is due to the fact that the insider has only one option, i.e. either to buy or sell stock, and has no effect on the value of the firm. When the manager is able to affect the value of the firm, and with the availability of a second signal to the market maker, the results change. This was particularly the case of [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] who show that, when the manager is a quantity-setting monopolist in the real market, the amount of information released by the insider is greater than half, and, depends on the parameters of the model. [START_REF] Daher | Market Structure and Insider Trading[END_REF] show that, when Cournot competition between the insiders exists, the stock price reveals even more information than [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF], since competition in the stock market makes the order flow more informative. The same is true in [START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF] who extend the Cournot real market structure of [START_REF] Jain | Effects of Insider Trading under Different Market Structures[END_REF] to incorporate Cournot competition in the financial market.

Following [START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] as well as the subsequent works, we adopt the same measure of information i.e., the conditional variance of z, given the information of the market maker.

Lemma 3

The effect of Cournot competition on information revelation relative to the monopoly case is given by, V ar(z|q, r) < V ar(z|q, r) M 1 (28) Lemma 3 shows that there is more information revealed in model II than in model I. Cournot competition in the stock market increases the amount of information revealed in the stock price. This result is consistent with the findings of Daher and[START_REF] Daher | Cournot Duopoly and Insider Trading with Two Insiders[END_REF](2007). Indeed, the expressions of the conditional variances measuring the amount of information revealed in the stock price in model I and in our model are:

V ar(z|q, r)

M 1 = σ 2 ε σ 2 z + 2σ 2 ε σ 2 z and V ar(z|q, r) = σ 2 ε σ 2 z + 3σ 2 ε σ 2 z
The key difference between these two conditional variances is the coefficient of σ 2 ε in the denominator. The greater the value of this coefficient, the greater the amount of information revealed in the stock price is. The origin of this value is the aggregate orders of the insiders. The aggregate order in the Cournot case is greater than in model I. 8 .

Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

Note that the monopoly structure adopted in the real market can be easily extended to Cournot oligopoly in the market of both the final and the intermediate goods.

See for example Batra et al. (2003) and Lin and Saggi (2007).

Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

Random variables are denoted with a tilde. Realized values lack the tilde. The mean of the random variable is denoted with bar.

Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

Following Jain and Mirman (2000), the existence of such compensation scheme is to ensure the second order condition of the manager's maximization problem. For more details, see[START_REF] Jain | Real and Financial Effects of Insider Trading with Correlated Signals[END_REF] and the extended works thereafter.

Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

It should be pointed out that the results can be automatically generalized to n insiders, but for the sake of comparison we restrict our attention to the two-insiders case. 11 Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

The superscript M1 refers to model I. 13 Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

x = x1 + x2 = √ 2σu σz (z -z), xM1 = σu σz (z -z)16 Documents de Travail du Centre d'Economie de la Sorbonne -2011.68

Insider's profits

In this section, we compare the manager's profits with model I.

Lemma 4

The effect of Cournot competition on the manager's profits relative to the monopoly case is given by,

Adding a second informed trader to model I does not always lower the manager's profits relative to the monopoly case. Indeed, Lemma 4 shows that the profits of the manager in this model are greater than in model 1 when σ 2 ε is small relative to σ 2 z , and they are less than model I when σ 2 ε is large relative to σ 2 z . The intuition for this result is that when σ 2 ε is large relative to σ 2 z , the unconditional net profits of the firm is less in this model than in model I. In contrast, when σ 2 ε is small relative to σ 2 z , the opposite is true, i.e. the net profits of the firm in model I is lower.

Finally, it is worth noting that the profits of firm 2 decrease with respect to model I:

This result is straightforward from Lemma 2. Indeed, we showed that competition in the financial sector decreases the level of output produced by firm 1 and the price of the intermediate good charged by firm 2. The lower price for the intermediate good explains why the profits of Firm 2 decrease with respect to model I.

Appendix A

We start this appendix by recalling the Theorem that we use to prove Lemma 1. Then we prove this Lemma.

Theorem 1 If the p × 1 vector Y is normally distributed with mean U and covariance V and if the vector Y is partitioned into two subvectors such that

are the corresponding partitions of Y * ,U and V, then the conditional distribution of the m × 1 (m < p) vector Y 1 given the vector

Proof : See Graybill, Theorem 3.10 pp 63.

Proof of Lemma 1: We apply Theorem 1 to the normal random vector B = (ṽ, q, r). First, in this case we have p = 3 and m = 1. Second by identification, we have Y 1 = ṽ

and

where D is the determinant of V 22 , that is D = σ 2 q σ 2 r -σ 2 qr .

So we obtain,

We start now by computing all the covariances in the last two expressions. Indeed, we have

Substituting for the variances and covariances in (32) and (33), we get

Computing ( 34) and (35), we obtain

Calculating for the expression of D, we get

Substituting the above expression of D in (35), we find

Solving (36) and (37), we get 2µ 1 (σ 2 ε + σ 2 z ) = σ 2 z (y + µ 1 ). Substituting for y to solve µ 1 , we obtain

To solve for µ 2 , we substitute the above value of µ 1 in (36) and taking the positive root, we get

where k = Appendix B: proof of Proposition 2

Applying Theorem 1 to the normal random vector (z, q, r), the conditional variance is given by V ar(z|q, r) = σ


M Lapert

J Salomon

D Sugny

Time-optimal monotonic convergent algorithms for the control of spin systems

Keywords: numbers: 32.80.Qk, 37.10.Vz, 78.20.Bh

HAL is

I. INTRODUCTION

The optimal control of quantum systems is a long-standing goal [1][2][3] which remains very attractive both from a practical and a fundamental point of views [START_REF] Shapiro | Principals of quantum control of molecular processes[END_REF][START_REF] Rice | Optimal control of molecular dynamics[END_REF][START_REF] Tannor | Introduction to quantum mechanics: A time-dependent perspective[END_REF]. By finding the optimal balance between the minimization of the distance to the target state and the minimization of the energy of the laser, the optimal solution allows to bring the system close to the target state while avoiding parasitic phenomena due to a too large laser fluence. In this context, the control duration is also a crucial parameter which has to be taken into account in the optimization process. For instance, a too long duration could be problematic if other concurrent physical or chemical processes with the same time scale occur during the control. This question is particularly interesting in quantum computing where coherence has to be preserved [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. Since a control field cannot generally fully compensate the dissipation effects [35], a too long interaction of the system with the environment can destroy its coherence and the quantum superposition or the entanglement produced by the control.

Solving time-optimal control problems remains however a challenging task. One way is to use the Pontryagin maximum principle and geometric optimal control theory [START_REF] Jurdjevic | Geometric control theory[END_REF][START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF].

However, such techniques can only be applied for the moment to small dimensional quantum problems with very few energy levels [12,[14][15][16][17]. On the other hand, monotonically convergent algorithms are an efficient way to solve optimal control problems and have been widely used in the control of chemical and physical processes since the pioneering papers by Tannor et al. [18] and Rabitz et al. [19] which were based on the work of Krotov [36]. This approach can be applied to very different and large quantum systems (See e.g. [20][21][22][23][24]) and to a variety of non-standard situations such as the nonlinear interaction between the system and the control field [25,26] or to take into account spectral constraints on the optimal solution [27,28]. Up to now, however, these algorithms have generally be used with a cost penalizing the laser fluence and a fixed control duration. By construction of these algorithms, a formulation in terms of a time-optimal control, i.e. with a duration which is not fixed, is a very difficult question since these methods imply the backward propagation of the adjoint state from the final time of the control. Other time-optimal control algorithms have already been proposed in [31,32], but these propositions differ from our approach in the sense that a second Lagrange multiplier on the control duration (in addition of the adjoint state) is added. This leads to a more complicated algorithm than the one proposed below.

We present in this paper a new formulation of monotonically convergent algorithms with a cost penalizing both the laser fluence and the control duration, which allows us to find the best compromise between these two parameters. Using a rescaling of time, we first rewrite the optimal equations on a fixed time interval independent on the control duration T , which appears as a new parameter in the time-dependent Schrödinger equation. We then consider a monotonic iterative algorithm whose each step is decomposed into two substeps consisting in an optimization of the energy of the field with T fixed and an optimization of the time T with a fixed control field. The first substep is realized through a standard monotonically convergent algorithm, while a gradient or another discrete optimization procedure is used for the second substep. We impose that each substep increases the cost functional leading thus to a monotonic algorithm.

To test the efficiency of this approach, we consider the control of spin systems [START_REF] Levitt | Spin dynamics: basics of nuclear magnetic resonance (John Wiley and sons[END_REF], and in particular the implementation of quantum gates in such systems. Different technologies have been developed so far to exploit the powerful of quantum computing. One of the most promising solution is Nuclear Magnetic Resonance (NMR) [30]. The control technology developed over the past fifty years allows the use of sophisticated control fields and permits the implementation of complex quantum algorithms such as the Deutsch-Jozsa and the Grover ones [START_REF] Chuang | [END_REF]. NMR is therefore an ideal testbed to experiment new ideas in quantum control. In this paper, we show how to implement two and four qubits CNOT gates. These different numerical computations allow to extract the main properties of our algorithm and to highlight the differences with respect to a standard approach.

The paper is organized as follows. In Sec. II, we describe the new monotonically algorithm for pure state quantum systems. The proof of its monotonic character is established. Section III is devoted to the application of this approach in a two and four spin systems in order to implement CNOT gates. We conclude in Sec. IV.

II. TIME-OPTIMAL CONTROL ALGORITHMS

We present in this section the algorithm in a general setting for pure quantum states. The formalism can be straightforwardly extended to mixed-state quantum systems [21] or to the control of evolution operators [29]. It is this latter generalization that will be used in Sec. III for the implementation of quantum gates. We consider the maximization of the projection onto a target state, but the algorithm could be equivalently defined for maximizing the expectation value of a given observable.

A. Methodology

Let |φ 0 and |φ f be the initial and target states of the dynamics. We consider the timeoptimal control problem of maximization of the cost functional J T (E T ) over the control duration T and the control field E T . Note that the subscript T is added in this paper to any quantity depending upon this time. The functional J T is defined by

J T (E T ) = 2 [ ψ T (T )|ψ f ] -α T 0 E 2 T (t)dt ( 1
)
where α is a positive parameter which weights the relative importance of the energy of the control field with respect to the projection onto the target state. [•] is the real part of a complex number. The state |ψ T (t) of the system satisfies the time-dependent Schrödinger equation which is written in units such that = 1:

i ∂ ∂t |ψ T (t) = (H 0 -µE T (t))|ψ T (t) (2)
with as initial condition |ψ T (0) = |φ 0 . The Hamiltonian H 0 is the field-free Hamiltonian and the operator µ describes the interaction between the system and the control field, which is assumed to be linear.

The first step of the approach is to define a fixed time interval, for instance [0, 1]. We consider for that purpose the time rescaling s = t/T . Introducing |ψ(s) = |ψ T (s • T ) and

E(s) = E T (s • T ), we obtain from Eq. (2) that i ∂ ∂s |ψ(s) = T (H 0 -µE(s))|ψ(s) (3)
with the initial condition |ψ(0) = |φ 0 . The cost functional is also changed by the time rescaling and becomes

J(E) = 2 [ ψ(1)|ψ f ] -αT 1 0 E 2 (s)ds. ( 4
)
The new optimal control problem consists now in maximizing the cost functional J with respect to the control field E and the time T which plays here the role of a parameter. The control duration is fixed to 1.

The algorithm is decomposed into two substeps. We alternatively optimize the functional J with respect to the control field E by a standard monotonic algorithm and with respect to the duration T by a discrete procedure such as a gradient method. We prove that the cost increases at each step of the algorithm.

a. Optimization of the control field. We introduce the triplets (|ψ(t) , E(t), T ) and

(| ψ(t) , Ẽ(t), T ) corresponding to the initial and final states of this substep of the algorithm.

The variation of the cost is given by:

∆J = J( Ẽ) -J(E) = 2 [ ψ(1) -ψ(1)|ψ f ] -αT 1 0 ( Ẽ2 (s) -E 2 (s)ds.
We introduce the adjoint state |χ(t) which satisfies

i ∂ ∂s |χ(s) = T (H 0 -µE(s))|χ(s) (5)
with the final condition |χ(1) = |φ f . We then have

[ ψ(1) -ψ(1)|φ f ] = [ ψ(1) -ψ(1)|χ(1) ] (6)
which can be transformed into

[ ψ(1) -ψ(1)|φ f ] = 1 0 ds[ ∂ ∂s χ| ψ -ψ + χ| ∂ ∂s ( ψ -ψ) ] . ( 7
)
Using Eqs. ( 3) and ( 5), one deduces that

[ ψ(1) -ψ(1)|φ f ] = 2T [ 1 0 ds χ|µ( Ẽ -E)| ψ ]. (8)
One finally arrives to b. Optimization of the control duration. At this stage of the algorithm, we consider the triplets (|ψ(s) , E(s), T ) and (| ψ(s) , E(s), T ). We recall that the state | ψ(s) satisfies

∆J = αT 1 0 ds(E -Ẽ)(E + Ẽ + 2 α [ χ|µ| ψ ]). (9)
i ∂ ∂s | ψ(s) = T (H 0 -µE(s))| ψ(s) . ( 10
)
We compute the variation of the cost functional ∆J which is equal to:

∆J = 2 [ ψ(1) -ψ(1)|ψ f ] -α( T -T ) 1 0 E 2 (s)ds. ( 11
)
Introducing the adjoint state |χ(s) whose dynamics is governed by Eq. ( 5), one obtains after similar computations as for the previous case that

∆J = α(T -T ) 1 0 dsE(s)( 2 α [ χ|µ| ψ ] + E(s)). ( 12
)
The parameter T has to be chosen such that ∆J ≥ 0. A solution consists in using a gradient method by noting that

∇ T J = -α 1 0 dsE(s)( 2 α [ χ|µ|ψ ] + E(s)). ( 13
)
We define the new time T from the preceding one as:

T = T -r∇ T J(E(s), T ), ( 14
)
where r is a real parameter. We choose numerically r small enough to ensure the monotonicity of the cost functional. The computation of the optimal value of r requires however several new propagations to determine the cost J since the evolution of | ψ (needed to calculate ψ(1)|ψ f ) depends on the value of T (see Eq. ( 10)). Other methods ensuring the monotonic behavior of the cost can be used for this substep as the following procedure. In this approach, we define the new duration Tk as a function of the old duration T k as follows:

Tk = (1 + a)T k
where a is a small positive or negative parameter. Practically, we can choose e.g. a = ±10 -3 , but this value can also be adjusted through the computation. This leads to two new costs J+ k and Jk . The final time at step k is the time associated to the maximum value between J+ k , Jk and Jk . This method has the advantage over the gradient approach to limit at each step the number of propagations of Eq. (3) to 2. This point can be interesting when very heavy computations are considered. This systematic procedure has been used in the numerical examples of Sec. III. In particular cases, we have checked that the gradient and this systematic approach give equivalent results.

The principles of control in NMR are detailed in different books and review articles.

Here we only give a brief account needed to introduce the model used [START_REF] Levitt | Spin dynamics: basics of nuclear magnetic resonance (John Wiley and sons[END_REF]. We consider the control of a system of coupled spins by different magnetic fields acting as local controls on each spin. This means that each field only controls one spin and does not interact with the others, i.e. the spins are assumed to be selectively addressable. This hypothesis has the advantage to render the system completely controllable. Similar models have been used in numerical studies analyzing the realization of quantum algorithms in NMR [33].

More precisely, we introduce a system of n coupled spins whose evolution is described by the following Hamiltonian:

H = H 0 + n j=1 (u jx H jx + u jy H jy ),
the couplings being given by:

H jx = σ jx , H jy = σ jy
where the operators (σ jx , σ jy ) are Pauli matrices which only act on the jth-spin. We assume that the free evolution Hamiltonian H 0 is associated to the topology of a chain of coupled spins with only nearest-neighbor interactions. The corresponding Hamiltonian is given by:

H 0 = n-1 j=1 σ jz ⊗ σ j+1; z ,
where the approximation is valid in heteronuclear spin systems if the coupling between the spins is small with respect to the frequency shifts [START_REF] Levitt | Spin dynamics: basics of nuclear magnetic resonance (John Wiley and sons[END_REF]. The coupling between the spins is taken to be uniform and equal to 1. The different equations being linear, other couplings could be considered from a standard rescaling of the time and of the amplitude of the control fields. Note that the algorithm could also be used with different couplings between the spins.

B. Optimal implementation of a CNOT gate

Our goal is to apply the time-optimal control algorithm to implement a C n-1 N OT gate (Controlled-Not) in a system of n qubits with n = 2 or 4. A C n-1 N OT gate is a gate in which the target qubit flips if and only if the (n -1) control qubits are equal to 1. For n = 2, the CN OT transformation is represented by the unitary operator U CN OT which can be written as:

U CN OT =        1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0        .
The logical states involved in a C n-1 N OT gate can be mapped onto the spin states in different ways. A straightforward and natural way used in this paper to map the logical states onto the spin states is to encode the first qubit in the first spin, the second qubit in the second spin, and so on if more than two spins are considered.

To implement quantum gates, we formulate the control problem in terms of evolution operators U (t). This means that the objective of the control is to reach the target state U CN OT , while optimizing the control duration and the energy of the fields. The time-optimal monotonically convergent algorithm for evolution operators can be sketched along the same way as for the wave function case of Sec. II. This algorithm can be obtained by substituting the wave function |ψ T (t) by U T (t), and the scalar product ψ(t)|χ(t) by Tr[U † V ] where V (t) is the adjoint propagator. The corresponding cost functional is given by:

J(E) = 2 Tr[U CN OT U T (T )] -α T 0 E 2 T (t)dt, ( 15
)
the evolution operator U T (t) satisfying the Schrödinger equation

i ∂U T (t) ∂t = (H 0 -µE T (t))U T (t). ( 16
)
Note that, in the example considered, 2 n fields are simultaneously optimized. The efficiency of the process is measured by the projection P = 1

2 n Tr[U CN OT U T (T )] .
From a numerical point of view, two different parameters, α 0 and E 0 (t), have to be adjusted when using this algorithm. These parameters do not play the same role since α is a parameter characteristic of the algorithm, while E 0 (t) is the initial field used to initiate the optimization process. More precisely, we assume that the parameter α depends on time and can be written as α(t) = α 0 sin 2 (πt/T ) where α 0 is a constant. This switching function is introduced to provide a smooth on and off switch of the field [13]. In order to not enforce the algorithm to follow a given pathway, we consider that the initial trial field E 0 (t) is zero over a given duration T 0 . The dependance of the final solution on the two parameters T 0 and α 0 will be analyzed in Sec. III C.

C. Numerical results

We first analyze the computational results for a system of two spins. Figure 1 displays the optimal solution computed by the algorithm for the values of parameters T 0 = 0.5 and α 0 = 0.08. The parameter a which describes the evolution of the control duration at each step of the algorithm is taken to be 5 × 10 -4 . Other values of a have been used leading either to worse results or to a slower convergence of the algorithm. As can be seen in Fig. 1, the evolution of the optimal control fields and of the probability is rather smooth with no rapid oscillation. A very good efficiency larger than 0.99 has been reached in 5000 iterations with a final duration of the order of T = 2.035. Note that a standard monotonically algorithm with this total duration leads to a solution very close to the ones obtained with this new algorithm. More precisely, for the standard algorithm, we have obtained a projection P larger than 0.999 for a control duration such that 1.9 < T < 2.4. This computation shows that the time-optimal control algorithm has found the best compromise between the duration, the minimization of the distance to the target state and the energy of the field. As could be expected, the modification of the control duration slows down the convergence of the algorithm since a projection larger than 0.99 is obtained respectively after 2512 and 700 iterations for the new and standard methods, respectively. The monotonic behavior of the algorithm can be checked in Fig. 2 together with the evolution of the duration T k . As for the cost J k , one sees that this parameter presents a rapid increase for the first 3000 iterations and then an approximatively constant behavior. This means qualitatively that the timeoptimal procedure first optimizes the control duration before decreasing the energy of the field without changing the total cost J. A crucial property that this algorithm must satisfy (at least locally) is the independence of the final solution with respect to the value of T 0 , i.e.

of the starting guess used to initiate the algorithm. This point is illustrated in Fig. 3 where two attraction points for the sequence (T k ) have been found when the time T 0 varies. We which are not represented in Fig. 3. Note that this attraction point characterizes not only the final control duration but also the final control fields and the final probability density as can been checked in Fig. 3. In Fig. 3, we see that a better efficiency is reached for T 0 = 0.9 with a longer and lower energetic optimal solution. This point indicates the importance of the control duration in the accuracy of the computation.

In Fig. 4, we study the evolution of the final time T f and of the probability density P as a function of the parameter α 0 . We observe that P increases and T f decreases as the parameter α 0 decreases. As could be expected, the smaller α 0 is, the more energetic the optimal solution is since α 0 controls the relative weight of the pulse energy in the cost J.

With a more energetic optimal solution, the algorithm can find an optimal solution with a lower duration and a better efficiency.

We extend these numerical results to the case of a four-spin system and a C 3 N OT gate.

Due to the complexity of this gate, a larger duration and a larger number of iterations are required to reach a sufficient efficiency. The parameter a is taken to be 5 × 10 -4 . As for the two-spin case, we find two possible optimal solutions according to the value of T 0 which are displayed in Fig. 5. The time evolution of the probability density shows that the structure of these two solutions is very close even if the duration is different. of iterations. For T 0 ≤ 0.7, the algorithm converges towards the same optimal duration close to the value 2. The parameter α 0 is taken to be 0.08. (bottom) Same as before but for the probability density P . A better efficiency is reached for T 0 = 0.9.

IV. CONCLUSION AND PERSPECTIVES

This work deals with the time-optimal control of spin systems in NMR. We propose a monotonically convergent algorithm which both optimizes the control duration and the energy of the field. We show that the change of the duration at each iteration of the algorithm leads to a more flexible algorithm and thus allows a better convergence with respect to a standard version of such algorithms. This algorithm has the advantage of simplicity and general applicability whatever the quantum optimal control problem considered. We have finally demonstrated the possibility of implementing quantum gates from the control fields computed by this algorithm. Since there exists no unique optimal solution, we have shown that we can select the control fields by changing the initial control duration.

  Knowing E(s), the choice Ẽ = -[ χ|µ| ψ ]/α ensures that ∆J ≥ 0 for this substep. This part of the algorithm can be summarized as follows. Starting from the quadruplet (|ψ(s) , |χ(s) , E(s), T ), we construct the quadruplet of the next sub-iteration by propagating backward the adjoint state | χ(s) from |φ f with the field E(s). We then propagate forward the state | ψ(s) from |φ 0 with the field Ẽ(s) which is computed at the same time by the relation Ẽ(s) = -[ χ(s)|µ| ψ(s) ]/α.

FIG. 1 :

 1 FIG.1: Evolution of the probability P and of the corresponding optimal fields acting on the first or second spin. Numerical values are taken to be α 0 = 0.08 and T 0 = 0.5. The final probability is P = 0.9964.

FIG. 2 : 4 FIG. 3 :

 243 FIG.2: Evolution of the cost J k (dashed line) and of the time T k (solid line) as a function of the number of iterations k of the algorithm. The same parameters (α 0 and T 0 ) as in Fig.1have been used.

FIG. 4 :

 4 FIG. 4: Evolution of T f (open circle) and P (cross) as a function of the parameter α 0 for T 0 = 0.5. The solid and dashed lines are just to guide the lecture.

5 FIG. 5 :

 55 FIG. 5: (top-middle) Evolution of the probability density P and of the control duration T k as a function of the number of iterations for different initial durations T 0 . (bottom) Time-evolution of the probability density P for the different optimal solutions. The parameter α 0 is taken to be α 0 = 0.01.


MD Lidewij M E Knook

MD Antoinette Y Konijnenberg

Joost Hoeven

MD Jan L L Kimpen

MD Jan K Buitelaar

Herman Engeland

Elisabeth R Graeff-Meeder

MD Joost Van Der Hoeven

MD Herman Van Engeland

MD Elisabeth R De Graeff-Meeder

Psychiatric disorders in children and adolescents presenting with unexplained chronic pain: what

Keywords: chronic pain, children, adolescents, psychiatry

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Unexplained chronic pain (UCP) in children, i.e. pain of unknown cause lasting 3 months or longer, is a common problem and a challenge to physicians [START_REF] Howard | Current status of pain management in children[END_REF]. It is often considered selflimiting and innocuous [START_REF] Malleson | Chronic musculoskeletal and other idiopathic pain syndromes[END_REF]. However referred children have impairments in daily life comparable to those seen in individuals with chronic somatic disorders [START_REF] Konijnenberg | Children with unexplained chronic pain: substantial impairment in everyday life[END_REF]. Moreover, recent population based and clinical studies have shown that the chronic pain is not selflimiting, despite treatment at specialized clinics [START_REF] El-Metwally | Predictors of abdominal pain in schoolchildren: a 4-year population-based prospective study[END_REF][START_REF] Martin | Children with chronic pain: impact of sex and age on long-term outcomes[END_REF][START_REF] Mirovsky | Non-specific back pain in children and adolescents: a prospective study until maturity[END_REF][START_REF] Perquin | The natural course of chronic benign pain in childhood and adolescence: a two-year population-based follow-up study[END_REF].

In adults with UCP, pain is associated with depressive and anxiety disorders, and this association is stronger than that observed in healthy controls or in patients with chronic pain of known organic pathology and is still present after correction for bodily symptoms of anxiety and depression [START_REF] Garcia-Cebrian | The association of depression and painful physical symptoms--a review of the European literature[END_REF][START_REF] Henningsen | Medically unexplained physical symptoms, anxiety, and depression: a meta-analytic review[END_REF]. It is generally recognized that psychosocial factors are involved in childhood UCP, for instance parental distress and maladaptive coping [START_REF] Eccleston | Adolescent chronic pain: patterns and predictors of emotional distress in adolescents with chronic pain and their parents[END_REF]37,38,[START_REF] Stanford | The frequency, trajectories and predictors of adolescent recurrent pain: a population-based approach[END_REF][START_REF] Walker | A typology of pain coping strategies in pediatric patients with chronic abdominal pain[END_REF], and there is increasing evidence that psychiatric symptoms and psychiatric disorders are also involved [START_REF] Campo | Recurrent abdominal pain, anxiety, and depression in primary care[END_REF][START_REF] Egger | Somatic complaints and psychopathology in children and adolescents: stomach aches, musculoskeletal pains, and headaches[END_REF][START_REF] Garber | Recurrent abdominal pain in children: psychiatric diagnoses and parental psychopathology[END_REF][START_REF] Konijnenberg | Psychiatric morbidity in children with medically unexplained chronic pain: diagnosis from the pediatrician's perspective[END_REF][START_REF] Mikkelsson | Psychiatric symptoms in preadolescents with musculoskeletal pain and fibromyalgia[END_REF][START_REF] Mongini | Accompanying symptoms and psychiatric comorbidity in migraine and tension-type headache patients[END_REF][START_REF] Powers | Headache and psychological functioning in children and adolescents[END_REF]. However, current evidence is diverse and does not allow clinically useful generalizations.

Most studies on psychiatric disorders have involved children in community, primary care or tertiary care (specialized pain clinics) samples, which are probably not representative of children with UCP visiting secondary care, like a paediatric outpatient clinic. Moreover, many of the studies of small clinical samples tended to focus on internalizing disorders, whereas psychiatric assessment in children with UCP should also include assessment of externalising disorders [START_REF] Egger | Somatic complaints and psychopathology in children and adolescents: stomach aches, musculoskeletal pains, and headaches[END_REF]. Another limitation is that different diagnostic measures have been used, mostly based on questionnaires. Relatively few studies have reported on psychiatric disorders assessed by structured clinical psychiatric interviews based on the Diagnostic and Statistical Manual of Mental Disorders, fourth edition (DSM-IV) -classification system. In studies in which these interviews were used, methodological shortcomings prevented appropriate estimation of the prevalence of clinically significant psychiatric disorders. First, most psychiatric disorders were assessed with respondent (parent or child)-based highly structured interviews and not by clinician assessment of the child. Second, impairment criteria were often not included in structured psychiatric interviews. When paediatricians encounter patients with UCP with psychiatric symptoms, they may request psychiatric assessment, because these symptoms may be distressing and treatment might improve pain and functioning, even if the symptoms are not associated with impairment, directly. However, when a child psychiatrist diagnoses psychopathology by means of the DSM-IV classification system, impairment, in addition to duration, is a key criterion for determining whether the psychopathology is clinically significant and need any treatment. In the general population, the prevalence of child psychiatric disorders decreases considerably when impairment is included [START_REF] Canino | The DSM-IV rates of child and adolescent disorders in Puerto Rico: prevalence, correlates, service use, and the effects of impairment[END_REF]. Therefore, although a variety of literature about psychopathology in children with UCP, little is known about its prevalence and clinical significance in referred children and whether psychiatrists should be involved in the diagnostic work-up.

In adults with medically unexplained symptoms, cognitive behavioural therapy is the only evidence-based treatment thus far found to reduce physical symptoms, psychological distress, and disability [16,[START_REF] Speckens | Cognitive behavioural therapy for medically unexplained physical symptoms: a randomised controlled trial[END_REF]. In children, general guidelines for the assessment and treatment of UCP advocate combination of psychological, physical, and pharmacological treatment [START_REF] Berger | Chronic abdominal pain in children[END_REF][START_REF] Howard | Current status of pain management in children[END_REF][START_REF] Merlijn | A cognitive-behavioural program for adolescents with chronic pain-a pilot study[END_REF], and evidence for a cognitive behavioural approach becomes apparent [10, [START_REF] Eccleston | Chronic pain in adolescents: evaluation of a programme of interdisciplinary cognitive behaviour therapy[END_REF][START_REF] Eccleston | Psychological therapies for the management of chronic and recurrent pain in children and adolescents[END_REF][START_REF] Palermo | Randomized controlled trial of an Internet-delivered family cognitive-behavioral therapy intervention for children and adolescents with chronic pain[END_REF]. Treatment results, however, are often limited to pain control, with little evidence for effect on mood and disability [START_REF] Eccleston | Psychological therapies for the management of chronic and recurrent pain in children and adolescents[END_REF]. On the other hand, there is strong evidence for psychological and pharmacological treatment of child psychiatric disorders. Proper identification of children with UCP at risk of psychiatric disorders is therefore required, because it justifies referral for psychiatric assessment and care. Moreover, the risk of repeated unproductive medical investigation and intervention might decrease [START_REF] Barsky | Somatization increases medical utilization and costs independent of psychiatric and medical comorbidity[END_REF][START_REF] Lindley | Consumerism in healthcare can be detrimental to child health: lessons from children with functional abdominal pain[END_REF][START_REF] Sharpe | Unexplained" somatic symptoms, functional syndromes, and somatization: do we need a paradigm shift?[END_REF].

The aims of our study were to estimate the prevalence of clinically relevant psychiatric disorders in children and adolescents visiting a secondary care paediatric outpatient clinic, using both parent responses to a diagnostic interview and an interview with the child by a child psychiatrist. Secondary, we investigated whether demographic parameters (age, sex, socioeconomic status) and pain parameters (severity, duration, and location) are predictors of psychiatric disorders in children referred for UCP.

Method

Setting, participants, and recruitment

The Pain of Unknown origin in Children study (PUC study) was conducted in the Wilhelmina Children's Hospital, University Medical Centre Utrecht, The Netherlands, between January 2000 and July 2002. Children who were referred to the outpatient clinic were eligible if they met the following criteria: age between 8-18 years; pain for at least 3 months prior to the visit and without an explanatory diagnosis made by the referring doctor; first visit to the outpatient clinic for this symptom; sufficient knowledge of Dutch. The Dutch healthcare system is organised such that a general practitioner first evaluates a child. If further specialised care is necessary the child can be referred to a regional or a university hospital. If specialists in the regional hospital need a second opinion, children can be referred to a university hospital.

In the Netherlands, therefore, a university hospital to some extent covers the same population as a regional clinic. Our clinic is organised such that ~50% of the general paediatric outpatient population and ~15% of the other paediatric departments consist of children who are evaluated only by a general practitioner before referral. Children with musculoskeletal pain, abdominal pain, and headache were recruited mainly from the department of general paediatrics, but also from the departments of paediatric neurology, orthopaedics, immunology, rheumatology, and gastroenterology. The three selected pain locations resemble the most frequent reported types of chronic pain in children and adolescents [START_REF] Perquin | Pain in children and adolescents: a common experience[END_REF] thereby covering most of the chronic pain population in children and adolescents. Each child was evaluated according to standard practice. Immediately after the first visit, the child and its parents were invited to participate in the study. After informed consent was given, additional research data were gathered. All study assessments were performed during the routine diagnostic work-up. The study protocol was approved by the Medical Ethics Committee of the University Medical Centre Utrecht.

The original study on Chronic Pain of Unknown origin in Children (PUC) was primarily focused on improving insight in the diagnostic process of general paediatricians encountering children referred for unexplained chronic pain. On the subject of psychiatric disorders the research question was if clinical characteristics or standardized questionnaires help to accurately diagnose psychiatric morbidity. For this purpose a clinician, based on child SCICA outcome, DISC outcome, and/or a combination of both, made a psychiatric diagnosis.

Secondary, the psychiatric DISC and SCICA data facilitated us to explore new hypotheses from a psychiatric point of view.

In total, 149 children were included. Fifteen children withdrew their consent because of lack of time and motivation (2 male and 6 female), or unwillingness to have a psychiatric assessment (1 male and 6 female), leaving 134 children (37 male and 97 female).

Assessments Psychiatric disorder

The Diagnostic Interview Schedule for Children (DISC) and the Clinical Interview for Children and Adolescents (SCICA) were used to assess the presence of psychiatric disorders. The DISC is a highly structured respondent-based interview for children [START_REF] Shaffer | NIMH Diagnostic Interview Schedule for Children Version IV (NIMH DISC-IV): description, differences from previous versions, and reliability of some common diagnoses[END_REF]. It assesses 34 different child-and adolescent psychiatric diagnoses according to DSM-IV diagnostic criteria.

The child version of the DISC-C is only reliable in children aged 12 years and older [START_REF] Schwab-Stone | Reliability of diagnostic reporting for children aged 6-11 years: a test-retest study of the Diagnostic Interview Schedule for Children-Revised[END_REF].

Since our study group consisted of children between 8 and 18, we used the parent version of the DISC. The interviews were conducted by three trained lay-interviewers who followed the guidelines of the DISC manual. The Semi-structured Clinical Interview for Children and Adolescents (SCICA) mimics clinical interviewing and observation by child psychiatrists [START_REF] Ferdinand | The role of the clinician: three-year predictive value of parents', teachers' and clinicians' judgment of childhood psychopathology[END_REF].

One experienced child psychiatrist (JvdH), who received extensive training in using this method, conducted the SCICA and was blinded to other information about the child. SCICA data were converted into 6 DSM-IV-based scales (affective problems, anxiety problems, somatic problems, attention-deficit hyperactivity problems, oppositional defiant problems, and conduct problems). We used clinical T scores of 55 or higher as cut-off, indicating a clear diagnosis as advocated by McConaughy [START_REF] Mcconaughy | Manual for the Semistructured Clinical Interview for Children and Adolescents[END_REF]. Since all our participants had chronic pain, we excluded the somatic problems category as an indication for psychiatric disorder. Child and parent interviews were conducted simultaneously in separate rooms, shortly after intake and before completion of the standard diagnostic work-up (routine care).

Clinical relevance of psychiatric diagnosis

The Children's Global Assessment Scale (CGAS) is a clinician scored scale of adaptive functioning that has good psychometric properties useful for incorporating a measure of impairment in the classification of childhood psychiatric disorders [START_REF] Bird | Further measures of the psychometric properties of the Children's Global Assessment Scale[END_REF][START_REF] Bird | Impairment in the epidemiological measurement of childhood psychopathology in the community[END_REF][START_REF] Shaffer | A children's global assessment scale (CGAS)[END_REF]. It is an adaptation of the Global Assessment Scale developed for adults. The CGAS has 10 anchor points which all contain behaviourally oriented descriptions that depict behaviours and life situations applicable to children. Scores on the CGAS range from 100 (no impairment) to 0 (severe impairment). A score of 61 is the best cut-off for definite psychiatric cases, identifying children who have received mental health services [START_REF] Bird | Impairment in the epidemiological measurement of childhood psychopathology in the community[END_REF]. A positive score on one or more DSM-IV scales combined with a CGAS lower than 61 was considered a clinically relevant psychiatric disorder.

Sociodemographic factors

Sex, age, and socioeconomic status, (SES) were assessed. SES was categorized according to the Dutch Standard Classification of Occupations, based on the father's occupation [START_REF]Statistics Netherlands; Standaard Beroepenclassificatie (Dutch Standard Classification of Occupations)[END_REF].

Pain

Children scored their pain on the Visual Analogue Scale (VAS) of the Varni/Thompson Pediatric Pain Questionnaire (PPQ) [START_REF] Varni | The Varni/Thompson Pediatric Pain Questionnaire[END_REF]. The PPQ-VAS assesses present pain and worst pain intensity for the previous week. Each VAS is a 10-cm horizontal line with no numbers, marks, or descriptive vocabulary words along the length of the line. The VAS is anchored with the pain descriptors "not hurting, no discomfort, no pain" at one end, and "hurting a whole lot, very uncomfortable, severe pain" at the other end. Subsequent research studies have supported the measurement properties of the PPQ -VAS in school-aged children [START_REF] Clancy | Pain in children and adolescents with spina bifida[END_REF][START_REF] Gragg | Assessing chronic musculoskeletal pain associated with rheumatic disease: further validation of the pediatric pain questionnaire[END_REF][START_REF] Graumlich | Multidimensional assessment of pain in pediatric sickle cell disease[END_REF][START_REF] Sawyer | The relationship between health-related quality of life, pain and coping strategies in juvenile idiopathic arthritis[END_REF][START_REF] Varni | Effects of perceived stress on pediatric chronic pain[END_REF][START_REF] Varni | The Varni/Thompson Pediatric Pain Questionnaire. I. Chronic musculoskeletal pain in juvenile rheumatoid arthritis[END_REF]. Pain location was determined from the medical history and was categorized into three groups: musculoskeletal pain, abdominal pain, and headache, resembling the most frequent reported types of pain in children and adolescents [START_REF] Perquin | Pain in children and adolescents: a common experience[END_REF]. Pain duration is expressed in months, from onset of pain to the day of assessment.

Statistical analysis

To assess statistical differences in pain location subgroups chi-quadrate analysis was used for proportional variables. Kruskal-Wallis one-way analyses of variance was used to assess age-differences in the 3 pain location groups. Spearman correlation was used to assess correlation between age and specific psychiatric diagnosis (anxiety disorder, affective disorder, and disruptive disorder). Univariate logistic regression analysis was performed to assess independent predictors of psychiatric morbidity. Results are expressed as odds ratios with corresponding 95% confidence intervals. Adjusted odds ratios were calculated using the same logistic model by adding possible confounding factors as covariates (age and gender of the child). For statistical analyses the Statistical Package for Social Sciences (SPSS, Inc, Chicago, Illinois, USA) version 17.0 was used. Statistical significance was considered for p values ≤ 0.05. 1. There were no age differences in the 3 pain location groups. Table 2a and 2b show the proportion of children with clinically relevant psychiatric disorders (CGAS < 61) and their psychiatric diagnosis, as assessed by parental interview (DISC, n=134) and child psychiatric assessment (SCICA n=131) distributed by pain location. Based on the parental interview, 40% of the children had a psychiatric disorder, but only 21% had a psychiatric disorder that was clinically relevant. The predominant disorder was anxiety disorder (specific phobias 10%, separation anxiety disorders 7%, social phobias 4%, generalized anxiety disorders 4%), which affected 18% of the children. Five percent of the children were diagnosed with an affective disorder, mainly depressive disorders (4.5%), and 5% were diagnosed with disruptive disorder, namely, oppositional defiant disorder (ODD; 4%) and attention-deficit/hyperactivity disorder (ADHD; 3%). Based on the child psychiatric assessment (SCICA), 53% of the children had a psychiatric disorder, but only 29% had a clinically relevant disorder. According to this assessment, 12% of the children had an anxiety disorder, 19% an affective disorder, and 9% a disruptive disorder. 3.

Results

Patient characteristics are shown in Table

Correlation analysis between age and specific psychiatric disorder (anxiety disorder, affective disorder, and disruptive disorder) revealed only a significant negative relation between age and disruptive disorder assessed by both DISC-P (r s -0.149, p = 0.043) and SCICA (r s -0.200, p = 0.011). This implies that with increasing age less disruptive disorders were diagnosed. Table 4 presents the predictors of clinical psychiatric disorder (yes /no), assessed with the DISC-P. Headache as main location of pain was a significant predictor of a clinical psychiatric diagnosis (OR = 3.10; 95 % CI 1.07 -8.92, p = 0.04). Pain duration showed a significant association with psychiatric diagnosis assessed by the DISC-P (OR = 1.014; 95 % CI 1.000 -1.028, p = 0.054, ORadj = 1.015; 95 % CI 1.001 -1.030, p = 0.036). Table 5 presents the predictors of clinical psychiatric disorder (yes /no), assessed with the child psychiatrist (SCICA). Again, headache was a significant predictor of clinical psychiatric disorder (OR = 2.85; 95 % CI 1.07 -7.61, p = 0.04). Age, sex, SES, and other pain parameters were not predictive of clinical psychiatric disorders.

Discussion

Our study is the first to systematically investigate the prevalence of clinically significant psychiatric disorders in children with UCP referred to a children's hospital. We found that psychiatric morbidity is common in children and adolescents with UCP referred to a children's hospital. Combining a parental psychiatric interview with an interview with the child by a child psychiatrist and using an impairment criterion for clinical relevancy, 35 % of the children with UCP were found to suffer from psychiatric disorder. Using a standardized psychiatric interview for parents (DISC-P) in combination with an impairment criterion, we found an overall prevalence of psychiatric disorder of 21%. This prevalence is three times higher than that among healthy Dutch adolescents assessed with the same instrument and impairment criterion [START_REF] Verhulst | The prevalence of DSM-III-R diagnoses in a national sample of Dutch adolescents[END_REF], and is also higher than the prevalence of psychiatric disorder, assessed with the same instrument, in children attending paediatric primary care [START_REF] Costello | Psychopathology in pediatric primary care: the new hidden morbidity[END_REF]. The high prevalence rate in our study group is in line with previous population studies of psychiatric morbidity in children with UCP [START_REF] Egger | Headaches and psychopathology in children and adolescents[END_REF][START_REF] Egger | Somatic complaints and psychopathology in children and adolescents: stomach aches, musculoskeletal pains, and headaches[END_REF] and studies performed in pain clinic [START_REF] Mongini | Accompanying symptoms and psychiatric comorbidity in migraine and tension-type headache patients[END_REF] and primary care [START_REF] Campo | Recurrent abdominal pain, anxiety, and depression in primary care[END_REF] settings.

The prevalence of psychiatric disorders was higher (28%) when a child psychiatrist interviewed the child (SCICA). Furthermore there was a discrepancy between symptoms reported by parents and this child psychiatrist assessment. More children reported depressive and disruptive disorders, while parents reported more anxiety symptoms.

Concerning the identification of psychiatric cases, more cases were identified using one informant (parent or child), than using both. In addition to the parent version of the DISC, the SCICA even resulted in an extra 15 % identification of children suffering from psychiatric disorder. Our results are in line with previous studies showing little overlap between subjects with child DSM diagnoses identified by parent report and child self-report [START_REF] Angold | Parent and child reports of depressive symptoms in children at low and high risk of depression[END_REF][START_REF] Foley | Informant disagreement for separation anxiety disorder[END_REF][START_REF] Verhulst | The prevalence of DSM-III-R diagnoses in a national sample of Dutch adolescents[END_REF]. Internalizing disorders are common comorbid disorders in children across the life span and have been explained by a common diathesis and genetic background [START_REF] Bittner | What do childhood anxiety disorders predict?[END_REF]. The higher prevalence of depressive disorders based on the child psychiatrist assessment is in contrast with the study of Garber et al, which showed that mothers tended to report more depressive symptoms than did their children with recurrent abdominal pain [START_REF] Garber | Recurrent abdominal pain in children: psychiatric diagnoses and parental psychopathology[END_REF]. Other research, however, has shown that girls, particularly as they grow older, report higher levels of depressive symptoms than do their mothers [START_REF] Kiss | Factors influencing mother-child reports of depressive symptoms and agreement among clinically referred depressed youngsters in Hungary[END_REF]. The presence of disruptive disorders, in our study ADHD and ODD, is in line with the population-based finding that disruptive disorders are part of the psycho-pathology associated with UCP [START_REF] Egger | Somatic complaints and psychopathology in children and adolescents: stomach aches, musculoskeletal pains, and headaches[END_REF]. The relatively low parent-rated prevalence of disruptive disorder (ADHD and ODD) was unexpected, because parent report is considered more valuable than child report for diagnosing symptoms of ADHD [START_REF] Loeber | Differences and similarities between children, mothers, and teachers as informants on disruptive child behavior[END_REF]. The mean age of the children was 12 years, an age at which children can be expected to reliably describe their behaviour and interference with daily activities, and this may explain why child report and observation by the child psychiatrist resulted in a higher prevalence of clinically relevant disruptive disorders than did diagnosis based on parent report.

To our knowledge we are the first study in children and adolescents with UCP showing that headache is distinctively predicting clinical psychiatric disorder. Compared to musculoskeletal pain, children referred for unexplained chronic headache had a 3-times higher risk of having an additional psychiatric disorder. Although not significant, possibly due to a lack of power, abdominal pain tended to predict psychiatric disorder assessed by a child psychiatrist. Egger also showed that pain location sites are associated with specific psychiatric disorders [START_REF] Egger | Headaches and psychopathology in children and adolescents[END_REF][START_REF] Egger | Somatic complaints and psychopathology in children and adolescents: stomach aches, musculoskeletal pains, and headaches[END_REF]. In this context a previous study on pain location in adults with chronic pain referred for psychiatric consultation, is also worthy of note. This study showed that patients with headache and abdominal pain were more likely to have anxiety, somatosensory amplification, and somatoform disorders whereas musculoskeletal pain patients showed stronger counter dependency traits [START_REF] Gregory | Pain location and psychological characteristics of patients with chronic pain[END_REF]. Furthermore the combination of headache and abdominal pain is the most reported combination of pain locations in children and adolescents with chronic pain [START_REF] Perquin | Pain in children and adolescents: a common experience[END_REF] , and children with persistent pain more often suffer from headache compared to other pain locations [START_REF] Perquin | The natural course of chronic benign pain in childhood and adolescence: a two-year population-based follow-up study[END_REF]. Therefore, although an abundance of research hypothesizes common aetiology in chronic pain conditions, distinct subgroups within the chronic pain population possibly exist. Future research however, is needed to elaborate the role of pain location in psychopathology of children with UCP. For pediatric practice, our results imply that for children with unexplained pain located in their head in particular, psychiatric work up is warranted.

Our findings should be interpreted in the context of several limitations. Focus on pain as presenting complaint may have lead to recall of less psychiatric symptoms, inducing a lower prevalence of psychiatric disorder. Absence of teacher-report questionnaires and a psychiatric assessment on pervasive developmental disorder and personality disorders might also result in missing common psychiatric disorders in our sample. The broad age range in our sample limits generalizabilty for clinical practice with respectively children and adolescents with UCP. A sample with exclusively adolescents with UCP, for example, would possibly have resulted in higher prevalence of depressive disorder. However analyses in our sample did not reveal relation between age and prevalence of psychiatric disorder in general, anxiety disorder, and affective disorder. The significant inverse relation between age and disruptive disorder cases we found implies that a sample without adolescents might have resulted in higher prevalence of disruptive disorder. The fact that the child psychiatrist was aware of the study hypothesis might have introduced bias, with the psychiatrist making more psychiatric diagnoses. However, we tried to ensure that these diagnoses were clinically relevant by using an impairment criterion, which resulted in a much lower prevalence of psychiatric diagnoses. We do not know whether the 15 children who refused to participate in the psychiatric assessment suffered from psychiatric morbidity, and therefore cannot estimate whether their withdrawal introduced bias. However, this group did not differ clinically from other non-participants, and therefore we do not think the non-response biased our results.

The high prevalence of clinically relevant psychiatric disorders in this referred sample with UCP has implications for diagnosis and treatment in daily clinical practice. When a child presents with UCP, it is essential to make a careful psychiatric assessment, including a professional child observation and interview, symptom severity and impairment. If psychiatric disorders are present, evidence-based child psychiatric treatment should be offered.

Although research on the effect of treatment of psychiatric disorders on UCP in children is lacking, there is evidence from adults to show that treatment of depression has a beneficial effect on the associated chronic pain [START_REF] Muller | Duloxetine in the treatment of major psychiatric and neuropathic disorders[END_REF][START_REF] Wise | Painful physical symptoms in depression: a clinical challenge[END_REF]. Because children with UCP frequently experience limitations and impairments in their daily activities, which interfere with their development, any physical, psychological, or psychiatric treatment able to reduce this interference should be incorporated in a tailored treatment plan.

Conclusion

We found that psychiatric assessment of children referred for UCP often revealed treatable psychiatric disorders. Child psychiatrist's ratings of child-reported symptoms and behavioural observation make an important contribution to the identification of specific psychiatric disorders. Children presenting with headache are particularly at risk.

Demographics

Figure 1

 1 Figure1shows the proportion of children identified as psychiatric cases by DISC and/or

  Furthermore, including observation of child psychiatric symptoms in the SCICA may result in distinctive information in the diagnostic process. A clinical child psychiatric assessment would thus seem to be of additional importance for diagnosing psychiatric disorders in children referred for UCP. When incorporating structured interviews in the diagnostic work-up of children with UCP, adding a SCICA to the parent version of the DISC considerably helps to identify children suffering from psychiatric disorders. Our results further show that in research on psychopathology in children with UCP, combining structured parent and child diagnostic interviews helps to determine reasonable prevalence of various psychiatric disorders.

  PainDuration of pain in months, median (range)VAS Current pain, mean (SD), n=130Pain previous days, mean(SD)

Figure 1 .

 1 Figure 1. Psychiatric cases identified by parental interview (DISC) and/or child psychiatrists' assessment (SCICA)

Table 2a . Prevalence of clinically relevant psychiatric disorder: parental interview (DISC-P) by pain location, n (%) 

 2a

		DISC-P
		total	MP	AP	HA	p -value
		n = 134	n =60	n = 43	n = 31
	Psychiatric disorder	28 (20.9)	8 (13.3) 10 (23.3) 10 (32.3)	0.098
	PD no impairment $	53 (39.6)	21 (35.0) 18 (41.9) 14 (45.2)	0.599
	Anxiety disorder	24 (17.9)	7 (11.7)	8 (18.6)	9 (29.0)	0.122
	Affective disorder	7 (5.2)	2 (3.3)	3 (7.0)	2 (6.5)	*
	Disruptive disorder #	7 (5.2)	3 (5.0)	2 (4.7)	2 (6.5)	*

Table 2b . Prevalence of clinically relevant psychiatric disorder: child psychiatrists' assessment (SCICA) by pain location, n (%) 

 2b

		SCICA
		total	MP	AP	HA	p-value
		n = 131	n = 58	n = 43	n = 30
	Psychiatric disorder	38 (29.0)	11 (19.0) 15 (34.9) 12 (40.0)	0.070
	PD no impairment $	69 (52.6)	23 (39.7) 28 (65.1) 18 (60.0)	0.027
	Anxiety disorder	16 (12.2)	4 (6.9)	6 (14.0)	6 (20.0)	*
	Affective disorder	25 (19.1)	7 (12.1) 11 (25.6)	7 (23.3)	0.185
	Disruptive disorder #	12 (9.2)	4 (6.9)	3 (7.0)	5 (16.7)	*
	MP = musculoskeletal pain; AP = abdominal pain; HA = headache
	 One or more psychiatric disorder per patient and CGAS < 61
	$ Psychiatric disorder without impairment criterion
	# ADHD, ODD, Conduct Disorder
	* Conditions for chi-quadrate analysis not met

Table 3 . Single and comorbid psychiatric cases by parental interview (DISC) and child psychiatrists' assessment (SCICA), n (%) Parental interview Child psychiatrist assessment

 3 AnxD = Anxiety Disorder, AffD = Affective Disorder, DisD = Disruptive Disorder

	Single	20 (71.4)	23 (60.5)
	AnxD	17 (61.7)	6 (15.8)
	AffD	2 (7.1)	12 (31.6)
	DisD	1 (3.6)	5 (13.2)
	Comorbid	8 (28.6)	15 (39.5)
	AnxD & AffD	2 (7.1)	8 (21.1)
	AnxD & DisD	3 (10.7)	2 (5.3)
	AffD & DisD	1 (3.6)	5 (13.2)
	AnxD & AffD & DisD	2 (7.1)	0 (0)
	Total	28 (100)	38 (100)

Table 4 . Predictors of psychiatric disorder based on parent report (DISC-P)

 4

		OR (95% CI)	p-value	OR adj* (95% CI)	p-value
	Demographics
	Sex:
	girls	reference
	boys	1.062 (0.421 -2.677)	0.898
	Mean age in years	0.884 (0.748 -1.046)	0.152
	SES:
	low	1.314 (0.442 -3.906)	0.623
	middle	0.756 (0.271 -2.107)	0.592
	high	reference
	Pain
	Duration of pain	1.014 (1.000 -1.028)	0.054	1.015 (1.001 -1.030)	0.036
	VAS (0-10):
	Current pain	1.058 (0.921 -1.217)	0.425	1.090 (0.939 -1.265)	0.256
	Pain previous days	1.037 (0.886 -1.214)	0.650	1.059 (0.897 -1.249)	0.500
	Location:
	Musculoskeletal	reference	-	reference	-
	Abdominal	1.970 (0.705 -5.500)	0.196	1.786 (0.629 -5.069)	0.276
	Headache	3.095 (1.074 -8.924)	0.036	2.993 (1.024 -8.743)	0.045
	* Adjusted for age and sex

Table 5 . Predictors of psychiatric morbidity based on child psychiatrists' assessment

 5

	(SCICA)
		OR (95% CI)	p-value	OR adj *(95% CI)	p-value
	Demographics
	Sex:
	girls	reference
	boys	0.920 (0.392 -2.158)	0.849
	Mean age in years	0.964 (0.833 -1.116)	0.622
	SES:
	low	2.466 (0.913 -6.662)	0.075
	middle	0.655 (0.248 -1.734)	0.395
	high	reference
	Pain
	Duration of pain	1.005 (0.991 -1.018)	0.477	1.006 (0.992 -1.020)	0.414
	VAS (0-10):
	Current pain	1.109 (0.974 -1.263)	0.119	1.125 (0.981 -1.291)	0.092
	Pain previous days	1.013 (0.881 -1.165)	0.854	1.012 (0.875 -1.170)	0.871
	Location:
	Musculoskeletal	reference	-	reference	-
	Abdominal	2.289 (0.923 -5.674)	0.074	2.235 (0.888 -5.622)	0.088
	Headache	2.848 (1.067 -7.606)	0.037	2.859 (1.064 -7.681)	0.037

*Adjusted for age and sex

Acknowledgements

Conflict of interest

The authors declare that they have no conflict of interest.