Frédéric Morlot
email: frederic.morlot@orange-ftgroup.com

Thomas Redon

Salah Eddine Elayoubi

Risk-hedging using

Keywords: data network, investment, real option, dynamic programming, least-squares

published or not. The documents may come

I. INTRODUCTION

Today, it is expected that the data traffic will be significantly growing in mobile networks. The total amount of transferred data is supposed to grow exponentially, as it has been the case since the middle 90' for the Internet. To face these soaring volumes of data to be transferred, mobile operators must periodically upgrade their equipments to offer higher throughputs and avoid blocking problems. However as the demand does not increase steadily and must be considered as partly random, the expected profit is difficult to be forecast.

In this article, we consider upgrade investments in a HSDPA cellular network. Note that, when demand increases, the data communication duration becomes longer and longer for each user until the network is saturated. The individual throughput experienced in the network may become very small. On the other hand, as the demand rises, the operator increases its profit. When the network starts experiencing saturation problems, throughput and profit may fall. The operator must then upgrade its network by adding new frequency carriers, facing the following trade-off:

• The later the investment, the lower individual throughputs and customer satisfaction. Permanent non-satisfaction will result into churn and additional loss of profit. • The sooner the investment, the more expensive the costs of upgrade elements. This article aims at modeling analytically the trade-off. We first derive analytical values for capacity, individual throughput and satisfaction as a function of the demand, and use them to calculate operator's profit, taking into account randomness of the rising demand, and decrease of network element costs according to time. We second introduce a real options method to hedge against the risk that demand evolves in an unexpected way leading to a premature investment decision or a too late one. To perform that, we introduce an American call that allows its owner (the mobile operator) to buy an equipment at a fixed price, possibly less than the real one, until a maturity date. Given the profit analytical model and the option's parameters, we propose a dynamic programming method to price the option. At the same time, we obtain the expected best investment date. Note that in a previous work [START_REF] Morlot | Optimizing the Date of an Upgrading Investment in a Data Network[END_REF], we already used the profit model to find the best investment date by an actualization algorithm. However, we did not use risk-hedging nor dynamic programming. Note that d'Halluin [START_REF] Halluin | Wireless Network Capacity Investment[END_REF] presented a work on a method to determine the best investment date in a wireless network. His approach was based on dynamic programming, but he did not introduce any risk hedging method nor an option pricing. Conversely, Longstaff & Schwartz [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF] introduced a pricing method for american options, but his work was not adapted to telecommunication networks investment.

The remainder of this paper is organized as follows: in a first section, we build an analytical model of the operator's profit based on an HSDPA network. Then we introduce in section II an american option to hedge against the risk aforementioned. We define the underlying asset and the option's payoff. To price the option we use a risk-neutral approach, whose mathematical justification lies in the appendix. In section III, we show how dynamic programming can help solving the pricing problem, and the best investment date problem as well. In section IV, we present the numerical results before concluding the paper.

II. THE BASIC MODEL: OPERATOR'S PROFIT AND INVESTMENT COST

The operator profit depends on the amount of data flowed by the network. It thus depends on the mean traffic demand per cell (in Mbits/sec/cell). We denote it by X t , where the time t = 0, 1... varies discretely, for example day by day. Note that profit does not necessarily increase when demand grows. It in fact also depends on the customer satisfaction, that we shall calculate hereafter.

A. Traffic Demand

We assume that the network is formed by circular cells of radius R, with a uniformly distributed demand. We have:

X t = λ t × E[ξ],
where λ t is the arrival rate per cell at the date t and E[ξ] is the mean size of a typical data flow. In the following we assume that E[ξ] remains stable during [0, T ], so that X t is proportional to λ t . This model is strictly equivalent to the case where the number of active users is constant, but they initiate connections more often.

To model the evolution of (X t ) t∈N , let us consider it as the daily sampling of a continuous stochastic process ( X(t, W t )) t∈R + . Usually, one monitors the 24 demands over one hour each, keeps the second or third highest, and multiply it by a given factor. As many random phenomena related to a social behavior (e.g. [START_REF] Salahaldin | Investing in Sustainable Transport to Relieve Air Pollution under Population-Growth Uncertainty[END_REF]), we assume that X(t, W t ) is a geometric brownian motion (see [START_REF] Karoui | Introduction au Calcul stochastique, première partie[END_REF] page 88):

X(t, W t ) = x 0 e (α-σ 2 /2)t+σWt , t ∈ R + ,
where W t is a standard brownian motion, α is the trend of the demand and σ is its volatility.

B. Recapitulation of the HSDPA model

To evaluate the customer satisfaction, let us first recall the flow throughput γ t (r) a user can expect at distance r from the center of the cell as carried out in [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF]. The resource of a single downlink data channel is time-shared between active users. Denote by φ u the fraction of time the BS transmits to user u, with u φ u = 1. The data rate of user u is then C u (r) = C(r)×φ u , where C(r) is the peak data rate, obtained in the absence of any other user in the cell, i.e., for φ u = 1. When there are x users, the "fair power" sharing is defined by φ u = 1

x . We have:

C(r) = min C 0 , Z b × Γ(r) η + I(r)
,

where:

• C 0 is the maximum peak rate (which depends on channel bandwidth and coding efficiency) • Z is the cell chip rate • b is a lower bound for energy-per-bit to noise density ratio (E b /N 0 ) • Γ(r) is the path loss between the BS and user u • η is the thermal noise to received power ratio • I(r) is the interference to received power ratio. Fig. 1 shows the peak data rate in a hexagonal cell. We only take interference from immediate adjacent cells in account, and we assume that the path loss Γ(r) decreases according to 1/r 4 .

We define the cell load as (see [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF]):

ρ t = X t πR 2 R 0 2πrdr C(r) .
• if ρ t > 1, the cell is overloaded and one can show ( [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF]) that the number of active users grows indefinitely; any individual data rate tends to zero, and the cell is saturated • if ρ t < 1, the cell is underloaded and the number of active users tends to a finite stationary regime.

We are naturally led to introduce:

X max = πR 2 R 0 2πrdr C(r) -1 , (1)
such that ρ t = X t /X max . Hence ρ t < 1 ⇐⇒ X t < X max .

Let us denote the flow throughput of users at distance r by γ t (r) (it is the ratio of the mean flow size to the mean flow duration). Then, it can be shown (see [START_REF] Bonald | Wireless Downlink Data Channels: User Performance and Cell Dimensioning[END_REF]) that, if ρ t < 1:

γ t (r) = C(r)(1 -ρ t ). (2)
By assumption, the mean flow size (the numerator of γ t (r)) does not vary significantly, whereas the mean flow duration (the denominator) does as the load increases. Thus, if we want to compute the mean flow throughput, the significant number to calculate is the harmonic mean of γ t (r) over the cell (balanced by the proportion of active users between r and r + dr). In other words, we have to calculate the arithmetic mean of 1/γ t (r) over the cell. From (2) we deduce:

γ t = R 2 R 0 2rdr γ t (r) -1 = X max -X t ,
and of course, in overload, γ t = 0. Finally, we can summarize the whole calculation by:

γ t = (X max -X t ) + , (3)
where x + = max(x, 0). Note that γ t = 0 if and only if the cell is saturated.

C. Customers satisfaction

Now we can compute the customer satisfaction, which can reasonably be supposed to depend on γ t . Since subjective satisfactions have been shown to be more sensitive to small variations at low throughputs than at high throughputs, Enderlé and Lagrange propose in [START_REF] Enderlé | User Satisfaction Models and Scheduling Algorithms for Packet-Switched Services in UMTS[END_REF] to model the customer satisfaction as a negative exponential function of the throughput:

S t = e -β/(Xmax-Xt) + .
For example, β can be chosen as: β = log(2) • γ 1/2 , where γ 1/2 is the throughput value ensuring a satisfaction of 50%. Once again, note that S t = 0 if and only if X t ≥ X max , i.e., if and only if the cell is saturated.

D. Daily Profit

Let us recall that X t is sampled day by day, for example during the second or the third highest peak hour. Within 24 hours, the operator transmits µ min(X t , X max ) to active users, where µ is a multiplicative factor between the peak hour and the whole day. Typically, we can consider that the peak hour represents 25 % of the total daily transfer. This leads to µ ≈ 4 • 3600 ≈ 14000 sec. Since taxation is applied to the volume of transfers and not to the duration, the gross daily profit per cell is given by:

π gross = δ µ min(X t , X max ), ( 4
)
where δ is the transfer price (say in $/Mbit). However the gross profit should be weighed by the customer satisfaction to account for the quality of the communications. The net profit is thus calculated as the product of π gross by S t :

π net = δ µ min(X t , X max )e -β/(Xmax-Xt) + .
If S t = 0, i.e. if the cell is saturated, the net profit is null. If the satisfaction is maximal, i.e., S t = 1, the net profit is equal to the gross profit (4). To sum up we have: Intuitively, as the demand rises, X t will increase as will the profit (Fig. 2). Then, the profit will decrease because the unsatisfaction effect becomes dominant.

π t = δ µ X t e -β/(Xmax-Xt) if X t < X max π t = 0 otherwise.

E. Upgrading Investment

As can be seen from Figure 2, if no upgrading action is taken, the profit will progressively tend to zero. Once the operator decides to upgrade, he can install additional transmitters operating on different frequency bands. In such a case we obtain a higher value of X max , so that:

π ′ t = δ µ min(X t , X ′ max )e -β/(X ′ max -Xt) + .
The upgrading cost is a decreasing function of time. The decrease of the cost is due to many factors, for example the R&D progress, and also the serialization in the manufacturing chain. Moore's law states that electronic devices' capacity doubles every 18 months. So in this paper we assume it decreases exponentially:

K(t) = K 0 e -ǫt ,
where ǫ is the depreciation rate.

F. Total Profit

Let us introduce date T , at which the investment becomes obsolete (in other words, the proposed investment cannot be undertaken after T ).If we denote the investment date by t 0 (0 < t 0 < T ), the total profit Π T (t 0 ) actualized at the date t = T is:

Π T (t 0 ) = t0-1 t=0 e ζ(T -t) π t + T t=t0 e ζ(T -t) π ′ t , ( 5
)
where ζ is the actualization rate. For simplicity, we assume that ζ is constant during the period [0, T ].

III. RISK-HEDGING USING AN AMERICAN OPTION

A. Externalizing the financial risk

As shown above, there is a tradeoff between the growth of the demand (encouraging to invest) and the depreciation of the equipment cost (encouraging to wait). Then the risk is to be led to invest while the equipment is still expensive. In this section we show how to hedge against this risk using an American option. This option, acquired from a third party like a bank, gives us the right to buy the equipment at price K ⋆ instead of K(t), until date t ⋆ = t(K ⋆ ) (see Fig. 3). Let us recall that the operator has the right but not the obligation to exercise this option, but has to pay in return a premium to the bank, denoted by P . If he has to invest before date t ⋆ , he will exercise the option, give K ⋆ to the equipment provider and the bank will pay the difference. Otherwise, he will not exercise the option and he will lose the premium, but he still can invest. In this section we will try to answer the two following questions:

• when is this option going to be exercised ?

• how much does it cost (i.e. calculate P ) ?

B. Introducing the American option

When is the option going to be exercised ? It depends on the additional profit expected from investing to upgrade the network: at least, this additional profit has to be greater than K ⋆ . At date t, it can be expressed as follows:

S t = E T t e -ζ(s-t) (π ′ (s, W s ) -π(s, W s )) ds F t . (6)
Facing the decision to invest or not, the operator's strategy is to compare the profit realized if investing with the value of waiting, typically to check that the traffic is not going to decrease unexpectedly which would make the upgrading expenditure a sunk cost. This appears to be the classical problem of finding the exercise strategy for an American option, with the following features:

• t ⋆ as the option's maturity • K ⋆ as the exercise price or strike • S t as the underlying asset • (S t -K ⋆ ) + as the option's payoff, denoted by Z(t):

Z(t) = max{S t -K ⋆ , 0} (7)

C. Pricing of the American option

The resolution of this problem appeals to classical stochastics theory and the risk-neutralization approach: see [START_REF] Broadie | Option Pricing: Valuation Models and Applications[END_REF] [START_REF] Karoui | Introduction au Calcul stochastique, première partie[END_REF].

1) Preliminaries: to detail this approach, let us introduce two progressively measurable processes µ t and κ t , respectively the expected total return on the asset and its volatility, so that:

dS t /S t = µ t dt + κ t dW t (8)
along with the market price of risk:

θ t = κ -1 t (µ t -ζ).
We obtain expressions of µ t , κ t and θ t in the appendices A to C, where we show that:

θ t = - π ′ (t, W t ) -π(t, W t ) ∂v ∂x (t, W t )
, with:

v(t, x) = T t E e -ζ(s-t) (π ′ (s, W s ) -π(s, W s ))|W t = x ds.
(9) Note that applying the risk-neutralization approach will also require Novikov's condition (see [START_REF] Karoui | Introduction au Calcul stochastique, deuxième partie: finance[END_REF], page 65), which states that:

E exp 1 2 T 0 θ 2 t dt < +∞. ( 10
)
In the appendix D, we show that Novikov's condition is verified in our specific case.

2) The risk-neutralization approach: under this condition, let S([t, t ⋆ ]) be the set of stopping times with values in [t, t ⋆ ] and define the following process known as the Snell envelope:

Y t = sup τ ∈S([t,t ⋆ ]) E Q ⋆ e -ζτ Z(τ ) F t . (11)
Here, Q ⋆ is the risk-neutral probability, whose density w.r.t. P, the historical probability, is:

dQ ⋆ dP = exp - 1 2 t ⋆ 0 θ 2 s ds - t ⋆ 0 θ s dW s .
In fact, since we will have to simulate trajectories of the asset beyond date t ⋆ (until date T ), we will rather choose the probability Q, whose density w.r.t. P is:

dQ dP = L T = exp - 1 2 T 0 θ 2 s ds - T 0 θ s dW s . (12)
Note that Q is indeed a probability measure, since E P [L T ] = 1, as θ t verifies Novikov's condition (see previous paragraph). Note also that Q ⋆ is the restriction of Q to F t ⋆ (see [START_REF] Karoui | Introduction au Calcul stochastique, première partie[END_REF], Theorem 9.1.2.), so that (11) still holds with

Q if t ≤ t ⋆ .
Then the premium of the option at any time t ∈ [0, t ⋆ ] is given by [START_REF] Broadie | Option Pricing: Valuation Models and Applications[END_REF]:

Π t = E Q e -ζ(τ (t)-t) Z(τ (t)) F t , (13)
where τ (t) is the solution of the maximization in [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF]. τ (t) is interpreted as the optimal exercise strategy of the option calculated at date t (1) .

IV. THE DYNAMIC PROGRAMMING SOLUTION

As stated above, the problem is to find the stopping time maximizing the option's payoff under risk neutrality (Eqn. [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF]). However, it is impossible to compute Z(t) analytically, so we make use of a dynamic programming approach, as in [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF]. We recall that it consists in dividing the problem into two binary decisions at the final date t ⋆ : the "immediate" one and its generated value, and the "delaying" one and its continuation value. Then moving backward, and repeating the same binary decision, we obtain the expected optimal time which lies in an expected interval in which the investment should be undertaken [START_REF] Salahaldin | A Numerical Algorithm for Decision-making in Sustainable Transport Projects Investment[END_REF]. We must then, at each moment, find two different values: the option's payoff in case of investment and the continuation value in case of waiting.

A. Monte-Carlo simulations to generate the underlying asset

Calculating S t involves a complex integration (Eqn. ( 6)) that cannot be performed analytically. We then use Monte-Carlo simulations as follows:

• first we compute v(t, x) with Eqn. (9) for t ∈ [0, t ⋆ ] and

x ∈ [w min , w max ]

( 2) .

• then we discretize time: t = t 0 . . . t N with t 0 = 0 and t N = t ⋆ = N δt. After that we simulate J trajectories of S t under Q: the j-th trajectory is denoted by (S j ) and has the value S j n at time t n = n δt. More precisely, we simulate (under Q) J trajectories of the historical brownian (W j ) (3 ) , and then we compute S j n = v(t n , W j n ) by interpolating v(t, x). This is far more efficient than computing directly the integral, especially if we want to simulate a large number of trajectories, since we do not have to compute v each time again. To know how we interpolate a surface, see appendix F.

B. Continuation value and decision tree algorithm

At time t ⋆ , the operator invests if Z N > 0. More generally, at a time t n < t ⋆ , the operator has two alternative choices: either invest now and get Z n , or wait and get the expected continuation value, denoted by C n . The generated cash-flow is then given by:

F n = max{Z n , C n }.
We already know Z n by [START_REF] Karoui | Introduction au Calcul stochastique, deuxième partie: finance[END_REF]. As for C n , we use the Least Squares Monte-Carlo (LSM) approach defined by Longstaff and Schwartz [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF]. This approach consists in writing the expected continuation value C n as a general function of S n (in our case we took a 2-degree polynom), taking information from the J cash-flows at t n+1 and using the fact that:

C n (S) = e -ζδt E[F n+1 |S n = S],
where F n+1 is the (random) cash-flow of the option at t n+1 . To obtain recursively C n , we can write the following algorithm:

• at t N , for each trajectory j = 1 . . . J, calculate the cashflow F j N = Z j N .

• move one period back to t N -1 . For each (S j ), check if the option is "in the money", i.e. if Z j N -1 > 0. If it is the case, calculate the continuation value C j N -1 using the cash-flow if investment is delayed: C j N -1 = e -ζδt F j N . Estimate then the general expression of C N -1 (S) by the LSM algorithm. This consists in regressing the found values C j N -1 on a constant, S and S2 , as in [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF] (see appendix E). Let us denote the estimated expression by ĈN-1 (S). The estimated cash-flow at N -1 is then given by:

F j N -1 = max{Z j N -1 , ĈN-1 (S j N -1 )}. ( 14
)
If it is optimal to exerce at t N -1 , then by convention F j N becomes 0 (because the option can only be exercised once).

• for each time t n , repeat the same process until n = 0.

Let us denote by O n the set of the j such that Z j n = 0, and by I n the set of the j such that Z j n > 0. Here is a summary of the whole algorithm:

1. simulate J trajectories (S j ) under Q 2. for j = 1 . . . J, put F j N = Z j N 3. for n = (N -1) . . . 1, 0 :

3.1. for j = 1 . . . J, calculate Z j n : -if Z j n = 0, j ∈ O n -if Z j n > 0, j ∈ I n 3.2.
process O n and I n separately:

∀j ∈ O n : ∀j ∈ I n : put F j n = e -ζδt F j n+1
-regress C j n = e -ζδt F j n+1 on 1, S and S 2 to obtain a 2-degree polynom Ĉn (S) -put F j n = max{Z j n , Ĉn (S)} -if Z j n > Ĉn (S), then n is the new investment date, so put F j m = 0 ∀m > n

C. Option premium

Averaging the F j 0 , and using (13) and the law of large numbers, we obtain the premium Π 0 of the option:

Π 0 ≈ 1 N J j=1 F j 0 .

D. Expected investing time

Investigating our decision tree, it can happen that for some j we do not decide to invest before t ⋆ . Then we will be lead to invest between t ⋆ and T (4) . For such trajectories, we do not know when the investment takes place. Furthermore, even for the other trajectories, additional information between t ⋆ and T can be useful to adjust the value of the investment date. For these two reasons, we decide to simulate S t further until T (5) . Thus we perform one more time a backward dynamic algorithm, that time between 0 and T , using:

Z(t) = (S t -K(t)) + if t > t ⋆ Z(t) = (S t -K ⋆ ) + otherwise . (15)
Finally, we obtain for each of the J trajectories a best investment date T j inv . If T j inv > t ⋆ , it means that we have invested without exercising the original option, whereas if T j inv ≤ t ⋆ , it means that we have exercised the option. Averaging the T j inv , we obtain the expected investing time under the risk-neutral probability E Q [T inv ]. But for us, it is more relevant to calculate E P [T inv ]. Using (12), we obtain:

E P [T inv ] = E Q T inv L T ≈ 1 N J j=1 T j inv L j T . (16)
V. NUMERICAL RESULTS

In order to illustrate our algorithm, we applied it using the free simulator Scilab (see [START_REF] Gomez | Engineering and Scientific Computing with Scilab[END_REF]). We considered a HSDPA pure data network with a random growing demand, as described in section II-A. We used the following parameters for our computation:

• the investment can take place until T = 150 days.

• the equipment can be purchased at the initial price K 0 = 300000 $, and its price decreases with a rate ǫ of 50% per year. • the actualization rate ζ is fixed to 5% per year.

• the traffic demand starts at x 0 = 1.2 Mbit/sec/cell, and increases with a drift fixed to α = 0.54% per day. Its volatility is fixed to 0.01 day -1/2 . Its maximal value is fixed to X max = 3 Mbit/sec/cell before the investment, and to X ′ max = 8 Mbit/sec/cell after the investment. • the data transfer price is fixed to δ = 0.1 $/Mbit.

• we take a satisfaction parameter of β = 0.7 Mbit/sec/cell.

• we simulate 10000 different trajectories of the asset. Recall that the price is obtained with equation ( 13), where t ⋆ implicitly appears in function Z (see equation [START_REF] Karoui | Introduction au Calcul stochastique, deuxième partie: finance[END_REF]). It appears that the price increases with t ⋆ . This was expected, since the longer the option's maturity is, the higher the risk for the bank is, and then the more expensive the option is. Recall that the date is obtained with equation ( 16), where t ⋆ appears in generalized function Z (see equation ( 15)), and may be prior to the investment's date. It appears that the investment date is very low for higher values of t ⋆ . This happens because K ⋆ is very low, thus it is all the more interesting to invest early. Theoretically, as K ⋆ decreases slowly toward 0, the investment date decreases accordingly until reaching 0 for t ⋆ = ∞. However, as investment can only occur on a daily basis, this cannot be obviously observed on Figure ( 5), unless we make computations for huge values of T , the final date. Unfortunately, this is not reasonable, since these computations will be extremely heavy.

A. Option's price

B. Investment date

However, the lower the option's maturity is, the later the investment takes place. The investment date may even be later than t ⋆ . In that case, the option is not exercised. This can be explained as follows: when t ⋆ is low, K ⋆ , the equipment's exercise price, is quite high. Thus, the option is not really interesting. Rapidly the equipment's real price will sink under K ⋆ , and within that short period it is better to take the risk of waiting.

VI. CONCLUSION

In this work, we proposed a model for risk hedging when dealing with investment under uncertainty in telecommunication networks. In such a case, the risk comes from the random evolution of the demand, possibly resulting in unexpected explosions of the trafic leading to network saturation. To hedge again this risk, the operator would buy an option from some financial parts that gives him the right but not the obligation of buying equipments at a given price, until a maturity date. We calculate, using backward dynamic programming and a least square approach, the premium of the option and the expected investment date. Our results show that the option price increases with the exercise date, whereas the mean investment date sinks. As a future work, we aim at considering the case where multiple investments are possible: adding more than one band, or implementing a more efficient technology (e.g. 3G LTE).

APPENDIX

The purpose of this technical annexe is to prove that the mathematical conditions for applying a risk neutralization approach to price the American option are fulfilled. Precisely, there will be three main steps: 1-study the regularity of the function v(t, x) from which the underlying asset is derived, and give a differential equation checked by its derivatives, 2deduce from Itō's lemma applied to v the expression of the market price of risk θ t , and 3-verify Novikov condition on θ t thanks to numerical simulations.

A. An explicit expression for v(t, x) Expression ( 6) of the underlying asset can be re-stated as follows:

S t = v(t, W t ),
where we have introduced the function:

v(t, x) = E T t φ(t, s, W s ) ds W t = x with: φ(t, s, w) = e -ζ(s-t) (π ′ (s, w) -π(s, w)).
In this section, we aim at giving a fully explicit expression for the function v(t, x), in order to study its properties in the following of the annexe. For this purpose, let us first swap sum and expectation in the expression of v (6) . We obtain:

v(t, x) = T t E [φ(t, s, x + W s -W t )|W t = x] ds,
and, since the increments of W s are independent:

v(t, x) = T t E [φ(t, s, x + W s -W t )] ds = T -t 0 E [φ(t, t + s, x + W t+s -W t )] ds.
Let us introduce another two functions:

f (t, s, w) = φ(t, t + s, w)
and:

u(t, s, x) = R f (t, s, w)g(x, s, w) dw, (17)
where g(x, s, .) is the gaussian density with mean x and variance s. We finally get:

v(t, x) = T -t 0 u(t, s, x) ds. (18)

B. Regularity and differential equation for v

Let us first recall that:

   f (t, s, w) = e -ζs (π ′ (t + s, w) -π(t + s, w)) g(x, s, w) = 1 √ 2πs e -(w-x) 2 2s
Lemma 1: π(s, w) and π ′ (s, w) are C ∞ (]0, +∞[×R), bounded, with bounded derivatives Proof: let us prove the property with π 1 for example.

• First we prove that π is C ∞ (]0, +∞[×R). For all s 0 ∈]0, +∞[, let us introduce w 0 = h(s 0 ), the number such that X(s 0 , w 0 ) = X max :

σw 0 = log(X max /x 0 ) -(α -σ 2 /2)s 0
The points (s 0 , w 0 ) define a line ∆ (see Fig. 6). Let us also introduce the two subsets of ]0, +∞[×R:

Ω 1 = {(s, < h(s)} Ω 2 = {(s, w)/w ≥ h(s)}
These two subsets are situated respectively under and above ∆. On Ω 1 , X(s, w) < X max and we have:

π(s, w) = δx 0 e (α-σ 2 2
)s+σw e -β/(Xmax-x0e (α-σ 2

2

)s+σw ) , and on Ω 2 , X(s, w) ≥ X max and π = 0.

On Ω 1 , since:

∂ ∂s X = (α -σ 2 /2)X ∂ ∂w X = σX ,
one can show by recursion over n = p + q that the derivatives of π can be written:

∂ n ∂s p ∂w q π = P p,q (X) (X max -X) 2n e -β/(Xmax-X) (19)
where P p,q is a polynom. Hence the denominator is counterbalanced by the second exponential term in the expression of the derivative of π, so that the derivatives all tend to 0 in the neighborhood of ∆ and the transition between Ω 1 and Ω 2 is C ∞ .

• Secondly we prove that each derivative of π is bounded. Expression (19) is a continuous function of X on [0, X max [, and is also continuous at X max . Hence, it is bounded for X ∈ [0, X max ]. Since:

X(Ω 1 ) =]0, X max ] ⊂ [0, X max ],
this achieves the proof.

Lemma 2: for any two compact sets C ⊂]0, +∞[, C ′ ⊂ R, we have the following majorations:

∀s ∈ C, ∂ i g ∂s i ≤ ϕ i x,C (w) ∀x ∈ C ′ , ∂ i g ∂x i ≤ ψ i s,C ′ (w)
where ϕ i x,C and ψ i s,C ′ are summable over R.

Proof: we only prove the first majoration, the second one is exactly similar. One can show by recursion that ∂ i g ∂s i can be written:

∂ i g ∂s i = g(x, s, w) di k=0 a i,k (s)w k , (20)
where each a i,k varies continuously with s. We want to bound (20) when s varies within a compact set C = [a, b] ⊂]0, +∞[. Since:

∂ i g ∂s i ≤ g(x, s, w) di k=0 |a i,k (s)||w| k ,
we deduce:

∂ i g ∂s i ≤ 1 √ 2πa e -(w-x) 2 2b di k=0 max s∈[a,b] |a i,k (s)| |w| k ,
from which the first majoration is immediate.

Lemma 3: f is C ∞ w.r.t. each of its variables t ∈ [0, T [, s ∈]0, T -t[, w ∈ R
, and its derivatives are bounded.

Proof: this comes directly from Lemma 1. In particular, there exist constants K i and K ′ i so that:

∂ i f ∂t i ≤ K i , ∂ i f ∂s i ≤ K ′ i Lemma 4: u is C ∞ w.r.t. each of its variables t ∈ [0, T [, s ∈]0, T -t[, x ∈ R.
For any compact set C ′ ⊂ R:

∂ i u ∂t i ≤ K i ∂ i u ∂x i ≤ K i,C ′ , ∀x ∈ C ′
and the following differential equation is verified by u:

u ′′ xx = 2(u ′ s - R f ′ s g)
Proof:

• Using Lemma 3, we obtain:

∂ i ∂t i (f g) = ∂ i f ∂t i g ≤ K i g
Then, the derivability of u w.r.t. t and the first majoration immediately come from expression (17) and the differentiation under the integral sign theorem.

• Using Lemmae 2 and 3, we obtain (for any x ∈ C ′ ):

∂ i ∂x i (f g) = f ∂ i g ∂x i ≤ K 0 ψ i s,C ′ (w)
Since ψ i s,C ′ is summable over R, we deduce the derivability of u w.r.t. x, along with the second majoration, taking:

K i,C ′ = K 0 R ψ i s,C ′ (w) dw.
• Each derivative ∂ i ∂s i (f g) is a sum of terms which can be written:

∂ p f ∂s p • ∂ q g ∂s q (p + q = i)
From Lemma 2 (taking C = [0, T ]) and Lemma 3, we get:

∂ p f ∂s p • ∂ q g ∂s q ≤ K ′ p ϕ q x,[0,T ] (w),
which is integrable over R. Hence, u is C ∞ w.r.t. s.

• By deriving under the integral sign, we have:

u ′′ xx (t, s, x) = R f (t, s, w)g ′′ xx (x, s, w) dw u ′ s (t, s, x) = R f (t, s, w)g ′ s (x, s, w) dw + R f ′ s (t, s, w)g(x, s, w) dw,
(splitting into two sums is allowed since f ′ s g is summable). Thanks to the heat equation verified by the gaussian kernel:

g ′ s = 1 2 g ′′ xx
we finally get third assertion of Lemma 4: Proof: the regularity of v is a direct consequence of equation ( 18) and Lemma 4. The differential equation checked by v is obtained as follows:

u ′′ xx = 2 R f (t, s, w)g ′ s (x, s, w) dw = 2 u ′ s - R f ′ s (t,
v ′ t = T -t 0 u ′ t ds -u(t, T -t, x) = T -t 0 R f ′ t g dw ds -u(t, T -t, x) = T -t 0 R (f ′ s g + ζf g) dw ds -u(t, T -t, x) = T -t 0 u ′ s - 1 2 u ′′ xx ds + ζv -u(t, T -t, x) = u(t, T -t, x) -lim s→0 u - 1 2 v ′′ xx + ζv -u(t, T -t, x).
f (t, ., w) being continuous at point s = 0, we have lim s→0 u(t, s, x) = R φ(t, t, w)g(x, 0, w) dw where the term g(x, 0, w) has to be understood as the Dirac distribution at x. Therefore:

v ′ t = -φ(t, t, x) - 1 2 v ′′ xx + ζv.
Lemma 6: ∂v ∂x (t,x) is null only on a curve x = x(t).

Proof: this lemma will not be rigorously proved, but instead inferred from numerical simulation of the surface ∂v ∂x (t, x), t ∈ [0, T [, x ∈ R. Fig. 7 shows this surface. Clearly, one can observe that on the left side of the red line, ∂v ∂x goes to zero only on a curve x(t). What can be proved analytically is that ∂v ∂x (t, x) < 0 on the right side of the line, where paradoxically the surface is very close to zero. Let us write ∂v ∂x in a new way: where W s is a standard brownian motion. φ(t, t+s, x+W s ) is always positive, and null iff t + s ≥ a(x + W s ) + b, where we have introduced two coefficients:

∂v ∂x (t, x) = T -t 0 R f (t, s, w) ∂g ∂x (x, s, w) dw ds = T -t 0 R f (t, s, w) w -x s g(x, s, w) dw ds = T -t 0 E φ(t, t + s, x + W s ) W s s ds,
a = - σ α -σ 2 /2 , b = log( X ′ max/x0) α -σ 2 /2 .
Hence:

∂v ∂x (t, x) = T -t 0 E φ(t, t + s, x + W s) W s s 1 {t+s<a(x+W s )+b} ds = T -t 0 E φ(t, t + s, x + W s) W s s W s < t+s-b a -x ×P t + s < a(x + W s) + b ds
If t-b a -x ≤ 0, it is immediate to obtain ∂v ∂x (t, x) < 0. The line t-b a -x = 0 being precisely the red line of Fig. 7, we have the result.

C. Expression of the risk premium θ t

In this section, we justify the existence of the market price of risk θ t and deduce its expression from Itō's lemma and the differential equation checked by v. Itō's lemma holds since v is regular. It gives:

dS t = ∂v ∂t (t, W t ) + 1 2 ∂ 2 v ∂x 2 (t, W t ) dt + ∂v ∂x (t, W t ) dW t
Provided X ′ max > X max , we have v(t, x) > 0 for any (t, x). Hence S t > 0 and we can write:

dS t S t = 1 v ∂v ∂t + 1 2 ∂ 2 v ∂x 2 dt + 1 v ∂v ∂x dW t .
By identifying this equation with the dynamics of the underlying asset (8), we get the expression of the expected total return on the asset µ t and the volatility κ t :

µ t = 1 v ∂v ∂t + 1 2 ∂ 2 v ∂x 2 κ t = 1 v ∂v ∂x
.

Lemma 6 ensures that κ t = 0 a.s., therefore the market price of risk θ t is well defined, and:

θ t = ∂v ∂x -1 ∂v ∂t + 1 2 ∂ 2 v ∂x 2 -ζv .
The final expression for θ t is a consequence of Lemma 5:

θ t = - φ(t, t, W t ) ∂v ∂x (t, W t ) = - π 2 (t, W t ) -π 1 (t, W t ) ∂v ∂x (t, W t )
.

D. Novikov's condition

Now we have to prove that:

E exp 1 2 T 0 θ 2 t dt < +∞.
Actually, is T 0 θ 2 t dt even finite ? The question is relevant, because Lemma 6 shows that on a certain line, ∂v ∂x is null, and so θ(t, w) is infinite (see Fig. 8). Now, could a trajectory (W t ) come close to this line during a time long enough so that:

E exp 1 2 T 0 θ 2 t dt = +∞ ?
Here we use a result of El Karoui and Gobet (see [START_REF] Karoui | Introduction au Calcul stochastique, première partie[END_REF], Proposition 1.3.8.):

P sup t≤T |W t | ≥ c ≤ 2 P(|W T | ≥ c),
which tends to 0 extremely rapidly when c → ∞. Hence, if we choose correctly our parameters so that the critic line lies far enough from the line w = 0, the probability to reach it during the experiment will be extremely low. Then, in practice, we will consider that θ t remains almost surely bounded by a constant θ; and a fortiori, Novikov's condition will be verified, since we will have:

E exp 1 2 T 0 θ 2 t dt ≤ e T θ2 /2 .

E. Regressing a set of points on a 2-degree polynom

Given a set of points (x i , y i ) 1≤i≤n in R 2 , the aim of the section is to find three real numbers a, b, c, such that It is equivalent to say that the vector y -ax 2 -bx -c is orthogonal to 1, x and x 2 . In other words, ax 2 + bx + c is the orthogonal projection of y on Vect(1,x,x 2 ), and thus we are sure that (21) has a solution. Is this solution unique ? If 1, x and x 2 were not independent, there would be three real numbers u, v and w such that: ∀i, u + vx i + wx 2 i = 0.

• as soon as there are more than two different values of x i , this is impossible, and so there is a unique solution.

• if x i takes only two different values, then the solution is not unique any more, but (1,x,x 2 ) has rank 2. So we choose to regress y on 1 and x for example (and we find a line which intersects the centroids of the two corresponding subsets). • if x i takes only one value, then (1,x,x 2 ) has rank 1. So we choose to regress y on 1 (and we find the mean of the y i s).

F. Interpolating a surface

Given a surface of equation: z = v(t, x), suppose we only know a discrete set of values of z: z i,j = v(t i , x j ). We want to compute v(t, x) for any value of t and x. The technique is very similar to a linear interpolation in 1 dimension:

• first find the intervals [t i , t i+1 [ and [x j , x j+1 [ in which t and x lie • then interpolate first w.r.t. t. Denoting t-ti ti+1-ti by α t , we obtain:

z 1 = α t z i,j
+(1 -α t )z i+1,j z 2 = α t z i,j+1 +(1 -α t )z i+1,j+1

• at least interpolate w.r.t. x. Denoting x-xi xi+1-xi by α x , we obtain:

z = α x z 1 + (1 -α x )z 2 .

Remark

We have: z = α t α x z i,j + (1 -α t )α x z i+1,j +α t (1 -α x )z i,j+1 + (1 -α t )(1 -α x )z i+1,j+1 , which is symmetrical in (t, i) and (x, j). Hence, we would have found the same result by interpolating first w.r.t. x, and then w.r.t. t (see Fig. 9).

Fig. 1 .

 1 Fig. 1. Peak data rate against distance to the center of the cell. The distance and the rate are normalized w.r.t. the cell's radius and to C 0 . Z/b = 3 Mbits/sec, η = 10%.

Fig. 2 .

 2 Fig. 2. Daily Profit generated by demand. Xmax = 3 Mbits/sec/cell, δ = 0.1 $/Mbit, µ = 14000 sec, β = 0.7 Mbit/sec (which corresponds to γ 1/2 ≈ 1 Mbit/sec).

Fig. 3 .

 3 Fig. 3. exponential decrease of the cost

Fig. 4 .

 4 Fig. 4. Price of the option On Figure (4), we represent the price of the option versus t ⋆ .Recall that the price is obtained with equation (13), where t ⋆ implicitly appears in function Z (see equation[START_REF] Karoui | Introduction au Calcul stochastique, deuxième partie: finance[END_REF]). It appears

Fig. 5 .

 5 Fig. 5. Investment date On Figure (5), we represent the investment date versus t ⋆ .Recall that the date is obtained with equation (16), where t ⋆ appears in generalized function Z (see equation (15)), and may be prior to the investment's date. It appears that the investment date is very low for higher values of t ⋆ . This happens because K ⋆ is very low, thus it is all the more interesting to invest early. Theoretically, as K ⋆ decreases slowly toward 0, the investment date decreases accordingly until reaching 0 for t ⋆ = ∞. However, as investment can only occur on a daily basis, this cannot be obviously observed on Figure (5), unless we make computations for huge values of T , the final date. Unfortunately, this is not reasonable, since these computations will be extremely heavy.However, the lower the option's maturity is, the later the investment takes place. The investment date may even be later than t ⋆ . In that case, the option is not exercised. This can be explained as follows: when t ⋆ is low, K ⋆ , the equipment's exercise price, is quite high. Thus, the option is not really interesting. Rapidly the equipment's real price will sink under K ⋆ , and within that short period it is better to take the risk of waiting.

Fig. 6 .

 6 Fig. 6. Partition of the plane into two subsets Ω 1 and Ω 2 .

Fig. 7 .

 7 Fig.7. The surface ∂v/∂x (we kept the previous parameters). On the right of the red line, we show in Lemma 6 that it is nonnull. On the left, we see that it is null only on the green line.

Fig. 8 .

 8 Fig. 8. Repartition of the peaks of θ(t, w). They all lie on the green line represented on Fig. 7. Normally, they should form a continuous crest, but due to discretization they show an uneven behavior.

4 i + b x 3 i + c x 2 i = x 2 i y i a x 3 i

 4323 n i=1 |y i -P a,b,c (x i )| 2is minimal, where:P a,b,c (x) = ax 2 + bx + c.After deriving w.r.t. a, b and c, we obtain respectively:+ b x 2 i + c x i = x i y i a x 2 i + b x i + c = y i(21)Has (21) a solution ? Let us consider the four vectors of R n :

  we can re-write the system as:  (x 2 |x 2 ) (x 2 |x) (x 2 |1) (x|x 2 ) (x|x) (x|1) (1|x 2 ) (1|x) (1|1)

Fig. 9 .

 9 Fig. 9. Interpolation of the surface.

Note that if the option were a european option, the price at date t would be:Πt = E Q e -ζ(t ⋆ -t) Z(t ⋆ ) Ft(see[START_REF] Karoui | Introduction au Calcul stochastique, première partie[END_REF] page 65). But here, our option is an American option, so we have to generalize this result and to use the Snell envelope.

to bind efficiently the brownian motion, see Appendix D.

to perform that, assuming that the probability of our random generator is Q, we simulate a standard brownian motion (W Q t ), and then using Girsanov's theorem (see[START_REF] Karoui | Introduction au Calcul stochastique, première partie[END_REF], Theorem 9.4.5.), we build by recursion a new brownian motion (Wt) under P, such that Wt = W Q t -0≤s<t θ(s, Ws) ds.

Note that in theory, it could happen that we never decide to invest, even after T . However, given the deterministic trend of the demand, this would mean that Wt remains extremely low. Considerations on the brownian motion (see appendix D) ensure that in practice it will not happen.

That is the reason why we chose Q instead of Q ⋆ .

This is possible because φ is positive, since 0 ≤ π ≤ π ′


Laura Di Giunta
email: laura.digiunta@uniroma1.it

Concetta Pastorelli

Nancy Eisenberg

Maria Gerbino

Valeria Castellani

Anna Silvia Bombi

Laura Di Giunta

TRAJECTORIES OF PHYSICAL AGGRESSION: COMPARING SELF AND MOTHER Developmental Trajectories Of Physical Aggression: Prediction Of Overt And Covert Antisocial Behaviors From Self And Mothers' Reports

Keywords: aggression, multiple informants, longitudinal analysis, overt antisocial behavior, covert antisocial behavior

Physical aggression declines for the majority of children from preschool to elementary school.

Although this desistance generally continues during adolescence and early adulthood, a small group of children maintains a high level of physical aggression over time and develops other serious overt and covert antisocial behaviors. Typically, researchers have examined relations of developmental changes in physical aggression to later violence with teachers' or mothers' reports on surveys. Little is known about the degree to which children's self-reported physical aggression predicts later antisocial behavior.

The longitudinal study in this article had a staggered, multiple cohort design. Measures of physical aggression were collected through self and mother reports from age 11 to 14, which were used to construct trajectory groups (attrition was 6% and 14% from age 11 to 14, respectively, for self-and mother-reports). Overt and covert antisocial behaviors were self-reported at age 18-19 (attrition was 36% from age 11 to 18-19). Four trajectory groups (low stable, 11%; moderate-low declining, 34%; moderate declining, 39%; high stable, 16%) were identified from self reports, whereas three trajectories (low declining, 33%; moderate declining, 49%; high stable, 18%) were identified from mothers' ratings. We examined the prediction of overt and covert antisocial behaviors in early adulthood from the high stable and the moderate declining trajectories.

According to both informants, higher probability of belonging to the high stable group was associated with higher overt and covert antisocial behavior, whereas higher probability of belonging to the moderate declining group was associated with higher covert antisocial behavior. Our results support the value of children's as well as mothers' reports of children's aggression for predicting different types of serious antisocial behavior in adulthood.

INTRODUCTION

A number of researchers have argued that it is important to differentiate physical aggression from other types of aggression [START_REF] Tremblay | The development of aggressive behavior during childhood: what we learned in the past century?[END_REF]. In recent decades, children's physical aggression (PA) has received more empirical attention than other modes of aggressive behavior, likely because it is the most predictive of risk for the development of maladjustment during childhood, adolescence, and early adulthood [START_REF] Brame | Developmental trajectories of aggression[END_REF][START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Cairns | Growth and aggression: 1. Childhood to early adolescence[END_REF][START_REF] Timmermans | Which forms of child/adolescent externalizing behaviors account for late adolescent risky sexual behavior and substance use?[END_REF]. Similar trajectories of PA typically have been identified in samples of elementary school aged children and pre-adolescents; these are a group with low stable aggression over time, a group (to which the majority of children belong) exhibiting a moderate level of aggression that declines over time, and a group with high stable aggression over time [START_REF] Brame | Developmental trajectories of aggression[END_REF][START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Côté | The development of physical aggression from toddlerhood to pre-adolescence: A nation wide longitudinal study[END_REF][START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF].

It has been found that aggressive children are prone to develop an overt antisocial pathway characterized by an escalation from minor aggressive (e.g., bullying, annoying others) to serious violent behaviors (e.g., gang fighting, physical assault) [START_REF] Dishion | The development and ecology of antisocial behavior in children and adolescents[END_REF]. However, the frequency of reported overt forms of antisocial behavior generally decreases during the emergence of adulthood [START_REF] Dishion | The development and ecology of antisocial behavior in children and adolescents[END_REF]. In addition, aggressive children can also develop a covert antisocial pathway (that is stable during childhood and increases from early adolescence to adulthood) characterized by minor covert antisocial behaviors (e.g. shoplifting, lying), followed by property damage (e.g. fire setting, vandalism), and finally by serious types of thefts [START_REF] Dishion | The development and ecology of antisocial behavior in children and adolescents[END_REF]. In particular, an increased risk for violent and nonviolent offending in adolescence has been noted for those individuals, mainly boys, who belong to the high stable PA trajectory throughout childhood and pre-adolescence [START_REF] Brame | Developmental trajectories of aggression[END_REF][START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF].

Accordingly, effective prevention strategies of antisocial behavior have been implemented in early phase of development [START_REF] Dodge | Community intervention and public policy in the prevention of antisocial behavior[END_REF][START_REF] Tremblay | Understanding development and prevention of chronic physical aggression: towards experimental epigenetic studies[END_REF], targeting multiple risk factors both at the individual and contextual level (school, peer, family, and community) [START_REF] Chamberlain | Treating chronic juvenile offenders: Advances made through the Oregon Multidimensional Treatment Foster Care model[END_REF][START_REF] Dodge | Community intervention and public policy in the prevention of antisocial behavior[END_REF][START_REF] Elliott | Issues in Disseminating and Replicating Effective Prevention Programs[END_REF][START_REF] Henggeler | Multisystemic treatment of antisocial behavior in children and adolescents[END_REF][START_REF] Scott | Randomised controlled trial of parent groups for child antisocial behaviour targeting multiple risk factors: the SPOKES project[END_REF]. However, further research efforts on aggressive behaviors within a developmental trajectory framework can be useful to identify subgroups of children who are the best candidates to behave antisocially over time and, thus, are most likely in need of intervention.

Initial information of relevance to planning interventions is available from the impressive

findings in existing longitudinal research on children's PA. However, there are some significant limitations in this body of work. To our knowledge, researchers previously have not examined if specific developmental pathways of PA are linked with specific antisocial pathways (i.e., overt and covert antisocial pathways) in young adulthood. In addition, investigations of developmental trajectories for nonaggregated types of aggressive behaviors (e.g., PA separated from other types of aggression) usually have not involved multiple raters. The majority of these studies have relied on teachers' or mothers' reports [START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF], whereas relatively little is known about PA trajectories based on self reports. To our knowledge, the only exception is the study of Martino and colleagues [START_REF] Martino | Multiple Trajectories of Physical Aggression among Adolescent Boys and Girls[END_REF], who identified PA trajectories from adolescents' reports.

Beyond the three typical physical aggression trajectories (low stable, desisting, high stable), Martino et al. [START_REF] Martino | Multiple Trajectories of Physical Aggression among Adolescent Boys and Girls[END_REF] identified a trajectory that started low but increased throughout adolescence and then declined in late adolescence. This result is in agreement with what has been suggested in other studies that not all individuals who become delinquent or violent have a history of early aggression [START_REF] Brame | Developmental trajectories of aggression[END_REF][START_REF] Loeber | Development of juvenile aggression and violence. Some common misconceptions and controversies[END_REF][START_REF] Schaeffer | Modeling Growth in Boys' Aggressive Behavior Across Elementary School: Links to Later Criminal Involvement, Conduct Disorder, and Antisocial Personality Disorder[END_REF]. Moreover, Martino et al. [START_REF] Martino | Multiple Trajectories of Physical Aggression among Adolescent Boys and Girls[END_REF] found that several self-reported indicators measured at Grade 7 (e.g., parental supervision, deviant peer association, academic orientation, impulsivity, and emotional distress) were all strongly associated with trajectory class membership, with higher maladjustment indicators being positively associated with the more problematic developmental pathways. However, Martino et al. [START_REF] Martino | Multiple Trajectories of Physical Aggression among Adolescent Boys and Girls[END_REF] did not examine the predictive validity of PA trajectories respect to adult outcomes. Moreover, these authors underlined the importance of determining whether their findings based on self-reports would replicate using data that incorporates different informants.

There is some disagreement among researchers about the validity of self report of PA. Some investigators have argued that self reports are advantageous because children can report episodes that mothers do not know about [START_REF] Ladd | Analysis of Cross-Informant Data for Concordance, Estimation of Relational Adjustment, Prevalence of Victimization, and Characteristics of Identified Victims[END_REF] and that teachers may underestimate [START_REF] Martino | Multiple Trajectories of Physical Aggression among Adolescent Boys and Girls[END_REF]. Others have argued that self reports are problematic because children may not recognize their own behavior as aggressive or are less willing than parents to report their aggressive behaviors [START_REF] De Los | Informant Discrepancies in the Assessment of Childhood Psychopathology: A Critical Review, Theoretical Framework, and Recommendations for Further Study[END_REF].

However, the use of multiple-informant strategies frequently has been recommended to improve the study of aggression [START_REF] Achenbach | Child/adolescent behavioral and emotional problems: implications of cross-informant correlations for situational specificity[END_REF][START_REF] Plewis | Modelling repeated ordinal reports from multiple informats[END_REF]. Some investigators have followed this advice in studies of either aggregated forms of children's aggression [START_REF] Caprara | Early emotional instability, prosocial behavior and aggression: some methodological aspects[END_REF][START_REF] Ladd | Analysis of Cross-Informant Data for Concordance, Estimation of Relational Adjustment, Prevalence of Victimization, and Characteristics of Identified Victims[END_REF][START_REF] Loeber | Development of juvenile aggression and violence. Some common misconceptions and controversies[END_REF][START_REF] Pastorelli | Measuring emotional instability, prosocial behavior and aggression in pre/adolescents: a cross/national study[END_REF][START_REF] Verhulst | Agreement between parents'reports and adolescents' selfreports of problem behavior[END_REF] or nonaggregated forms of adolescents' aggression [START_REF] Raine | The Reactive-Proactive Aggression Questionnaire: Differential Correlates of Reactive and Proactive Aggression in Adolescent Boys[END_REF]. Others have used repeated measurements, relying on different informants over many years, to obtain information on either inter-individual [START_REF] Caprara | Prosocial foundations of children's academic ahievement[END_REF] or intra-individual change in aggression [START_REF] Cairns | Growth and aggression: 1. Childhood to early adolescence[END_REF][START_REF] Timmermans | Which forms of child/adolescent externalizing behaviors account for late adolescent risky sexual behavior and substance use?[END_REF]. Overall, moderate inter-informant agreement has been found for externalizing behaviors because they are directly observable by different informants [START_REF] Achenbach | Child/adolescent behavioral and emotional problems: implications of cross-informant correlations for situational specificity[END_REF][START_REF] Verhulst | Agreement between parents'reports and adolescents' selfreports of problem behavior[END_REF]; however, relatively low adult-youth agreement on reports of child psychopathology is typical [START_REF] Achenbach | Child/adolescent behavioral and emotional problems: implications of cross-informant correlations for situational specificity[END_REF][START_REF] De Los | Informant Discrepancies in the Assessment of Childhood Psychopathology: A Critical Review, Theoretical Framework, and Recommendations for Further Study[END_REF].

Because of the need to better understand the validity of preadolescents' self-reported PA and the predictive relation of PA to serious externalizing problems in adulthood, the goals of the present study were [START_REF] Achenbach | Manual for the Youth-Self Report and 1991 Profile[END_REF] to determine the similarity of the developmental trajectories of PA during preadolescence using self-reports and mothers' reports; [START_REF] Achenbach | Child/adolescent behavioral and emotional problems: implications of cross-informant correlations for situational specificity[END_REF] to examine the predictive validity of the identified trajectories based on both informants in respect to overt and covert antisocial behaviors in early adulthood. This study goes beyond the study of Martino et al. [26] by replicating similar PA pathways with data that incorporated multiple reporters' perspectives and validating the usefulness of self reports in identifying children with chronic physical aggression and in predicting antisocial tendencies in early adulthood. Such results would be helpful for practitioners working in prevention programs with limited resources.

METHODS

Participants and Procedure

Participants were 439 children (55% boys) from a longitudinal study on social adjustment.

Children were drawn from two regular public schools in Genzano, a residential community located the larger society, composed of families of skilled workers, farmers, professionals, local merchants and their service staff. In particular, the occupational socio-economic distribution of the children's families matched the Italian national profile [START_REF]Annuario statistico italiano 2002[END_REF]. Families with different socioeconomic stata are integrated within this community in terms of both residence in which they live and schools that their children attend. The socioeconomic diversity of the sample and high residential integration adds to the generalizability of the findings. Consistent with national statistics, most subjects were from intact families (94%) and two-children families (64%).

The longitudinal project followed a staggered, multiple cohort design, with four cohorts assessed initially at age 11 in 1992, 1993, 1994, and 1995. They were reassessed yearly at ages 12, 13, and 14. All of the children enrolled in these grades participated in the study unless they happened to be absent from school when the measures were administered (on average 11% at age 11). Finally, due to funding limitations, the last follow-up took place in two different years (2000 and 2002) and included most of the original sample at the age of 18 (cohorts that started in 1993 and 1995) and 19 (cohorts that started in 1992 and 1994). Data collections' schema differentiated by cohort is reported in Table 1; the sample size over time is reported in Table 2.

Participants attended 6 th grade at age 11 and junior high school between ages 12 and 14. At age 18, 81% of the participants attended high school, 11% were college students, and 8% were workers; at age 19, 66% of the participants attended high school, 16% were college students, and 18% were workers (mostly unskilled).

Preliminary analyses indicated that children who participated all four time lags were rated as no different in aggression and in demographic variables than children who missed either one, or two data collections. Children who missed more than two data collections were not included in the present paper. Of the 439 children who participated in the present study, 294 of their mothers agreed to participate. Mothers provided yearly ratings when children were between the ages of 11 Insert Table 1 here and 14. Those mothers who did not participate in the research (30%) were traced by the researchers, they said they would participate in the research but never did so.

Those mothers who did not participate in the research did not differ in marital status, occupation, or years of education from those mothers who participated (respectively,  2 (3) = 1.027, p = .79;  2 (8) = 13.274, p = .10;  2 (3) = 1.027, p = .79;  2 (4) = 5.089, p = .28). In addition, children whose mothers participated did not differ in self-reported aggression from children whose mothers did not participate, PA at age 11: F(1,437) = .096, p = .76; at age 12: F(1,400) = 3.173, p = .076; at age 13: F(1,429) = 3.267, p = .071; and at age 14 F(1,401) = 2.259, p = .134. Children whose mothers participated at all four assessments were rated as more aggressive at 11 years old than children whose mothers missed one of the four assessments, whereas there were no significant differences at 12, 13, and 14 years old (PA at age 11: F(1,216) = 4.25, p = .040; PA at age 12:

F(1,245) = .238, p = .626; PA at age 13: F(1,252) = .016, p = .899; PA at age 14: F(1,149) = .804, p = .371). Attrition of children and mothers from age 11 to 14 was 6% and 14%, respectively, and was mainly due to relocation from the area or absence from school at the time of the assessments.

Participants were administered the measures by three female researchers during specially scheduled sessions in a school. Mothers completed the questionnaire at their child's school in a group setting. After junior high school, youths were invited to participate in the study by phone and they received paper versions of the questionnaires via mail to complete at home. They were asked to return the completed questionnaires to the researchers during specially scheduled sessions at a school. They received a small payment for their participation in the research (about 25 €).

During the entire research project, consents/assents were obtained, experimenters offered explanations as needed, and confidentiality was guaranteed for all participants. The entire study was approved by Sapienza University of Rome's Human Subjects Institutional Review Board.

Measures

Physical aggression (PA). Items pertaining to PA (6 for children, 5 for mothers) from the Physical and Verbal Aggression Scale 1 (PVA) [START_REF] Caprara | Early emotional instability, prosocial behavior and aggression: some methodological aspects[END_REF] were rated by children and mothers (1= never; 3 = often; e.g., I kick and hit or punch; items were in the third person for mothers; mean alphas for self-and mother reports of PA from age 11 to 14 = .80 and .81, respectively).

Overt antisocial behavior (OAB)

. OAB was assessed with self ratings (1 = never, 5 = often) on 4 items of the Violence Scale [START_REF] Caprara | Una scala per la misura dell'atteggiamento verso la violenza [A scale for the measurement of attitude toward violence[END_REF] (i.e., Have you participated in violent gangs actions?; Have you participated in violent actions of supporters groups?; Have you been involved in fights between people or rival groups?; Have you ever used violence during a quarrel?; mean alpha across ages 14, 18, and 19 =.78).

Covert antisocial behavior (CAB)

. CAB was assessed with self ratings (0 = not true, 2 = very true or often true) on 5 items from the Youth Self-Report (YSR) [START_REF] Achenbach | Manual for the Youth-Self Report and 1991 Profile[END_REF] (I lie or cheat; I set fires; I steal things at home; I steal things from places other than home; I use alcohol or drugs other than for medical conditions; mean alpha at ages 14, 18, and 19 =.70).

Analytic Approach

To explore if the frame of reference of PA for each reporter was stable over time [START_REF] Vandenberg | A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research[END_REF], the longitudinal factorial invariance of the PA factor structure was analyzed separately for self and mother reports. The PA items were considered ordered categorical variables because of their threepoint response format. With categorical data, two models are recommended [START_REF] Muthèn | Mplus users guide[END_REF]: the unconstrained model (i.e., thresholds and factor loadings are freed across groups) and the constrained model (i.e., thresholds and factor loadings are constrained to be equal across groups). Evidence for longitudinal invariance was examined by comparing the overall fit of the models and testing for the significance of difference in the χ² value, based on WLSMV χ² (i.e., weighted least square meanand varianceadjusted χ²), using Mplus 4.21 [START_REF] Muthèn | Mplus users guide[END_REF]. Partial longitudinal invariance occurs when significant difference in the χ² value between unconstrained and constrained models occurs [START_REF] Muthèn | Mplus users guide[END_REF]. Evaluation of the goodness of fit was based on indices that are less sensitive to sample size. Comparative Fit Index (CFI) values of at least .95 [START_REF] Hu | Cut-off criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives[END_REF] and Root-Mean-Square Error of Approximation (RMSEA) values lower than .05 are considered good [START_REF] Hu | Cut-off criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives[END_REF], whereas RMSEA values between .05 and .08, reflect an acceptable error of approximation [START_REF] Browne | Alternative ways of assessing model fit[END_REF].

In order to identify distinct developmental patterns of PA, we used the group-based semiparametric approach recommended by Nagin [START_REF] Nagin | Analyzing developmental trajectories: A semiparametric, group-based approach[END_REF][START_REF] Nagin | Group-based modeling of development[END_REF] with SAS Proc-Traj [START_REF] Jones | A SAS procedure based on mixture models for estimating developmental trajectories[END_REF]. For each participant, Proc Traj produces an explicit metric (i.e., the posterior probability of group membership in each trajectory group) for evaluating the accuracy of group assignments [START_REF] Nagin | Analyzing developmental trajectories: A semiparametric, group-based approach[END_REF]. The term trajectory probability is used when referring to the probability of an individual's membership in each of the trajectory groups (continuously distributed probabilities). In addition, Proc Traj assigns a categorical score based on each individual's highest trajectory probability. When referring to the categorical group score that is assigned to each participant, the term trajectory membership is used.

We conducted analyses to identify the best fitting trajectory models (censored normal models using continuous variables) for self or mother reports, including sex as a time independent covariate [START_REF] Jones | A SAS procedure based on mixture models for estimating developmental trajectories[END_REF]. We report the results of the multivariate logit regressions used to examine if sex affected the trajectory membership. Then we investigated inter-informant agreement of the trajectory groups by performing cross-tabulations between the trajectory membership related to the trajectories modeled from self and mothers' reports of PA using a χ² test and estimates of the standardized residuals. In addition, correlations across reporters were computed for yearly PA scores. Children who missed more than two data collections were not included in the aforementioned longitudinal analyses.

Finally, in order to explore the association of mother-and self-reported PA trajectories with OAB and CAB in young adulthood, a path analysis was used to test PA predicting age 18-19 OAB and CAB. Specifically, due to the longitudinal design of our study, part of our sample was followed up at age 18 (for those who were 11 years old in 1993 and in 1995), part was followed up at age 19 (for those who were 11 years old in 1992 and in 1994). We computed two new variables called OAB at age 18-19 and CAB at age 18-19 by collapsing into a single variable the data from the corresponding variables at ages 18 or 19. Then we created a control variable, called age 18-19, to take into account the age at which OAB and CAB were assessed.

We computed a multigroup path analysis (the self-reported and mother-reported information were the two levels of the group variable) in which the trajectory probabilities were the potential predictors of long-term OAB and CAB, while controlling for sex, the initial level of both antisocial behaviors (age 14) 2 , and the age 18-19 covariate.

To examine differences in the estimated parameters based on self and mother reports, we constrained all parameters to be equal across groups and used the χ² difference test to compare nested models [START_REF] Muthèn | Mplus users guide[END_REF]. The model fit was assessed with the same indices discussed previously in relation to longitudinal invariance.

Maximum likelihood estimation under missing at random assumption was specified in order to optimally take into account the available data [START_REF] Muthèn | Mplus users guide[END_REF].

RESULTS

Means and standard deviations for the major variables are presented in Table 2.

Longitudinal invariance

For self reports, the χ² difference test did not support the viability of the full longitudinal invariance hypothesis. Inspection of the modification indices suggested that one item, and its corresponding threshold, was not invariant across time points. In the partially constrained model, beyond the significant χ², all the fit indices satisfied the recommended criteria and the difference in χ² test supported partial longitudinal invariance,  2 (n = 439 ; df = 76) = 102.222, p = .024; CFI = .991; RMSEA = .031,  2 (df = 22) = 31.166, p = .093. According to Steenkamp and Baumgartner [START_REF] Steenkamp | Assessing measurement invariance in cross-national consumer research[END_REF], the comparison of factor means (computed with all the PA items) across time points may be considered meaningful because at least one item besides the marker item had invariant intercepts in the PA dimension. In our case, longitudinal invariance of the PA dimension for self report supported the invariance of 5 out of 6 items across the four examined time lags; thus, we kept the one item that was not invariant in the composite score of PA.

For mother reports, χ² difference test supported full longitudinal invariance,  2 (n = 294; df = 30) = 26.619, p = .485; CFI = 1.00; RMSEA = 0.00,  2 (df = 15) = 12.123, p = .670 Insert Table 2 here

Trajectory Models for PA.

For self reports, the 4-group model comprised of two stable trajectory groups and two linear trajectory groups was the ideal solution. The average group assignment probabilities were between .80 and .87. For mothers' reports, the 4-group model was the ideal solution, but the smallest group had only 7 individuals (2.5% of sample); thus, the 3 group model with two linear and one stable trajectory was selected based on parsimony [START_REF] Nagin | Group-based modeling of development[END_REF]. The average group assignment probabilities were between .84 and .89 (see Figure 1 for models).

Table 3 presents the results of the multivariate logit regression examining the capacity of sex to distinguish membership in the trajectories membership. For self reports, relative to the low/moderate declining group, children either in the moderate declining group or in the high stable group were more likely to be boys. For mothers' reports, relative to the moderate declining group, children in the low stable group were less likely to be boys, whereas children in the high stable group were more likely to be boys.

Inter-informant Agreement

Table 4 presents the cross-tabulation between trajectory membership based on self and mother reports, χ² (6) = 128.583, p <.001. Children assigned on the basis of self-ratings to the low stable group were significantly more likely than expected by chance to be assigned to the low declining group based on mothers' reports and less likely to be assigned to the mother-reported moderate declining and high stable groups. Children assigned on the basis of self-ratings to the low/moderate declining group were more likely than expected by chance to be assigned to the mother-reported low declining group and also less likely to be assigned to the mother-reported high stable group.

Children assigned to the self-reported moderate declining group were more likely to be assigned to the mother-reported moderate declining group and the low declining group. Finally, children assigned to the self-reported high stable group were less likely to be assigned to the mother-reported low declining group and more likely to be assigned to the high stable group.

Insert Figure 1 here Insert Table 3 here Insert Table 4 here For both self and mother reports, longitudinal correlations from age 11 to 14 (i.e., within and across time) attested to moderate stability, with lower correlations the longer the distance of time.

The zero-order correlations were all significant and ranged from .23 (across 4 years) to .49 (all but 2 of 16 across-reporter within-and across-time correlations were above .33, and the mean acrossreporter correlation was .37).

Path analyses

We computed a multi-group path analysis in which we included as potential predictors of longterm OAB and CAB the trajectory probabilities of belonging to the moderate declining group and to the high stable group. We create a "stacked" data file in which each case had two records, one for each informant. Due to the nonnormality of the OAB and CAB measures (the skewness varied from 2.25 to 3.12, the kurtosis varied from 5.11 to 9.90), the path model was estimated in MPlus using the multiple linear regression (MLR) estimator setting (maximum likelihood parameter estimates with standard errors that are robust to nonnormality and nonindependence of observations) [START_REF] Muthèn | Mplus users guide[END_REF].

The equivalence between the multiple groups was evaluated by constraining the estimates for the model's parameters to be the same for self and mother reports. Both the unconstrained and the constrained models represented a good model-data fit, χ 2 (4) = 1.116, p = .89, CFI = 1.00, RMSEA <.001 for the unconstrained model; χ 2 (15) = 5.917, p = .98, CFI = 1.00, RMSEA < .001, for the constrained model. Also, the χ² difference test, χ 2 (11) = .912, p = .99, supported the viability of an invariant association between the trajectory probabilities of belonging to both moderate declining and high stable developmental pathways with long-term OAB and CAB when assessed with self versus mother reports.

Figure 2 presents a diagrammatic representation of the constrained model and the path coefficients only for significant paths. In addition to the contribution either of sex to OAB at age 18-19 (being male was associated with higher OAB) or of earlier OAB to OAB at age 18-19, the trajectory probability for belonging to the high stable group, based on both self and mother reports, significantly predicted OAB and CAB in early adulthood. The probability of being in the moderate declining group, based on both self and mother reports, was positively related to only CAB at age 18-19. Finally, the significant path of the age covariate on both outcomes reflected higher values of both OAB and CAB at age 19 than at age 18.

DISCUSSION

Several investigators have argued that in order to better understand aggression, researchers need to investigate nonaggregated forms of aggressive behavior [START_REF] Tremblay | The development of aggressive behavior during childhood: what we learned in the past century?[END_REF] and use multiple informant strategies [START_REF] Achenbach | Child/adolescent behavioral and emotional problems: implications of cross-informant correlations for situational specificity[END_REF][START_REF] De Los | Informant Discrepancies in the Assessment of Childhood Psychopathology: A Critical Review, Theoretical Framework, and Recommendations for Further Study[END_REF]. Historically, investigators focusing on PA have not relied on multiple informants, whereas those who have used different informants typically have examined aggregates of aggression [START_REF] Caprara | Early emotional instability, prosocial behavior and aggression: some methodological aspects[END_REF][START_REF] De Los | Informant Discrepancies in the Assessment of Childhood Psychopathology: A Critical Review, Theoretical Framework, and Recommendations for Further Study[END_REF][START_REF] Pastorelli | Measuring emotional instability, prosocial behavior and aggression in pre/adolescents: a cross/national study[END_REF].

Taking into account the distinct role of PA as a risk factor for later violence [START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF], we investigated the specific construct of PA from late childhood to adolescence according to two privileged points of view: the self and mothers. As a prerequisite to modelling change over time with latent growth curve analysis, we established partial longitudinal invariance of self-reported PA from age 11 to 14 and full longitudinal invariance for mothers' reports (i.e., we established that the frame of reference of PA for each reporter was stable over time).

In addition, consistent with previous studies [START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF], four trajectory groups were identified for self reports of PA (high stable, moderate declining, low/ moderate declining, and low stable), whereas three trajectories were identified for mothers' reports (high stable, moderate declining, low declining). This is the first study of PA trajectories in Italy and the similarities in the development of PA with other samples are notable.

The fact that more developmental trends were identified by self reports than by mothers' reports can be attributed to a greater ability of children, compared to mothers, either to be aware of their own behavior or to better discriminate the severity of their own behavior. Furthermore, this difference may be connected to the fact that in community samples, children tend to report more problematic behaviors than do parents [START_REF] Waaktar | Youth-parent consistencies on ratings of difficulties and prosocial behavior: Exploration of an at-risk sample[END_REF].

Insert Figure 2 here Similar to Card et al. [START_REF] Card | Direct and Indirect Aggression During Chidlhood and Adolescence: A Meta-Analytic Review of Gendere Differences, Intercorrelations, and relations to Maladjustment[END_REF], according to both informants, girls were more likely to be assigned to the trajectory groups characterized by the lowest levels of PA over time, whereas boys were more likely to be assigned to groups displaying higher levels of PA. Moreover, in accordance with others' findings [START_REF] De Los | Informant Discrepancies in the Assessment of Childhood Psychopathology: A Critical Review, Theoretical Framework, and Recommendations for Further Study[END_REF], we found a moderate level of agreement between raters on PA. Importantly, these findings support the validity of children's self-reported PA in childhood and early adolescence.

Consistent with other previous studies in which aggression was disaggregated [START_REF] Brame | Developmental trajectories of aggression[END_REF][START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF], those who belonged to the high stable trajectory based on either self-or mothers' reports were more likely to be at risk for both OAB and CAB later in development. Moreover, as discussed by different authors [START_REF] Dishion | The development and ecology of antisocial behavior in children and adolescents[END_REF], our results are consistent with findings that OAB and CAB may have a diametrically opposite development cycle. In fact, those who belonged to the moderate declining physically aggressive group, as assessed by both self and mother reports, tended not to be significantly related to OAB but were significantly related to CAB.

Finally, consistent with others who have not found the so-called "late onset" group [START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF], we did not find evidence of a group of preadolescents who increased their level of aggression over time in our identified trajectories based on both informants. This is in contrast with Martino et al. [START_REF] Martino | Multiple Trajectories of Physical Aggression among Adolescent Boys and Girls[END_REF] who examined self-reported PA trajectories and found an increasing trend. This may be due to the fact that their trajectories did not end at 14 years old as did ours and those as in other studies [START_REF] Broidy | Developmental trajectories of childhood disruptive behaviors and adolescent delinquency: A six site, cross national study[END_REF][START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF].

Or it may be due to the fact that the construct of PA they referred to focused not only on physically aggressive behavior as in this study, but also on threats of physical aggression. However, we did find some support for the existence of young adults who behave antisocially even though they did not belong to an increasing pathway of PA during adolescence. In fact, we found a significant association between the moderate decreasing PA pathway (based on both informants) and CAB at age 18-19. This result is in agreement with Brame et al. [START_REF] Brame | Developmental trajectories of aggression[END_REF] who did not examine the trajectories of PA with self-reports and did not find an increasing trajectory for PA, but did find a significant association between belonging to a desisting PA pathway from 6 to 13 years old and belonging to an increasing delinquency pathway from 13 to 17 years old. Moreover, our results may indirectly support the declining developmental pathway of OAB and the increasing developmental pathway of CAB from adolescence until adulthood [START_REF] Dishion | The development and ecology of antisocial behavior in children and adolescents[END_REF].

The use of self-reports in our study can be viewed as a limitation if we consider the fact that self-reports can be biased due to imperfect memory and to socially desirable responding [START_REF] Martino | Multiple Trajectories of Physical Aggression among Adolescent Boys and Girls[END_REF].

However, mother/teacher-reported aggression can be similarly susceptible to such distortions and the continuity and the predictive validity of self-reported violent and delinquent offences have been supported [START_REF] Hoeve | Trajectories of delinquency and parenting styles[END_REF]. Thus, overall, the resemblance between the trajectories we identified based on selfreports and mothers' reports and their similar predictive validity in respect to antisocial behavior in early adulthood gives us confidence in the veracity of our participants' reports and, more generally, provide additional support for the validity for self-reports.

A limitation of this study is that, although we did look at sex differences in the identified models (accounting for sex as a covariate), we did not identify PA trajectories using separate samples for boys and girls as in some previous studies [START_REF] Harachi | Aggressive Behavior Among Girls and Boys During Middle Childhood: Predictors and Sequelae of Trajectory Group Membership[END_REF][START_REF] Nagin | Analyzing developmental trajectories: A semiparametric, group-based approach[END_REF]. This was due to our smaller sample size in comparison to other studies relying upon growth mixture modeling approaches. Moreover, caution should be used in generalizing results from the present study to other populations because participants in this study were Italians, predominantly middle-class, educated, and from two-parent families.

CONCLUSION

Most of what is known about children's PA is what children's teachers or mothers report on questionnaires. In our study, we clearly found that what children report, as well as mother report, is useful when studying the relation between disaggregated forms of aggressive behaviors and the risk for later OAB and CAB. This suggests that additional information is needed before drawing the conclusion that self-reported PA is not very reliable prior to adolescence [START_REF] De Los | Informant Discrepancies in the Assessment of Childhood Psychopathology: A Critical Review, Theoretical Framework, and Recommendations for Further Study[END_REF]. The findings suggest preadolescents' self-reported PA may be useful in identifying children at-risk for OAB and CAB who are candidates for intervention programs. Finally, in line with our findings, practitioners should focus their attention not only on those children who are chronically aggressive, but also on those who exhibit an initial moderate level of PA that decreases over time. The latter can be candidates for endorsing more CAB (rather than OAB).

FOOTNOTES 1 Previous psychometric studies on the PVA scale have not explored its multidimensionality but sustained its monolithic dimensionality based on exploratory factor analyses [START_REF] Caprara | Early emotional instability, prosocial behavior and aggression: some methodological aspects[END_REF][START_REF] Pastorelli | Measuring emotional instability, prosocial behavior and aggression in pre/adolescents: a cross/national study[END_REF]. In order to update these results, confirmatory factor analyses were conducted to examine the factor structure of the PVA scale for self and mother reports, accounting for the order categorical nature of its items. A two-factor model provided a better fit to the PVA scale for both self and mother reports: one factor for physical aggression and another one for verbal aggression. Details on these analyses are available from the first author upon request. Only PA was used in this study.

Figure 1 .Figure 2 .

 12 Figure 1. Physical Aggression Trajectories Based on Self and Mothers' Reports From Age 11 To 14

Table 2 .

 2 Descriptive Statistics of the Measures

		PHYSICAL AGGRESSION
		Self-Report	Mother Report
	AGE	N (% boys)	N (% boys)
	11	439 (55%)	219 (53%)
	12	397 (55%)	248 (52%)
	13	426 (55%)	291 (58%)
	14	399 (55%)	188 (62%)
		OVERT BEHAVIOR
		Children sample of Physical Aggression	Mother Sample of Physical Aggression
		N (% boys)	N (% boys)
	14	398 (55%)	277 (58%)
	18 -19	282 (54%)	208 (57%)
		COVERT BEHAVIOR
		Children sample of Physical Aggression	Mother Sample of Physical Aggression
		N (%boys)	N (%boys)
	14	400 (55%)	279 (58%)
	18-19	283 (54%)	209 (57%)

Table 3 .

 3 The Relation of Sex to Group Membership Probabilities, Separately for Self and Mother Reports Note. Males were coded as 1, females as 0. P-value <.01** <.05*

	SELF REPORT

Table 4 .

 4 Cross-Tabulations of Trajectories Based on Self and Mother Reports

	MOTHER-REPORTS

In each model, the initial level of the outcome referred to the unique contribution of the outcome at age 14 that was not shared with PA at age 14. Thus, we obtained residual variables by regressing PA at age 14 based on self report onto OAB and CAB at age 14 (explained variances were, respectively,

38% and 15%).

This study was partially supported by grants from the Spencer Foundation and W. T. Grant Foundation to Albert Bandura, from the Ministero dell'Istruzione dell'Università e della Ricerca (MIUR) to Gian Vittorio Caprara (COFIN 1998(COFIN , 2000)), and to Eugenia Scabini (COFIN 2000(COFIN -2002)), and from the National Institute of Mental Health to Nancy Eisenberg.