Jiatu Cai

Masaaki Fukasawa

Mathieu Rosenbaum

Peter Tankov

Optimal discretization of hedging strategies with directional views

Keywords: Discretization of hedging strategies, delta hedging, hitting times, asymptotic optimality, expectation-error criterion, semi-martingales, limit theorems, linearquadratic optimal control

We consider the hedging error of a derivative due to discrete trading in the presence of a drift in the dynamics of the underlying asset. We suppose that the trader wishes to find rebalancing times for the hedging portfolio which enable him to keep the discretization error small while taking advantage of market trends. Assuming that the portfolio is readjusted at high frequency, we introduce an asymptotic framework in order to derive optimal discretization strategies. More precisely, we formulate the optimization problem in terms of an asymptotic expectation-error criterion. In this setting, the optimal rebalancing times are given by the hitting times of two barriers whose values can be obtained by solving a linear-quadratic optimal control problem. In specific contexts such as in the Black-Scholes model, explicit expressions for the optimal rebalancing times can be derived.

Introduction

In order to manage the risks inherent to the derivatives they buy and sell, practitioners use continuous time stochastic models to compute their prices and hedging portfolios. In the simplest cases, notably in that of the so-called delta hedging strategy, the hedging portfolio obtained from the model is a time varying self financed combination of cash and 1 the underlying of the option. We denote the price at time t of the underlying asset by Y t and assume it to be a one-dimensional semi-martingale. Hence, in such situations, the outputs of the model are the price of the option together with the number of underlying assets to hold in the hedging portfolio at any time t, denoted by X t (the weight in cash is then deduced from the self financing property). Therefore, assuming zero interest rates, the theoretical value of the model based hedging portfolio at the maturity of the option T is given by T 0 X t dY t .

Typically, the process X t derived from the model is a continuously varying semi-martingale, requiring continuous trading to be implemented in practice. This is of course physically impossible and would be anyway irrelevant because of the costs induced by microstructure effects. Hence practitioners do not use the strategy X t , but rather a discretized version of it. This means the hedging portfolio is only rebalanced at some discrete times and thus is held constant between two rebalancing times. Let us denote by (τ n j ) j≥0 an increasing sequence of rebalancing times over [0, T] (the meaning of the parameter n will be explained below). With respect to the target portfolio obtained from the model, the hedging error due to discrete trading Z n T is therefore given by

Z n T = +∞ j=0 X τ n j (Y τ n j+1 ∧T -Y τ n j ∧T ) - T 0 X t dY t .
Thus, some important questions in practice are:

• What is the order of magnitude of Z n T in the case of classical discretization strategies ? • For a given criterion, how to optimize the rebalancing times ?

The most classical rebalancing scheme is that of equidistant trading dates of the form τ n j = jT/n, j = 0, . . . , n,

where n represents the total number of trades on the period [0, T]. In this setting, the first question has been addressed in details. There are two popular approaches to quantify the hedging error Z n T , both of them being asymptotic, assuming the rebalancing frequency n/T tends to infinity (that is n tends to infinity since T is fixed). A first possibility is to use the L 2 norm, where one typically looks for asymptotic bounds of the form

E[(Z n T ) 2 ] ≤ cn -θ , n → ∞.
Many authors have explored various aspects of this problem in this deterministic rebalancing dates framework. For European call and put options in the Black-Scholes model, it is shown in [START_REF] Bertsimas | When is time continuous[END_REF] and [START_REF] Zhang | Couverture approchée des options Européennes[END_REF] that the L 2 error has a convergence rate θ = 1. For other options, the convergence rate depends on the regularity of the payoff. For example, it is shown in [START_REF] Gobet | Discrete time hedging errors for options with irregular payoffs[END_REF] that for binary options, the convergence rate is θ = 1/2. However, in this context, the convergence rate θ = 1 can be achieved by choosing a suitable non equidistant deterministic rebalancing grid defined by τ n j = T -T 1j/n 1/β , with β ∈ (0, 1] being the fractional smoothness in the Malliavin sense of the option payoff, see [START_REF] Geiss | Quantitative approximation of certain stochastic integrals[END_REF]. An asymptotic lower bound for the L 2 error is given in [START_REF] Fukasawa | Asymptotically efficient discrete hedging[END_REF][START_REF] Fukasawa | Efficient discretization of stochastic integrals[END_REF] for a general class of rebalancing schemes.

The second way to assess the hedging error is through the weak convergence of the sequence of the suitably rescaled random variables Z n T . When X and Y are It ô processes, the case of equidistant rebalancing dates has been investigated in this approach in [START_REF] Bertsimas | When is time continuous[END_REF][START_REF] Hayashi | Evaluating hedging errors: an asymptotic approach[END_REF][START_REF] Rootzen | Limit distributions for the error in approximations of stochastic integrals[END_REF], where the following convergence in law is proved:

√ nZ n T L -→ T 2 T 0 σ X t σ Y t dB t , (1.1)
where σ X and σ Y are the volatilities of X and Y and B is a Brownian motion independent of the other quantities. The case where X and Y are processes with jumps is treated in [START_REF] Tankov | Asymptotic analysis of hedging errors in models with jumps[END_REF].

This asymptotic approach has also been recently used in the context where the rebalancing times are random stopping times. Some specific hitting times based schemes derived from a microstructure model are investigated in [START_REF] Robert | On the microstructural hedging error[END_REF]. In [START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF], the author works with quite general sampling schemes based on stopping times. More precisely, for a given parameter n driving the asymptotic, one considers an increasing sequence of stopping times 0 = τ n 0 ≤ τ n 1 ≤ . . . ≤ τ n j ≤ . . . so that almost surely, lim j→∞ τ n j = T (meaning in fact that the stopping times are all equal to T for large enough j) and sup j (τ n j+1 -τ n j )

tends to 0 in a suitable sense as n goes to infinity. Under some regularity conditions on the (random) rebalancing times, a general limit theorem for the hedging error is obtained in [START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF].

It is shown that after suitable renormalization (specified in the next sections), the hedging error converges in law to a random variable of the form

1 3 T 0 s t dY t + 1 √ 6 T 0 a 2 t - 2 3 s 2 t 1/2 σ Y t dB t . (1.2)
Here B is a Brownian motion independent of all the other quantities and the processes s and a can be interpreted as the asymptotic local conditional skewness and kurtosis of the increments of the process X between two consecutive discretization dates (see next sections for details).

One can remark a crucial difference between the deterministic discretization schemes associated to (1.1) and the random stopping times case leading to (1.2). For deterministic dates, the discretization error asymptotically behaves as a stochastic integral with respect to Brownian motion. Therefore, it is (essentially) centered. In the case of random discretization dates, one may obtain a "biased" asymptotic hedging error because of the presence of the term

T 0 s t dY t .
Hence, if s does not vanish and Y has non zero drift, the asymptotic hedging error is no longer centered.

From a practitioner viewpoint, this is quite an interesting property. Indeed, it shows that in the presence of market trends, the trader may actually be compensated for the extra risk arising from discrete trading, provided that the rebalancing dates are chosen in an appropriate way. Of course one may say this is not the option trader's job to try to get a positive expected return with the hedging strategy. However, knowing that there is anyhow a hedging error, it seems reasonable to optimize it to the trader's benefit.

Hence we place ourselves in the asymptotic high frequency regime where n is large and therefore sup

j (τ n j+1 -τ n j )
is small, meaning that the hedging error should be small. In this setting we address the second question raised above, that is finding the optimal times to rebalance the portfolio. To do so, we simply use an asymptotic expectation-error type criterion. More precisely, we wish to maximize the expectation of the hedging error under a constraint on its L 2 norm. This is quite in the spirit of [START_REF] Sepp | When you hedge discretely: Optimization of sharpe ratio for delta-hedging strategy under discrete hedging and transaction costs[END_REF], where the author aims at finding an optimal hedging frequency to optimize the Sharpe ratio. Remark that in our context, the L 2 norm is more meaningful than the variance since the primary goal of the trader is to make the hedging error small. Our asymptotic approach goes as follows. First, we approximate the law of the renormalized hedging error by that in Equation (1.2). Then we find the processes a * t and s * t which correspond to optimality in terms of our expectation-error criterion for the family of laws given by (1.2). Finally, we show that we can indeed build a discretization rule which leads to the optimal a * t and s * t in the limiting distribution of the hedging error.

Note that using an asymptotic framework to design optimal discretizations of hedging strategies has been a quite popular approach in the recent years. Such method (although in a slightly different context) is in particular used in [START_REF] Fukasawa | Asymptotically efficient discrete hedging[END_REF][START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF][START_REF] Gobet | Almost sure optimal hedging strategy[END_REF] in the continuous setting whereas the case with jumps is investigated in [START_REF] Rosenbaum | Asymptotically optimal discretization of hedging strategies with jumps[END_REF]. All these works aim at minimizing some form of transaction costs (typically the number of trades) under some constraint on the L 2 norm of the hedging error. Here we also put a constraint on the L 2 norm of the hedging error. However, instead of minimizing transaction costs, we maximize the expectation of the hedging error. Thus our viewpoint is that of a trader giving himself a lower bound on the quality of his hedge (the L 2 norm of the hedging error), but allowing himself to try to take advantage of market drifts provided the constraint is satisfied.

In practice, our work should probably only be considered as a benchmark. Indeed, we somehow make the highly unreasonable assumption that practitioners observe the drift. This is of course not realistic at all since any kind of statistical estimation of the drift is irrelevant in this high frequency setting. However, some practitioners still have views on the market and our work gives them a way to incorporate their beliefs in their hedging strategies.

The paper is organized as follows. In Section 2, we investigate the set of admissible dis-cretization rules, that is those leading to a limiting law of the form (1.2). In particular, we extend the examples provided in [START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF] by showing that the discretization rules based on hitting times of stochastic barriers are admissible. In Section 3, we consider a first criterion for optimizing the trading times: the modified Sharpe ratio. It enables us to carry out very simple computations. However, the relevance of the modified Sharpe ratio being in fact quite arguable, a more suitable approach in which we consider an expectation-error type criterion is investigated in Section 4. Using tools from linear-quadratic optimal control theory, explicit developments are provided in the Black-Scholes model in Section 5. Finally, the longest proofs are relegated to an appendix.

Assumptions and admissible strategies

In this section we detail our assumptions on the processes X and Y together with the admissibility conditions for the sampling schemes.

Assumptions on the dynamics and admissibility conditions

Let (Ω, F , F, P) be a filtered probability space. We write Y for the underlying asset. Let T > 0 stand for the maturity of the derivative to be hedged. We assume that the benchmark hedging strategy deduced from a theoretical model simply consists in holding a certain number of units of the underlying asset, denoted by X, and some cash in a self financed way, under zero interest rates. Throughout the paper, we assume both Y and X are It ô processes of the form

dY t = b Y t dt + σ Y t dW Y t , dX t = b X t dt + σ X t dW X t (2.1)
on [0, T], where W X and W Y are F-Brownian motions which may be arbitrarily correlated, and the coefficients of X and Y satisfy the following technical assumptions.

Assumption 2.1.

• The processes b Y , b X , σ Y and σ X are adapted and continuous on [0, T] almost surely.

• The volatility process σ Y of Y is positive on [0, T] almost surely.

• The volatility process σ X of X is positive on [0, T) almost surely.

• The instantaneous Sharpe ratio ρ = b Y /σ Y satisfies E T 0 ρ 2 t dt < +∞.
Example 2.1 (The Black-Scholes model). The case that b Y t = bY t and σ Y t = σY t with constants b and σ > 0 corresponds to the Black-Scholes model. The instantaneous Sharp ratio ρ = b/σ is a constant. To hedge a call option with payoff (Y T -K) + and strike K > 0, the standard theory suggests to use the so-called Delta hedging strategy:

X t = Φ(d 1 (t, Y t )), d 1 (t, y) = log(y/K) + σ 2 (T -t)/2 σ √ T -t ,
where Φ stands for the distribution function of a standard Gaussian random variable. By It ô's formula, we see that X is an It ô process of the form (2.1) with

W X = W Y and b X t = φ(d 1 (t, Y t )) ∂d 1 ∂t (t, Y t ) + σ 2 2 ∂ 2 d 1 ∂y 2 (t, Y t )Y 2 t + b ∂d 1 ∂y (t, Y t )Y t + σ 2 2 ∂d 1 ∂y (t, Y t ) 2 φ ′ (d 1 (t, Y t ))Y 2 t , σ X t = σφ(d 1 (t, Y t )) ∂d 1 ∂y (t, Y t )Y t ,
with φ the density of a standard Gaussian random variable. Almost surely, Y T K and therefore both b X and σ X are continuous on [0, T] and b X T = σ X T = 0. Furthermore σ X is positive on [0, T). Hence Assumption 2.1 is satisfied.

As explained in the introduction, in practice, the trader cannot realize the theoretical strategy X t which typically implies continuous trading. Hence the quantity T 0 X s dY s only represents a benchmark terminal wealth and X t is a benchmark hedging strategy. Thus, we consider that this strategy is discretized over the stopping times

0 = τ n 0 ≤ τ n 1 ≤ • • • ≤ τ n j ≤ • • •
, so that for given n, almost surely, τ n j attains T for j large enough. Such array of stopping times is called a discretization rule. Consequently, if we define the discretized process X n by

X n t = X τ n j , t ∈ [τ n j , τ n j+1
), the hedging error Z n T with respect to the benchmark strategy writes

Z n T = T 0 (X n s--X s )dY s .
We now define the admissibility conditions for our discretization rules which we comment in the next subsection.

Condition 2.1 (Admissibility conditions).

A discretization rule (τ n j ) is admissible if there exist continuous F-adapted processes a and s satisfying

E T 0 1 + (ρ t ) 2 (a 2 t + s 2 t )(σ Y t ) 2 dt < ∞, (2.2)
and a positive sequence ε n tending to zero such that:

• The first two moments of the renormalized hedging error ε -1 n Z n T converge to those of a random variable of the form

Z * a,s = 1 3 T 0 s t dY t + 1 √ 6 T 0 a 2 t - 2 3 s 2 t 1/2 σ Y t dB t , (2.3) that is, E[ε -1 n Z n T ] → E[Z * a,s ], E[(ε -1 n Z n T ) 2 ] → E[(Z * a,s ) 2 ], (2.4)
where B is a Brownian motion independent of all the other quantities.

• Almost surely, the processes a t and s t satisfy a 2 t ≥ s 2 t , for all t ∈ [0, T].

Comments on the admissibility conditions

Equation 2.2 is simply a technical integrability condition. We now give the interpretation of the sequence ε n . Recall that for fixed n, we deal with an increasing sequence of stopping times (τ n j ) over [0, T]. Typically, ε 2 n will represent the order of magnitude of the interarrival time τ n j+1 -τ n j . For example, in the case of equidistant trading times with frequency n/T, ε n can simply be taken equal to n -1/2 . In the case of the hitting times based scheme consisting in rebalancing the portfolio each time the process X has varied by ν n , where ν n is a deterministic sequence tending to zero, one can choose ε n = ν n (since the order of magnitude of the time interval between two hitting times is ν 2 n ).

The specific form (2.3) may appear rather ad hoc at first sight. However, it is in fact quite natural. Indeed, Proposition 2.1 below, which is proved in Appendix and used to show the main result of the next subsection, indicates that as soon as the quadratic covariations ε -2 n Z n and ε -1 n Z n , Y have regular limits, the form (2.3) appears for the weak limit of the renormalized hedging error. So the idea for this admissibility condition is that in our asymptotic approach, we want to work in regular cases where the renormalized hedging error can be approximated by a random variable of the form (2.3). However, our asymptotic optimality criterion will be based on the first two moments of the renormalized hedging error only. Therefore, we just require these first two moments to be asymptotically close to those of a random variable of the form (2.3) (in particular we do not impose the convergence in law of the renormalized hedging error towards Z * a,s , although this is the underlying idea behind this admissibility condition). We now give Proposition 2.1. Proposition 2.1. If there exist a sequence ε n → 0 and continuous processes s and a such that

ε -2 n Z n → 1 6 • 0 a 2 u (σ Y u ) 2 du, (2.5) ε -1 n Z n , Y → 1 3 • 0 s u (σ Y u ) 2 du, (2.6)
uniformly in probability on [0, T], then ε -1 n Z n converges in law to

1 3 • 0 s t dY t + 1 √ 6 • 0 a 2 t - 2 3 s 2 t 1/2 σ Y t dB t (2.7) in C[0, T]. In particular the convergence in law of ε -1 n Z n T to Z * a,s defined by (2.3) holds. If in addition, ε -4/3 n sup j≥0 (τ n j+1 ∧ T 0 -τ n j ∧ T 0 ) → 0 (2.8)
in probability, for all T 0 ∈ [0, T), then almost surely a 2 t ≥ s 2 t for all t ∈ [0, T]. We now consider the processes a 2 t and s t appearing in the admissibility conditions. We place ourselves in the situation where Proposition 2.1 can be applied. In that case, an inspection of the proof of this lemma shows that the inequality a 2 t ≥ s 2 t essentially follows from the elementary fact that 2 for a general random variable ∆. Indeed, a 2 t and s t are respectively related to the local third and fourth conditional moments of the increments of X. Proposition 2.2 below, which is proved in Appendix and used to show the main result in the next subsection, somehow illustrates the connections between a 2 t and s t and the conditional moments. Thus we give it now. Let ∆ j,n = X τ n j+1 -X τ n j be the increment of X between τ n j and τ n j+1 and N n t be the number of rebalancing times until time t:

E[∆ 2 ]E[∆ 4 ] ≥ E[∆ 3 ]
N n t = max j ≥ 0|τ n j ≤ t .
The following proposition holds.

Proposition 2.2. Let ε n be a positive sequence tending to 0 and s and a be continuous processes.

Assume the following:

• The family of random variables

ε -4 n sup t∈[0,T] |X n t -X t | 4 (2.9)
is uniformly integrable.

• The following uniform convergences in probability on [0, T 0 ] hold for all T 0 ∈ [0, T):

ε -1 n N n • j=0 κ τ n j E ∆ 3 j,n F τ n j → - • 0 s u (σ Y u ) 2 du, ε -2 n N n • j=0 κ τ n j E ∆ 4 j,n F τ n j → • 0 a 2 u (σ Y u ) 2 du, (2.10)
where

κ u = (σ Y u /σ X u ) 2 .
Then the convergences (2.5), (2.6) and (2.8) hold.

Proposition 2.2 is useful to obtain the convergences (2.5) and (2.6) for a given discretization rule since it is usually easy to have approximate values of the conditional moments of the increments. We actually apply this approach in the proof of the main result of the next subsection.

Examples of admissible discretization rules

We show in this section that the most common discretization rules are admissible. We start with hitting times based schemes. We have the following result.

Proposition 2.3 (Hitting times based discretization rule). Let ε n be a positive sequence tending to zero and l and l be two adapted processes which are positive and continuous on [0, T] almost surely with

E T 0 1 + (ρ t ) 2 (l t ∨ l t ) 2 (σ Y t ) 2 dt < ∞. (2.

11)

The discretization rule based on the hitting times of ε n l t or ε n l t by the process X:

τ n j+1 = inf t > τ n j : X t (X τ n j -ε n l t , X τ n j + ε n l t ) ∧ T (2.12)
is admissible. Moreover, we can take

s t = l t -l t , a 2 t = (s t ) 2 + l t l t (2.13)
and we also have the convergence (2.7) and therefore the convergence in law of Z n T towards Z * a,s defined by (2.3).

It is interesting to note here that the limit Z * a,s does not depend on the structure of X.

This result is particularly important since many traders monitor the values of the increments of their so-called delta (which corresponds to the process X) in order to decide when to rebalance their portfolio. Thus they are indeed using hitting times based strategies. This proposition notably extends the weak convergence results in [START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF] since it shows that not only constant barriers (between τ n j and τ n j+1 ) but also time varying stochastic barriers can be considered. This will be very useful in the next sections since our optimal discretization rules will correspond to hitting times of such barriers. Furthermore, in the proofs, the assumption that the time varying barriers satisfy (2.11) will enable us to deduce quite easily some relevant integrability properties for the hedging error (which would be harder to obtain with locally constant barriers). Now remark that under the condition a 2 t > s 2 t , we can always find some positive processes l t and l t such that (2.13) is satisfied. Indeed, it is easy to see that the real numbers l t and -l t can be taken as the roots of the quadratic equation Consequently, if one has some processes a t and s t as targets, Lemma 2.1 implies that a strategy giving rise to these processes in the limiting distribution (2.3) can be found. We will work in this framework in Section 4. Remark that there are infinitely many strategies for which the hedging error converge in law to some Z * a,s with the same a and s as limiting processes. The hitting time strategy is an efficient one among them, in the sense that it requires the least number of rebalancing in an asymptotic sense, see [START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF] for the detail.

x 2 + s t x + s 2 t -a 2 t = 0.
Another classical discretization rule is given by equidistant trading times. Here, the integrability property (2.4) in the admissibility conditions does not hold in full generality. Compared to the hitting times setting, this is because the deviations of the benchmark strategy are not explicitly controlled by the barriers. Nevertheless, the following example describes a reasonable framework under which such discretization rule is admissible.

Proposition 2.4 (Equidistant sampling discretization rule). Consider the hedging strategy of a European option with payoff h(Y T ) and replace Assumption 2.1 by that the underlying Y t follows a diffusion process of the form

dY t = b(t, Y t )Y t dt + σ(t, Y t )Y t dW t ,
with b, σ and h some deterministic functions satisfying the regularity assumptions p.21-23 in [START_REF] Zhang | Couverture approchée des options Européennes[END_REF] (allowing in particular for call and put in the Black-Scholes model). Define the delta hedging portfolio:

X t = ∂P ∂y (t, Y t ), with P(t, y) = E Q (t,y) [h(Y T )],
where E Q denotes the expectation operator under the risk neutral probability. Let ε n be a positive sequence tending to zero. Then the equidistant trading times discretization rule:

τ n j = jε 2 n , j = 0, . . . , n, . . .
is admissible (under the original measure). Moreover, we can take

s t = 0, a 2 t = 3(σ X t ) 2 .
The proof of Proposition 2.4 follows easily from previous works. We can first obtain the convergence in law towards Z * a,s using for example the results of [START_REF] Hayashi | Evaluating hedging errors: an asymptotic approach[END_REF]. Indeed, up to localization, we can assume that σ X , σ Y , b X and b Y are bounded. Then the integrability conditions in the mentioned reference are obviously satisfied and the convergence follows. For (2.4), it suffices to use Theorem 2.4.1 in [START_REF] Zhang | Couverture approchée des options Européennes[END_REF] where the convergence of the L 2 norm of the normalized error under the original measure is provided.

Finally, note that the discretization rule based on equidistant trading times will not be of interest for us since the associated s t process vanishes and so the expectation of the limiting variable is zero.

Asymptotic optimality: a preliminary approach

Our viewpoint is that the trader's priority is to get a small hedging error. However, once this error is suitably controlled, he may try to take advantage of the directional views he has on the market. Hence, adopting the asymptotic approximation under which the first two moments of the renormalized hedging error are given by those of Z * a,s , we aim at maximizing E[Z * a,s ] while keeping E[(Z * a,s ) 2 ] reasonably small. This very problem is treated in Section 4.

Here, as a first step, we consider the approximation for

E[(Z * a,s ) 2 ] given by E[(Z * ,c a,s ) 2 ]
, where Z * ,c a,s denotes the sum of the two integrals with respect to the Brownian motions W Y and B in the definition of Z * a,s in Equation (2.3), that is

Z * ,c a,s = 1 3 T 0 s t σ Y t dW Y t + 1 √ 6 T 0 a 2 t - 2 3 s 2 t 1/2 σ Y t dB t .
To do so, we place ourselves in this section under the additional admissibility condition that the renormalized hedging error weakly converges in the sense of (2.7) and we take s t and a 2 t as the processes in the limit (2.7) (so s t and a 2 t are uniquely defined). Replacing E[(Z * a,s ) 2 ] by E[(Z * ,c a,s ) 2 ] is technically very convenient but in practice quite arguable since this approximation is meaningful only when the drift is small. However, our aim here is only to have a first rough idea about the form of the optimal discretization rules. Since we wish to get the moment of order one large while that of order two remains controlled, we consider that we want to maximize the so-called modified Sharpe ratio S defined by

S = S(a, s) = E[Z * a,s ] E[(Z * ,c a,s ) 2 ]
.

This ratio is said to be modified since we use E[(Z * ,c a,s ) 2 ] instead of the variance of Z * a,s .

Hence we are looking for strategies which maximize S. To do so, we now introduce the notion of nearly efficient (modified) Sharpe ratio.

Definition 3.1 (Nearly efficient Sharpe ratio). The value S * ∈ R is said to be a nearly efficient Sharpe ratio if:

• For any admissible discretization rule with associated limiting processes a and s, the associated modified Sharpe ratio S(a, s) satisfies S(a, s) ≤ S * .

• For any η > 0, there exists a discretization rule with associated limiting processes a and s such that S(a, s) ≥ S * -η.

We only consider nearly efficient ratios since our strategies will not enable us to attain exact efficiency (which would corresponds to η = 0 in the previous definition). Of course the slight difference between efficient and nearly efficient ratios has no importance in practice.

In our setting, for any limiting variable Z * a,s , we have

S(a, s) = E 1 3 T 0 s t b Y t dt E 1 9 T 0 s 2 t (σ Y t ) 2 dt + 1 6 T 0 a 2 t -2 3 s 2 t (σ Y t ) 2 dt 1/2 .

Now, the admissibility condition

a 2 t ≥ s 2 t implies S(a, s) ≤ √ 6 3 E T 0 s t b Y t dt E T 0 s 2 t (σ Y t ) 2 dt 1/2
and Cauchy-Schwarz inequality gives

S(a, s) ≤ √ 6 3 E T 0 b Y t σ Y t 2 dt 1/2
.

This provides an upper bound for the modified Sharpe ratio. We now wish to find a discretization rule enabling to (almost) attain this upper bound. To achieve this, our rule must be so that for the associated processes a t and s t , the inequalities used above (a • the product l t l t should be negligible compared to (l tl t ) 2 ,

• the term (l tl t )b Y t should be essentially positive. From these remarks together with Proposition 2.3, we easily deduce the following theorem.

Theorem 3.1. Suppose that for all t ≤ T, b Y t 0. Then the value √ 6 3 E T 0 b Y t σ Y t 2 dt 1/2
is a nearly efficient Sharpe ratio. It is approximately attained by the discretization rule defined for λ > 0 by

τ n,λ j+1 = inf t > τ n,λ j ; X t -X τ n,λ j = - b Y t (σ Y t ) 2 e λ ε n or b Y t (σ Y t ) 2 e -λ ε n , τ n 0 = 0. (3.1)
Indeed,

lim λ→+∞ S(λ) = √ 6 3 E T 0 b Y t σ Y t 2 dt 1/2
, where S(λ) denotes the modified Sharpe ratio obtained for the law of the variable Z * a,s associated to the discretization rule (3.1) with parameter λ.

This result provides simple and explicit strategies for optimizing the modified Sharpe ratio. It is also very easy to interpret. Indeed, we see that in order to take advantage of the drift, one needs to consider asymmetric barriers. The limitation is that we do not really control accurately the magnitude of the hedging error at maturity. The asymptotic setting simply means that we require λ to be quite large while e λ ε n is small. When using such discretization rule in practice, it is reasonable to consider that the trader fixes a maximal value for the asymmetry between the barriers controlled by λ. This way he can choose the parameter λ. Then ε n is set to match the bound on E[(Z * ,c a,s ) 2 ] that the trader does not want to exceed.

Asymptotic expectation-error optimization

In this section, we now consider a natural expectation-error type criterion in order to optimize our discretization rules. To do so, we work in an asymptotic setting where we are looking for discretization rules which are optimal in the expectation-error sense for their associated limiting random variable Z * a,s . Before giving our main result, we introduce some definitions inspired by classical portfolio theory. A couple (m, v) ∈ (R + ) 2 is said to be nearly efficient if it is non dominated and for any η > 0, there exists an admissible discretization rule such that its associated limiting random variable Z * a,s satisfies

E[Z * a,s ] = m, E[(Z * a,s ) 2 ] ≤ v + η.
It is efficient if we can take η = 0.

We introduce the set Z T of random variables of the form

Z T,s = 1 3 T 0 s t dY t + 1 3 √ 2 T 0 s t σ Y t dB t , (4.1)
where B is a Brownian motion independent of F and s t is an adapted continuous process such that

E T 0 1 + (ρ t ) 2 s 2 t (σ Y t ) 2 dt < ∞. (4.2)
We also define the notions of non dominated and efficient couples with respect to Z T . The definitions are the same as Definition 4.1 and Definition 4.2 except that we replace "admissible discretization rule" by "process s satisfying (4.2)" and "its associated limiting random variable Z * a,s " by Z T,s .

We can now state our main result which enables us to compute efficient discretization rules.

Theorem 4.1. The following results hold:

• The non domination domain coincides with the non domination domain with respect to Z T .

• Let (m * , v * ) be an efficient couple with respect to Z T , with associated optimal process s * . Then (m * , v * ) is a nearly efficient couple. More precisely, let δ > 0 and (l δ t , l δ t ) be defined by

l δ t -l δ t = s * t , (l δ t ) 2 -l δ t l δ t + (l δ t ) 2 = (s * t ) 2 + 6δ (σ Y t ) 2
, that is

l δ t = (s * t ) 2 4 + 6δ (σ Y t ) 2 + s * t 2 , l δ t = (s * t ) 2 4 + 6δ (σ Y t ) 2 - s * t 2 . ( 4

.3)

Then the hitting times based discretization rule specified through the barriers

(l δ t , l δ t ) satisfies E[Z * a,s ] = m * , E[(Z * a,s ) 2 ] = v * + δT.
We have therefore reduced the impulse control problem of finding the optimal rebalancing times to a classical expectation-error optimization with continuous dynamics. The solutions of this problem can be obtained by solving for µ > 0 inf

(s t ) -E[Z T,s ] + µE[(Z T,s ) 2 ] ,
for which we can apply the theory of linear-quadratic optimal control, see for example [START_REF] Lim | Mean-variance portfolio selection with random parameters in a complete market[END_REF][START_REF] Zhou | Continuous-time mean-variance portfolio selection: A stochastic LQ framework[END_REF]. As shown in the next section, we can even obtain closed formulas in the case where the underlying has deterministic drift and volatility. Note that again, our barriers strategies enable us to attain only nearly efficient couples. Indeed, reaching efficient couples would lead to the use of degenerate barriers with δ = 0. This does not make sense in practice, however δ can of course be selected small.

In practice, once he has chosen the target nearly efficient couple he wants to reach, the trader needs to select δ and ε n . Two ideas enabling to avoid microstructure effects seem natural and easy to implement:

• Fix a minimal time between two rebalancings t min . After a rebalancing at a random time say τ, wait t min and then apply the strategy with δ = 0 (that is rebalance immediately

if at t = τ + t min , X t -X τ is not inside the interval (-ε n l 0 t , ε n l 0 t
) and wait for the exit time otherwise). The parameter ε n can be chosen according to the average number of transactions the trader is willing to make.

• Fix (roughly) a minimal distance for the closest barrier after a rebalancing. Then compute δ and ε n according to the general level of volatility σ Y t so that they (approximately) lead to this bound and the average number of transactions the trader is willing to make.

One explicit example : Black-Scholes model with time varying coefficients

In this section, we explain how our method can be applied in practice through the simple example of the Black-Scholes model with time varying coefficients. So we assume the underlying follows the dynamics

dY t = Y t (b t dt + σ t dW t ),
where b t and σ t are continuous deterministic functions. We also assume b t and σ t do not vanish. Using the theory of linear-quadratic optimal control, we give an explicit solution for the problem of designing optimal rebalancing times in this specific setting.

Explicit formulas

We aim at finding the efficient couples for the controlled random variables of the form Z T,s as in (4.1). Following [START_REF] Zhou | Continuous-time mean-variance portfolio selection: A stochastic LQ framework[END_REF], such problem is classically recast as follows: solving for any µ > 0 the optimization problem inf

(s t ,0≤t≤T) -E[Z T,s ] + µE[(Z T,s ) 2 ] = inf (s t ,0≤t≤T) µE Z T,
P t = µ L 1 2 exp T t ρ 2 s ds + 1 2 ,
with L is the inverse function of x → xe x . Moreover, the optimal process Z * satisfies

d Z * t Z * t = - Ṗt P t (dt + 1 ρ t dW t ), Z * 0 = - 1 2µ
.

Therefore, we obtain

E[ Z * T ] = - 1 2µ P 0 P T .
Using Theorem B.1, we get

E ( Z * T ) 2 + 1 2 T 0 (s * t Y t σt ) 2 dt = 1 2µ 2 P 0 P T .
Consequently, we have that the optimal variable Z T,s * satisfies

E[Z T,s * ] = 1 2µ 1 - P 0 P T and E Z T,s * - 1 2µ 2 = 1 2µ 2 P 0 P T . Hence E[(Z T,s * ) 2 ] = 1 2µ 2 (1 - P 0 P T ).
We have thus proved the following proposition. with m > 0 (remark that the ratio P T P T -P 0 does not depend on µ). Furthermore, the associated process s * t enabling to compute optimal rules according to Theorem 4.1 is explicitly given by

1 3 s * t Y t = - 1 b t Ṗt P t Z * t , with d Z * t Z * t = - 1 b t Ṗt P t dY t Y t , Z * 0 = - 1 2µ .
Note that in practice, Z * is not observable. However, it can of course be approximated by a process Z( * ) thanks to historical data, using for example a scheme of the form Z( * )

t i+1 = Z( * ) t i 1 - 1 b t i Ṗt i P t i Y t i+1 -Y t i Y t i , Z( * ) 0 = - 1 2µ ,
where the t i are the observation times of market data.

Appendix A Proofs

In the following C denotes a constant which may vary from line to line. Note that we use several localization procedures in the proofs. We often give them in details since some of them are slightly unusual, in particular because of the fact that σ X may vanish at maturity.

A.1 Proof of Proposition 2.1

We start by proving in a very standard way the stable convergence of ε -1 n Z n in C[0, T], which is stronger than the weak convergence. More precisely, we show that for any bounded continuous function f on C[0, T] and bounded random variable U defined on (Ω, F , F, P),

lim n→∞ E[U f (ε -1 n Z n • )] = E[U f (Z * • )],
where Z * is defined by

Z * t = 1 3 t 0 s u dY u + 1 √ 6 t 0 a 2 u - 2 3 s 2 u 1/2 σ Y u dB u ,
on an extension of (Ω, F , F, P) on which B is a Brownian motion independent of all the other quantities. For K > 0, we set

α K = inf{t > 0; |ρ t | ≥ K} ∧ T.
Since ρ is continuous on [0, T] almost surely,

lim K→∞ P[α K < T] = 0. Now remark that |E[U f (ε -1 n Z n • )] -E[U f (Z * • )]| ≤ |E[U f (ε -1 n Z n • )] -E[U f (ε -1 n Z n •∧α K )]| + |E[U f (ε -1 n Z n •∧α K )] -E[U f (Z * •∧α K )]| + |E[U f (Z * •∧α K )] -E[U f (Z * • )]| ≤ 4 f ∞ U ∞ P[α K < T] + |E[U f (ε -1 n Z n •∧α K )] -E[U f (Z * •∧α K )]|.
Consequently, it suffices to show that for any K > 0,

lim n→∞ |E[U f (ε -1 n Z n •∧α K ) -E[U f (Z * •∧α K )]| = 0. Let E = exp - α K 0 ρ t dW Y t - 1 2 α K 0 ρ 2 t dt . Since E[E] = 1, the measure Q defined by dQ dP = E
is a probability measure under which Z n •∧α K is a local martingale. Under Q, the uniform convergences in probability (2.5) and (2.6) on [0, T] remain true. Therefore by Theorem IX.7.3 of [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], we have the stable convergence of ε

-1 n Z n •∧α K to Z * •∧α K under Q. Note that Ũ = U/E is a Q-integrable
positive random variable and moreover, for all A > 0,

|E[U f (ε -1 n Z n •∧α K )] -E[U f (Z * •∧α K )]| ≤|E Q [ Ũ f (ε -1 n Z n •∧α K )] -E Q [ Ũ f (Z * •∧α K )]| ≤|E Q [( Ũ ∧ A) f (ε -1 n Z n •∧α K )] -E Q [( Ũ ∧ A) f (Z * •∧α K )]| + 2 f ∞ E Q [ Ũ½ { Ũ≥A} ].
The second term tends to 0 uniformly in n as A → ∞. The first term converges to 0 due to the stable convergence under Q since ( Ũ ∧ A) is a bounded random variable. Now, we prove a 2 ≥ s 2 under the additional condition (2.8). Since a and s are continuous, it suffices to show a 2 t ≥ s 2 t for all t ∈ [0, T). Fix T 0 < T and let

αK = inf{u > 0; |b X u | ∨ σ X u ∨ σ Y u ≥ K or σ X u ≤ 1/K} ∧ T 0 (A.1)
for K > 0. Since σ X is positive and continuous on [0, T 0 ], we have

lim K→∞ P[ αK < T 0 ] = 0. (A.2)
Therefore, it suffices to show

a 2 u∧ αK ≥ s 2 u∧ αK (A.3)
for all u ≥ 0 and K > 0. Fix K and define the probability measure Q by

d Q dP = exp - αK 0 b X u σ X u dW X u - 1 2 αK 0 b X u σ X u 2 du .
Under Q, X •∧ αK is a martingale with bounded quadratic variation. Since Q is equivalent to P, it suffices to show (A.3) under Q.

By (2.8), there exists a subsequence {n(k)} such that

Q ε -4/3 n(k) sup j≥0 (τ n(k) j+1 ∧ T 0 -τ n(k) j ∧ T 0 ) > 1 k < 1 k . Let T k = inf u > 0, ε -4/3 n(k) sup j≥0 (τ n(k) j+1 ∧ u -τ n(k) j ∧ u) > 1 k ∧ αK . Then lim k→∞ Q[T k < αK ] = 0
and so,

ε -1 n(k) Z n(k) , Y t∧T k → 1 3 t∧ αK 0 s u (σ Y u ) 2 du, ε -2 n(k) Z n(k) t∧T k → 1 6 t∧ αK 0 a 2 u (σ Y u ) 2 du, (A.4)
in probability as k → ∞ for all t ≥ 0. Let

τk j = τ n(k) j ∧ T k
for j ≥ 0. We now give three technical lemmas.

Lemma A.1. Let κ u = (σ Y u /σ X u ) 2 .
We have

1 3 ε -1 n(k) N n(k) t∧T k j=0 κ τk j ∧t E Q (X τk j+1 ∧t -X τk j ∧t ) 3 F τk j ∧t -ε -1 n(k) Z n(k) , Y t∧T k → 0,
in probability as k → ∞ for all t ≥ 0.

Proof. By It ô's formula,

1 3 E Q (X τk j+1 ∧t -X τk j ∧t ) 3 F τk j ∧t = E Q τk j+1 ∧t τk j ∧t (X u -X τk j )d X u F τk j ∧t .
We now show that

ε -1 n(k) N n(k) t∧T k j=0 κ τk j ∧t E Q τk j+1 ∧t τk j ∧t (X u -X τk j )d X u F τk j ∧t -ε -1 n(k) N n(k) t∧T k j=0 κ τk j ∧t τk j+1 ∧t τk j ∧t (X u -X τk j )d X u → 0 (A.5)
and

ε -1 n(k) N n(k) t∧T k j=0 κ τk j ∧t τk j+1 ∧t τk j ∧t (X u -X τk j )d X u -ε -1 n(k) Z n(k) , Y t∧T k → 0, (A.6)
in probability.

By Lenglart inequality for discrete martingales (see e.g., Lemma A.2 of [START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF]), a sufficient condition for (A.5) is the fact that

ε -2 n(k) N n(k) t∧T k j=0 κ 2 τk j ∧t E Q τk j+1 ∧t τk j ∧t (X u -X τk j )d X u 2 F τk j ∧t → 0, (A.7)
in probability. To get this convergence, first use successively H ölder inequality, It ô's formula and Burkholder-Davis-Gundy inequality to obtain that

N n(k) t∧T k j=0 κ 2 τk j ∧t E Q τk j+1 ∧t τk j ∧t (X u -X τk j )d X u 2 F τk j ∧t ≤ C N n(k) t∧T k j=0 κ 2 τk j ∧t E Q X τk j+1 ∧t -X τk j ∧t 3/2 τk j+1 ∧t τk j ∧t (X u -X τk j ) 4 d X u 1/2 F τk j ∧t ≤ C N n(k) t∧T k j=0 E Q X τk j+1 ∧t -X τk j ∧t 3 F τk j ∧t 1/2 N n(k) t∧T k j=0 E Q τk j+1 ∧t τk j ∧t (X u -X τk j ) 4 d X u F τk j ∧t 1/2 = C N n(k) t∧T k j=0 E Q X τk j+1 ∧t -X τk j ∧t 3 F τk j ∧t 1/2 N n(k) t∧T k j=0 E Q (X τk j+1 ∧t -X τk j ∧t ) 6 |F τk j ∧t 1/2 ≤ C N n(k) t∧T k j=0 E Q X τk j+1 ∧t -X τk j ∧t 3 F τk j ∧t ≤ C N n(k) t∧T k j=0 E Q |τ k j+1 ∧ t -τk j ∧ t| 3 F τk j ∧t . Note also that j ≤ N n(k) t∧T k = τ n(k) j ≤ t ∧ T k ∈ F τk j ∧t . Then (A.7) follows since ε -2 n(k) E Q ∞ j=0 |τ k j+1 ∧ t -τk j ∧ t| 3 ≤ ε -2 n(k) ε 8/3 n(k) k 2 E Q ∞ j=0 |τ k j+1 ∧ t -τk j ∧ t| ≤ ε 2/3 n(k) k 2 t → 0.
We now turn to (A.6). Note that

ε -1 n(k) N n(k) t∧T k j=0 κ τk j ∧t τk j+1 ∧t τk j ∧t (X u -X τk j )d X u = ε -1 n(k) ∞ j=0 κ τk j ∧t τk j+1 ∧t τk j ∧t (X u -X τk j )d X u and ε -1 n(k) Z n(k) , Y t∧T k = ε -1 n(k) ∞ j=0 τk j+1 ∧t τk j ∧t (X u -X τk j )κ u d X u .
Therefore, the absolute value of the left hand side of (A.6) is dominated by

ε -1 n(k) ∞ j=0 τk j+1 ∧t τk j ∧t |X u -X τk j | |κ u -κ τk j ∧t |d X u ≤ ε -2 n(k) ∞ j=0 τk j+1 ∧t τk j ∧t |X u -X τk j | 2 |κ u -κ τk j ∧t | 2 d X u 1/2 ∞ j=0 τk j+1 ∧t τk j ∧t d X u 1/2 ≤ sup u≥0 1 κ u∧T k sup u≥0, j≥0 κ τk j+1 ∧u -κ τk j ∧u ε -1 n(k) Z n(k) 1/2 t∧T k X 1/2 t∧T k
, which converges to 0 due to (A.4) and the uniform continuity of κ.

Lemma A.2. We have

1 6 ε -2 n(k) N n(k) t∧T k j=0 κ τk j ∧t E Q X τk j+1 ∧t -X τk j ∧t 4 F τk j ∧t -ε -2 n(k) Z n(k) t∧T k → 0, in probability as k → ∞, for all t ≥ 0.
Proof. The proof is very similar to the previous one. By It ô's formula,

1 6 E Q X τk j+1 ∧t -X τk j ∧t 4 F τk j ∧t = E Q τk j+1 ∧t τk j ∧t (X u -X τk j ) 2 d X u F τk j ∧t .
We now show that

ε -2 n(k) N n(k) t∧T k j=0 κ τk j ∧t E Q τk j+1 ∧t τk j ∧t (X u -X τk j ) 2 d X u F τk j ∧t -ε -2 n(k) N n(k) t∧T k j=0 κ τk j ∧t τk j+1 ∧t τk j ∧t (X u -X τk j ) 2 d X u → 0 (A.8) and ε -2 n(k) N n(k) t∧T k j=0 κ τk j ∧t τk j+1 ∧t τk j ∧t (X u -X τk j ) 2 d X u -ε -2 n(k) Z n(k) t∧T k → 0, (A.9)
in probability.

By Lenglart inequality for discrete martingales, a sufficient condition for (A.8) is

ε -4 n(k) N n(k) t∧T k j=0 κ 2 τk j ∧t E Q τk j+1 ∧t τk j ∧t (X u -X τk j ) 2 d X u 2 F τk j ∧t → 0, (A.10)
in probability. To get this convergence, first use successively H ölder inequality, It ô's formula and Burkholder-Davis-Gundy inequality to obtain that

N n(k) t∧T k j=0 κ 2 τk j ∧t E Q τk j+1 ∧t τk j ∧t (X u -X τk j ) 2 d X u 2 F τk j ∧t ≤ N n(k) t∧T k j=0 κ 2 τk j ∧t E Q X τk j+1 ∧t -X τk j ∧t 4/3 τk j+1 ∧t τk j ∧t (X u -X τk j ) 6 d X u 2/3 F τk j ∧t ≤ C N n(k) t∧T k j=0 E Q X τk j+1 ∧t -X τk j ∧t 4 F τk j ∧t 1/3 N n(k) t∧T k j=0 E Q τk j+1 ∧t τk j ∧t (X u -X τk j ) 6 d X u F τk j ∧t 2/3 = C N n(k) t∧T k j=0 E Q X τk j+1 ∧t -X τk j ∧t 4 F τk j ∧t 1/3 N n(k) t∧T k j=0 E Q (X τk j+1 ∧t -X τk j ∧t ) 8 |F τk j ∧t 2/3 ≤ C N n(k) t∧T k j=0 E Q X τk j+1 ∧t -X τk j ∧t 4 F τk j ∧t ≤ C N n(k) t∧T k j=0 E Q |τ k j+1 ∧ t -τk j ∧ t| 4 F τk j ∧t .
Then, observe that

ε -4 n(k) E Q ∞ j=0 |τ k j+1 ∧ t -τk j ∧ t| 4 ≤ t k 3 → 0,
which gives (A.8). The proof for (A.9) is obtained in the same way as that for (A.6).

We finally give the following almost straightforward result, which is easily deduced from simplified versions of the proofs of the previous lemma.

Lemma A.3. We have

N n(k) t∧T k j=0 κ τk j ∧t E Q X τk j+1 ∧t -X τk j ∧t 2 F τk j ∧t -Y t∧T k → 0,
in probability as k → ∞ for all t ≥ 0.

We are now ready to complete the proof of Proposition 2.1. From (A.4) and Lemmas A.1, A.2 and A.3, we have for all 0 ≤ v ≤ t the following convergences in probability as k → ∞:

N n(k) t∧T k j=N n(k) v∧T k +1 κ τk j ∧t E Q X τk j+1 ∧t -X τk j ∧t 2 F τk j ∧t → t∧ αK v∧ αK (σ Y y ) 2 du, ε -1 n(k) N n(k) t∧T k j=N n(k) v∧T k +1 κ τk j ∧t E Q X τk j+1 ∧t -X τk j ∧t 3 F τk j ∧t → t∧ αK v∧ αK s u (σ Y y ) 2 du, ε -2 n(k) N n(k) t∧T k j=N n(k) v∧T k +1 κ τk j ∧t E Q X τk j+1 ∧t -X τk j ∧t 4 F τk j ∧t → t∧ αK v∧ αK a 2 u (σ Y y ) 2 du. Since E Q (X τk j+1 ∧t -X τk j ∧t ) 3 F τk j ∧t 2 ≤ E Q (X τk j+1 ∧t -X τk j ∧t ) 2 F τk j ∧t E Q (X τk j+1 ∧t -X τk j ∧t ) 4 F τk j ∧t ,
we have

ε -1 n(k) N n(k) t∧T k j=N n(k) v∧T k +1 κ τk j E Q (X τk j+1 ∧t -X τk j ∧t ) 3 F τk j ∧t 2 ≤ ε -2 n(k) N n(k) t∧T k j=N n(k) v∧T k +1 κ τk j E Q (X τk j+1 ∧t -X τk j ∧t ) 4 F τk j ∧t N n(k) t∧T k j=N n(k) v∧T k +1 κ τk j E Q (X τk j+1 ∧t -X τk j ∧t ) 2 F τk j ∧t .
This implies that for all 0 ≤ v ≤ t, t∧ αK

v∧ αK s u (σ Y u ) 2 du 2 ≤ t∧ αK v∧ αK a 2 u (σ Y u ) 2 du t∧ αK v∧ αK (σ Y u ) 2 du.
Thus we obtain (A.3).

A.2 Proof of Proposition 2.2

In this proof, using a classical localization procedure together with Girsanov theorem, we can assume that b X = 0 and that σ X and σ Y are bounded on [0, T]. We start with two technical lemmas and their proof.

Lemma A.4. For any p ∈ [0, 2),

ε -p n sup j≥0 ( X τ n j+1 -X τ n j ) → 0, in probability.
Proof. Let K > 0 and

γ n K = inf{t > 0; ε -1 n |X t -X n t | ≥ K} ∧ T. (A.11)
Using the tightness of the family (2.9), we get lim

K→∞ sup n∈N P[γ n K < T] = 0. (A.12)
Therefore, it is enough to show that for any K > 0,

ε -p n sup j≥0 ( X τ n j+1 ∧γ n K -X τ n j ∧γ n K ) → 0, in probability. Take an integer m > 2/(2 -p). Since sup j≥0 ( X τ n j+1 ∧γ n K -X τ n j ∧γ n K ) ≤ ∞ j=0 ( X τ n j+1 ∧γ n K -X τ n j ∧γ n K ) m 1/m ,
the statement of the lemma follows from the fact that

E ∞ j=0 ( X τ n j+1 ∧γ n K -X τ n j ∧γ n K ) m ≤ CE ∞ j=0 sup t≥0 |X τ n j+1 ∧γ n K ∧t -X τ n j ∧γ n K ∧t | 2m ≤ Cε 2m-2 n E ∞ j=0 sup t≥0 |X τ n j+1 ∧γ n K ∧t -X τ n j ∧γ n K ∧t | 2 ≤ Cε 2m-2 n E ∞ j=0 ( X τ n j+1 ∧γ n K -X τ n j ∧γ n K ) ≤ Cε 2m-2 n .
Here we have used that E[ X T ] < ∞. The result follows using H ölder inequality.

Lemma A.5. For any p ∈ [0, 2) and T 0 ∈ [0, T),

ε -p n sup j≥0 (τ n j+1 ∧ T 0 -τ n j ∧ T 0 ) → 0,
in probability. In particular, the convergence in probability (2.8) holds for all T 0 ∈ [0, T).

Proof. Let T 0 ∈ [0, T), K > 0 and

γK = inf{t > 0; σ X t ≤ 1/K} ∧ T 0 . (A.13)
Using the continuity and the positivity of σ X on [0, T), we get

lim K→∞ P[ γK < T 0 ] = 0.
Therefore, it is enough to show that for any K > 0,

ε -p n sup j≥0 (τ n j+1 ∧ γK -τ n j ∧ γK ) → 0,
in probability. This follows from Lemma A.4 since

sup j≥0 (τ n j+1 ∧ γK -τ n j ∧ γK ) ≤ C sup j≥0 ( X τ n j+1 -X τ n j
).

We now give the end of the proof of Proposition 2.2. Define γ n K by (A.11). It suffices to show that for any K > 0

sup t≥0 ε -2 n Z n t∧γ n K - 1 6 t∧γ n K 0 a 2 u (σ Y u ) 2 du → 0, sup t≥0 ε -1 n Z n , Y t∧γ n K - 1 3 t∧γ n K 0 s u (σ Y u ) 2 du → 0, in probability. Since ε -1 n sup t≥0 |X t∧γ n K -X n t∧γ n K | ≤ K, (A.14) the families ε -2 n Z n •∧γ n K and ε -1 n Z n , Y •∧γ n K are equicontinuous.
So we just need to prove that for any t ∈ [0, T),

ε -2 n Z n t∧γ n K - 1 6 t∧γ n K 0 a 2 u (σ Y u ) 2 du → 0, ε -1 n Z n , Y t∧γ n K - 1 3 t∧γ n K 0 s u (σ Y u ) 2 du → 0, in probability. Let β n M = inf{u > 0; (1/σ X u ) ≥ M} ∧ t ∧ γ n K for M > 0. Since t < T, Lemma A.4 gives that lim M→∞ sup n∈N P[β n M < t ∧ γ n K ] = 0.
Therefore it is enough to show that for any M > 0,

ε -2 n Z n β n M - 1 6 β n M 0 a 2 u (σ Y u ) 2 du → 0, ε -1 n Z n , Y β n M - 1 3
β n M 0 s u (σ Y u ) 2 du → 0,
in probability. From the assumptions of Proposition 2.2, we have

ε -1 n N n β n M j=0 κ τ n j E ∆ 3 j,n F τ n j + β n M 0 s u (σ Y u ) 2 du → 0, ε -2 n N n β n M j=0 κ τ n j E ∆ 4 j,n F τ n j - β n M 0 a 2 u (σ Y u ) 2 du → 0, in probability. Moreover, by It ô's formula, 1 3 ε -1 n N n β n M j=0 κ τ n j E ∆ 3 j,n F τ n j = ε -1 n N n β n M j=0 κ τ n j E τ n j+1 τ n j (X u -X n τ n j )d X u F τ n j , 1 6 ε -2 n N n β n M j=0 κ τ n j E ∆ 4 j,n F τ n j = ε -2 n N n t∧β K j=0 κ τ n j E τ n j+1 τ n j (X u -X n τ n j ) 2 d X u F τ n j .
Now remark that the following convergences in probability hold:

sup t≥0 ε -1 n N n β n M j=0 κ τ n j τ n j+1 τ n j (X u -X n τ n j )d X u -ε -1 n N n β n M j=0 κ τ n j E τ n j+1 τ n j (X u -X n τ n j )d X u F τ n j → 0, sup t≥0 ε -2 n N n β n M j=0 κ τ n j τ n j+1 τ n j (X u -X n τ n j ) 2 d X u -ε -2 n N n β n M j=0 κ τ n j E τ n j+1 τ n j (X u -X n τ n j ) 2 d X u F τ n j → 0. (A.15)
Indeed, as seen in the proofs of Lemmas A.1 and A.2, the convergences in probability in (A.15) are deduced from the following ones:

ε -2 n N n β n M j=0 κ 2 τ n j E τ n j+1 τ n j (X u -X n τ n j )d X u 2 F τ n j → 0, ε -4 n N n β n M j=0 κ 2 τ n j E τ n j+1 τ n j (X u -X n τ n j ) 2 d X u 2 F τ n j → 0. (A.16) Since Q n = ε -4 n sup t∈[0,T] |X n t -X t | 4 is uniformly integrable and ∞ j=0 ( X τ n j+1 -X τ n j ) 2
is bounded and converges to 0 in probability by Lemma A.4, we have

E N n β n M j=0 κ 2 τ n j τ n j+1 τ n j ε -k n (X u -X n τ n j ) k d X u 2 ≤ CE Q k/2 n ∞ j=0 ( X τ n j+1 -X τ n j ) 2 → 0 for k = 1, 2, which gives (A.16).
We also have

sup t≥0 ε -1 n N n β n M j=0 κ τ n j τ n j+1 τ n j (X u -X n τ n j )d X u -ε -1 n ∞ j=0 κ τ n j τ n j+1 ∧β n M τ n j ∧β n M (X u -X n τ n j )d X u → 0, sup t≥0 ε -2 n N n β n M j=0 κ τ n j τ n j+1 τ n j (X u -X n τ n j ) 2 d X u -ε -2 n ∞ j=0 κ τ n j τ n j+1 ∧β n M τ n j ∧β n M (X u -X n τ n j ) 2 d X u → 0, (A.17) as δ → 0. Let T n K = inf{t > 0; w t (ε n ) ≥ KE[w T 0 (ε n )]} ∧ T 0 . Note that sup n∈N P[T n K < T 0 ] ≤ sup n∈N P[w T 0 (ε n ) ≥ KE[w T 0 (ε n )]] ≤ 1 K → 0,
as K → ∞. On the set {T n K < T 0 }, we can replace l and l by l •∧T n K and l •∧T n K respectively. This means that we can assume without loss of generality that w T 0 (ε n ) ≤ KE[w T 0 (ε n )]. Now in order to apply Proposition 2.2, it remains to show (2.10).

Part 1: Technical lemma

We give here a first technical lemma.

Lemma A.6. The sequence ε

2 n N n T 0 is tight. Proof. Since |X τ n j+1 ∧T 0 -X τ n j ∧T 0 | 2 ≥ ε 2 n K 2 , we have ε 2 n N n T 0 ≤ K 2 N n T 0 j=0 (X τ n j+1 ∧T 0 -X τ n j ∧T 0 ) 2 → K 2 X T 0 ,
in probability by Lemma A.5.

Part 2: Approximation lemma

We give here an important result. Let τn j+1 be the exit time of fixed barriers defined by

τn j+1 = inf t > τ n j : X t (X τ n j -ε n l τ n j , X τ n j + ε n l τ n j ) ∧ T 0 . (A.19)
We have the following lemma.

Lemma A.7. We have

N n T 0 j=0 E (τ n j+1 -τ n j+1 )|F τ n j → 0, in probability. Proof. Since the sequence ε 2 n N n T 0 is tight, it is enough to show that 1 ε 2 n sup j≤N n T 0 E τn j+1 -τ n j+1 F τ n j → 0. We write 1 ε 2 n sup j≤N n T 0 E τn j+1 -τ n j+1 F τ n j = R 1 + R 2 , with R 1 = 1 ε 2 n sup j≤N n T 0 E (τ n j+1 -τ n j+1 )½ {τ n j+1 ∨τ n j+1 ≥τ n j +ε n } F τ n j , R 2 = 1 ε 2 n sup j≤N n T 0 E (τ n j+1 -τ n j+1 )½ {τ n j+1 ∨τ n j+1 <τ n j +ε n } F τ n j .
We first treat R 1 . We have

R 1 ≤ T ε 2 n sup j≤N n T 0 P τn j+1 ∨ τ n j+1 ≥ τ n j + ε n F τ n j .
Since l, l and σ X are bounded from below by 1/K, using the Dambis, Dubins-Schwartz theorem we get that there exists some C > 0 such that

P τn j+1 ∨ τ n j+1 ≥ τ n j + ε n F τ n j ≤ P[ρ n ≥ Cε n ],
with ρ n the first exit time of [-ε n /K, ε n /K] by a Brownian motion starting from zero. Using the well-known bound E (X u -X n u )(σ X u ) 2 du F τ n j .

[(ρ n ) k ] ≤ Cε 2k n for k ∈ N, Markov's inequality gives the convergence to zero of R 1 . We now turn to R 2 . Recall that w T 0 (ε n ) ≤ KE[w T 0 (ε n )] = δ n → 0.
Since ε -1 n sup t |X t -X n t | ≤ K and σ X ≤ K, R and R ′ converge to 0 uniformly in probability on [0, T 0 ] by Lemma A.7. Using that for b 1 > 0 and b 2 > 0, the probability that a Brownian motion starting from zero hits level b 1 before level -b 2 is equal to b 2 /(b 2 + b 1 ), we get

ε -2 n E (X τn j+1 -X τ n j ) 4 F τ n j E (X τn j+1 -X τ n j ) 2 F τ n j = a 2 τ n j , ε -1 n E (X τn j+1 -X τ n j ) 3 F τ n j E (X τn j+1 -X τ n j ) 2 F τ n j = -s τ n j ,
where a 2 = l 2 + l 2ll, s = ll.

Then, to complete the proof, it suffices to show that the convergences

N n • j=0 κ τ n j a 2 τ n j E (X τn j+1 -X τ n j ) 2 |F τ n j → • 0 a 2 u (σ Y u ) 2 du, N n • j=0 κ τ n j s τ n j E (X τn j+1 -X τ n j ) 2 |F τ n j → • 0 s u (σ Y u ) 2 du
hold uniformy in probability on [0, T 0 ]. This follows from Lemma A.4 in [START_REF] Fukasawa | Discretization error of stochastic integrals[END_REF] together with Lemma A.4.

A.3.2 Proof of (2.4)

Here we prove a moment convergence result. Thus the localization procedure does not apply here. We set

A n = ε -1 n T 0 (X n t -X t )b Y t dt, B n = ε -1 n T 0 (X n t -X t )σ Y t dW Y t .
We have

(ε -1 n Z n T ) 2 = (A n + B n ) 2 ≤ 2(A 2 n + B 2 n
). Thus it is enough to prove the uniform integrability of (A The right hand side of the last inequality being integrable, this gives the result for (A n ) 2 .

We now turn to (B 2 n ). The sequence (B 2 n ) is non negative integrable and converges in law towards an integrable limit. Thus the uniform integrability is equivalent to the convergence in expectation, see for example [START_REF] Billingsley | Convergence of Probability Measures[END_REF]. Since

ε -2 n Z n T → 1
E[B 2 n ] = E[ε -2 n Z n T ] → 1 6 E[ T 0 a 2 t (σ Y t ) 2 dt],
which concludes the proof.

A.4 Proof of Theorem 4.1

We start with the first part of Theorem 4.1. Let (m, v) be a non dominated couple. Suppose it is a dominated couple with respect to Z T . This means there exists a process s * t such that the associated expectation, say m ′ = E[Z T,s * ], is larger than m and the expected error, say v ′ = E[(Z T,s * ) 2 ], is strictly smaller than v. From Lemma 2.

Definition 4 . 1 (Definition 4 . 2 (

 4142 Non dominated couple). A couple (m, v) ∈ (R + ) 2 is said to be non dominated if there exists no admissible discretization rule such that its associated limiting random variable Z * a,s satisfies E[Z * a,s ] ≥ m, E[(Z * a,s ) 2 ] < v. The set of non dominated couples is called the non domination domain. Nearly efficient couple).

Proposition 5 . 1 .

 51 In the Black-Scholes model with time varying coefficients, the efficient points are the couples of the form (m, m 2 P T P T -P 0 ),

  Under the condition a 2 t > s 2 t , this equation admits two nonzero roots with different signs. Therefore, another interesting property of hitting times based schemes is the following.

	Lemma 2.1. For any pair of limiting processes s and a satisfying (2.2) and a 2 t > s 2 t , we can always build a corresponding admissible discretization rule based on hitting times as in (2.12)-(2.13).

  2 t ≥ s 2 t and Cauchy-Schwarz) become almost equalities. This means that a t should be close to s t and s t essentially proportional to b Y t /(σ Y t ) 2 . Furthermore, we want the product s t b Y t to be essentially positive in order to get a positive modified Sharpe ratio. If we look for this rule among the hitting times based schemes specified by two processes (l t , l t ), Lemma 2.1 implies that • the difference l tl t should be essentially proportional to b t /(σ Y t ) 2 ,

  Let us define the family of processes of the form d Žt = s t Y t ( bt dt + σt dW t ), Ž0 = 0, with bt = b t /3, σt = σ t /3 and s t adapted continuous. Using obvious computations, the independence between the process B in Equation (4.1) and F , and the fact that s t is Fadapted, we get E[ ŽT ] = E[Z T,s ] and

	µE ŽT -	1 2µ	2	= µE Z T,s -	1 2µ	2	-	µ 18	E	0	T	(s t σ t Y t ) 2 dt .
	Hence, we can equivalently solve
		inf (s t ,0≤t≤T)	E µ Z2 T + (s t σt Y s * T µ 2 0 t Y t = -1 bt Ṗt P t Z * t ,
	where P t is the solution of the (ordinary) differential equation
				Ṗt = ρ 2 t	P 2 t P t + µ	, P T = 2µ,
	with ρ
													s -	1 2µ	2	-	1 4µ	.

t )

2

dt ,

with d Zt = s t Y t ( bt dt + σt dW t ), Z0 = -1 2µ .

Using the results of

[START_REF] Zhou | Continuous-time mean-variance portfolio selection: A stochastic LQ framework[END_REF]

which are summarized in Theorem B.1 in Appendix B, the optimal control s * t and optimally controlled process Z * t satisfy t = b t /σ t . The solution of this equation is given by

  Using again the Dambis, Dubins-Schwarz theorem and the various boundedness assumptions, we get E E Ĵn Here we prove (2.10), which completes the proof of the convergence in law of ε -1 n Z n T with the help of Proposition 2.1 and Proposition 2.2. As already seen, by It ô's formula, we have

	Therefore, we obtain
		N n t				N n t
	ε -2 n	j=0	κ τ n j A j = ε -2 n	j=0	κ τ n j E (X τn j+1 -X τ n j ) 4 |F τ n j + R t
		N n t				N n t
	ε -1 n	j=0	κ τ n j B j = ε -1 n	j=0	κ τ n j E (X τn j+1 -X τ n j ) 3 |F τ n j + R ′ t
	where
	R t = 6ε -2 n	N n t j=0	κ τ n j E	τ n j+1 τn j+1	(X u -X n u ) 2 (σ X u ) 2 du F τ n j ,
	R ′ t = 3ε -1 n	N n t j=0	κ τ n j E	τ n j+1 τn j+1
						Then, we have
		(τ n j+1 -τ n j+1 )½ {τ n j+1 ∨τ n j+1 <τ n
	Consequently,
	which gives the result.					E[R 2 ] ≤ Cδ n ,
	Part 3: Proof of (2.10)
	E[∆ 4 j,n |F τ n j ] = 6E	τ n j +1 τ n j	(X t -X τ n j ) 2 d X t F τ n j = A j ,
	E[∆ 3 j,n |F τ n j ] = 3E	τ n j +1 τ n
						29

j +ε n } ≤ Ĵn j+1 -Jn j+1 , with Ĵn j+1 = inf t ≥ τ n j ; X τ n j +t -X τ n j -ε n (l τ n j + δ n ), ε n (l τ n j + δ n ) , Jn j+1 = inf t ≥ τ n j ; X τ n j +t -X τ n j -ε n (l τ n j -δ n ), ε n (l τ n j -δ n ) . j+1 -Jn j+1 F Jn j+1 F τ n j ≤ Cε 2 n δ n . j (X t -X τ n j )d X t F τ n j = B j .

  [START_REF] Geiss | Quantitative approximation of certain stochastic integrals[END_REF]

						T
						0	a 2 t (σ Y t ) 2 dt
	and	ε -2 n Z n	T ≤	0	T	(l t ∨ l t ) 2 (σ Y t ) 2 dt,
	we readily obtain

  1, for any η we can find an admissible strategy with limiting variable Z * s * +η,s * . Clearly, we can find η small enough, such that E|Z * s * +η,s *] = m ′ and v ′ ≤ E[(Z * s * +η,s * ) 2 ] < v. Consequently (m, v) is a dominated couple, which is absurd. Conversely, any point which is non dominated with respect to Z T is non dominated since a 2 t ≥ s 2 t .For the second part, it remains to show that the proposed discretization rules indeed lead to nearly efficient couples. The fact that they are admissible is clear from Proposition 2.3. Recall now that for the suggested rule

	and
	E[(Z * a,s ) 2 ] =	1 9	E	0	T	s * t dY t	2	+	1 6	E	0	T	a 2 t -	2 3	(s * t ) 2 (σ Y t ) 2 dt
					=	1 9	E	0	T	s * t dY t	2	+	1 18	E	0	T	(s * t ) 2 (σ Y t ) 2 dt + δT.
	The couple
	1 3	E	0	T	s * t dY t ,
										a 2 t = (s * t ) 2 +	6δ (σ Y t ) 2
											1 3	E		0	T	s * t dY t

. This equality gives that the limiting variable Z * a,s associated to this discretization rule satisfies

E[Z * a,s ] =

Acknowledgements

We thank Philippe Amzelek and Joe Bonnaud from BNP-Paribas for inspiring discussions.

in probability. These two convergences follow using that sup j≥0,t∈[0,T]

in probability for i = 1, 2, which is deduced from (A.12) and the fact that sup j≥0,t≥0

in probability, by Lemma A.4.

Finally, remark that the uniform continuity of κ and (A.14) imply

in probability. Then Proposition 2.2 is eventually obtained from (A.15) together with (A.17) and (A.18).

A.3 Proof of Proposition 2.3 A.3.1 Proof of the convergence in law to (2.3)

We start with the stable convergence in law of the renormalized hedging error. Such convergence being stable against localization procedures, we can assume without loss of generality that |b

By Lemma A.5, we have (2.8) for all T 0 ∈ [0, T). Furthermore ε -1 n Z n , Y and ε -2 n Z n are equicontinuous. Therefore the uniform convergences in probability (2.5) and (2.6) follow from the corresponding convergences in probability at each t ∈ [0, T).

Fix T 0 ∈ [0, T) and define αK by (A.1). Then we have (A.2) and so, we can assume without loss of generality that 1/σ X ≤ K in order to show the convergences (2.5) and (2.6) on [0, T 0 ]. Also, thanks to the Girsanov-Maruyama transformation, we can assume b X = 0. Define for δ > 0 and t ∈ [0, T 0 ]

Since l and l are continuous and bounded, we have

Appendix B Linear-quadratic optimal control

We give here a summary of useful formulas from [START_REF] Zhou | Continuous-time mean-variance portfolio selection: A stochastic LQ framework[END_REF]. Consider a controlled system governed by the following linear SDE:

where x is the initial state and

) is a m-dimensional Brownian motion on a given filtered probability space (Ω, F , P,

We suppose that all the parameters are deterministic and continuous on [0, T] and H belongs to S n + the set of n × n symmetric positive matrices. We introduce the following matrix Riccati equation

Then following result is given in [START_REF] Zhou | Continuous-time mean-variance portfolio selection: A stochastic LQ framework[END_REF].


Agorram Boujemaa

Clement Pierre

Selmaoui Sabah

Khzami Salaheddine

Chafik Jamal

Chiadli Abdellatif

University students' conceptions about the concept of gene: Interest of historical approach

Keywords: gene concept, gene functions, genetic determinism, historical models, conceptions

Concepts of genetics are often difficult to teach, specifically the central concept of gene. Even the scientists disagree when defining this concept. This paper investigates university students' understanding about the gene and its functions. The results show the dominance of two conceptions of the gene: the Neoclassical model and the Mendelian model. The existence of hybrid conceptions and the lack of the modern model show that students are unable to mobilize the knowledge taught in biology. These results suggest to improve the teaching methods of genetics, for instance, by developing activities that bring students face to face with their conceptions.

Introduction

During the second half of the 20th century, genetics progressively became an essential field of biology also feeding controversial ethical, social and economical debates. The multiplicity and availability of products and applications of genetic technology (GMOs, DNA fingerprinting, screening of genetic diseases, gene therapy, cloning, ... ) are more and more daily present, requiring us a high level of scientific literacy and understanding of these issues for a citizenship control [START_REF] Dawson | Western Australian school students' attitude of biotechnology[END_REF][START_REF] Marbach-Ad | Attempting to break the code in students' comprehension of genetic concepts[END_REF].

Genetics is also one of the most difficult subjects in the biology curricula at the secondary school [START_REF] Banet | Teaching of biological inheritance and evolution of living beings in secondary school[END_REF]Bahar, et al., 1999a;[START_REF] Chattopadhyay | Understanding of genetic information in higher secondary students in Northeast India and the implications for genetics education[END_REF][START_REF] Kindfield | Understanding a basic biological process: Expert and novice models of meiosis[END_REF]Lewis & Wood-Robinson, 2000;Lewis, et al., 2000aLewis, et al., , 2000bLewis, et al., , 2000c;;[START_REF] Longden | Genetics-Are their inherent learning difficulties[END_REF][START_REF] Marbach-Ad | Student's cellular and molecular explanation of genetic phenomena[END_REF][START_REF] Marbach-Ad | Attempting to break the code in students' comprehension of genetic concepts[END_REF]Scriver, 1993;Wood-Robinson, 1 994) and university levels (Bahar, 1999b;[START_REF] Johnstone | Isolating topics of high perceived difficulty in school biology[END_REF][START_REF] Kindfield | Understanding a basic biological process: Expert and novice models of meiosis[END_REF].

The term "gene" was introduced by Wilhelm L. Johannsen in 1909. This central concept in genetics was initially defined as an entity of calculations to account for the transmission of hereditary traits. It became a material entity firstly as a part of chromosome [START_REF] Morgan | The origin of five mutations in eye color in Drosophila and their mode of inheritance[END_REF] and then, with the development of molecular biology, as a segment of DNA. More recently, three types of genes were defined related to their functions: genes coding for proteins, genes specifying the untranslated RNA (tRNA, RNAs ... ) and the regulatory genes. The more recent conceptions of gene have never totally replaced earlier conceptions: Multiple scientific conceptions of the gene are coexisting [START_REF] Morange | Redefinition of the concept of gene[END_REF]. Several authors tried to categorize them.

Griffiths distinguished two different conceptions of the gene: The molecular gene is "the molecular process underlying the capacity to express a particular polypeptide product", and the evolutionary gene is "a theoretical entity with a role in a particular, atomistic approach to the selection of phenotypic traits" (Griffiths & Neumann, 1999). Later, Griffiths and Stotz outlined three conceptions of the gene: instrumental, nominal, and postgenomic.

"

The instrumental gene has a critical role in the construction and interpretation of experiments in which the relationship between genotype and phenotype is explored. The nominal gene is a critical practical tool, allowing stable communication between bioscientists in a wide range of fields grounded in welldefined sequences of nucleotides, but this concept does not embody major theoretical insights into genome structure or function. The post-genomic gene embodies the continuing project of understanding how genome structure supports genome function, but with a deflationary picture of the gene as a structural unit". [START_REF] Griffiths | Genes in the postgenomic era?[END_REF] Gericke and Hagberg (2007) defined five different historical models of gene function:

the Mendelian model, the classical model, the biochemical-classical model, t he neoclassical model and the modern model.

(1) In the Mendelian model, the gene is a hypothetical construct and its main purpose is to explain genetic transmission, no connection was however made to a material unit in the cell.

(2) In the classical model, the gene is a particle, an indivisible unit of genetic transmission, recombination, mutation and function. The gene determines a characteristic. Definite characteristics were the product of genes, which were located at well-defined loci on the chromosomes.

(3) In the biochemical-classical model, the gene is a particle of transmission, function, mutation and recombination. The gene produces a substance that determines a characteristic. Tatum proposed in 1941 the one-gene-one-enzyme hypothesis for genetic function.

(4) In the neoclassical model, the gene is a materiel unit consisting of a DNAsegment. In this model, structure and function coincide and the gene codes for the production of a polypeptide. The neoclassical view of the gene peaked at about 1970 and stated that the gene (cistron) is a contiguous stretch of DNA that is transcribed as one unit into messenger RNA, coding for a single polypeptide.

(5) The modern model of gene function considers the gene as a hypothetical construct with a diverse material base consisting of DNA segments that take part in a developmental process. The gene is a producer of molecules in a developmental system. There are a number of categories of genes such as enzyme-producing genes, genes producing structural (nonsoluble) proteins, regulatory genes, and genes coding for RNA-molecules. No direct entities representing environmental aspects are present in this model.

In the present work, we will use these five categories.

Aims of the study

In Morocco, genetics is taught in the 3rd year of upper secondary school: Mendelian genetics, the molecular basis of heredity, human genetics and some basic principles of population genetics. At the university, the teaching of genetics differs from one university to another: Sometimes the genetic is treated in one module at the second year (semester 3), in other cases, it is treated in two modules at the second and third years (semester 3 and semester 5 or 6). Similarly, the main parts of the genetics are treated with more detail than in the upper secondary school : classical genetics, molecular and population genetics. A few research has analysed until now knowledge and understanding of genetics among Moroccan secondary school students [START_REF] Agorram | Analysis of students' conceptions among genetics[END_REF][START_REF] Elaboudi | Study of secondary students' conceptions among heredity and hereditary disease[END_REF] and among upper secondary school teachers [START_REF] Agorram | Moroccan primary and secondary teachers and futures teachers on the genetic determinism[END_REF], but, no study has focused on university students.

The purpose of this study is to identify the university students' conceptions of the gene concept.

Methodology

Questionnaire design

We use here only a part of the questionnaire containing three open questions: (1) The first related to the definition of gene; (2) The second to its biological functions; and (3) The third aimed to identify the concepts that can associate to the word "gene". We used open questions to capture the various gene conceptions in order to compare them to the five categories of conceptions identified in the literature [START_REF] Gericke | Definition of historical models of gene function and their relation to students' understanding of genetics[END_REF].

Sample of students

The study was conducted with a sample of 94 university students. They studied several years in the university: 59 obtained the license degree and 35 the Master's degree. A license degree is awarded to students completing educationally broad based post secondary programs requiring at least three years of full-time study. A Master's degree is awarded to students completing two years of university studies full time made following a license degree (license+2 years).

The written questionnaire was administered by the teacher to the students within a class period of one hour and was filled out anonymously.

Results

Definition of gene

The responses of students were categorized (see Table 1). The majority of students knew that genes are parts of DNA (75 students: 79.8%), but only one student

(1.3%) mentioned that the genes could also, in some viruses, be parts of RNA.

Among these 75 students, only 18 indicated that DNA is responsible for the synthesis of proteins, which govern hereditary characters (active view of genes), 14 indicated that DNA is responsible for the synthesis of protein but they did not mention a link between these proteins and hereditary traits. 28 students indicated that the DNA is responsible for hereditary characters but they did not mention protein. 15 students (16%) indicated that gene is a DNA segment but they make no reference to their products (proteins via mRNA, others RNA) or the characters they control. 19 students (20.2%) did not make any connection between the gene and its chemical nature (DNA or RNA).

Nine students (9.6%) defined the gene by its relationship to a phenotype regardless of the specific molecular sequence and the whole developmental mechanisms involved. Very few students indicated that the genes are located on chromosomes (9/9.6%) or are located in the nucleus (2/2.1%). 16 students indicated that gene is composed of alleles (17%), one of which indicated that the gene is composed of only two or three alleles. Only one student mentioned that the gene is composed of introns and exons. On the other hand, many students mentioned in the definition of gene, which it carries genetic information (21/22.3%) and responsible for its transfer from one generation to another (3/3.2%).

What is the biological function of a gene?

We identified several categories of conceptions of gene functions cited by students (analytical categories were not mutually exclusive; each student response may be assigned to one or more categories):

(1) Genes are active particles that controls characters (55.3%) (coding for protein synthesis, but only 7 students indicate that gene codes for primary structure of protein);

(2) Genes determine characters (33%);

(3) Genes play a role in transmission of hereditary information (34%);

(4) Genes carry hereditary information (26.6 %).

Some students cited other functions of the gene such as: genes are responsible of variation (4.2%), genes play a role in regulation (7.4%) and conservation of hereditary characters (6.4%) or in conservation of species (7.4%), in evolution (2.1%) and conservation of breed and lineage determination (2.1%), and genes are involved in the duplication of DNA (3.2%). Also, some students cited that genes cause diseases (without mentioning by which mechanisms) (8.5%). Some of these conceptions are not in accordance with the scientific knowledge, for instance, genes are carrier of characters (5.3%).

Gene's related concepts

The aim of this question is to identify all concepts related to "gene" concept spontaneously mobilized by students.

The 94 students cited 915 concepts in total with an average of 9 concepts by student. We have classified it into 7 categories (see Table 2).

The concepts most commonly cited by students are related to genetic information and its transmission (35.4%) and the chemical nature of the gene (nucleotides, DNA ... ) (25.6%). Only 7.6% of citations concern the cellular environment in which the gene is expressed and only 4.6% of citations are linked to genetic engineering and the applications of genes in different fields (medicine, agriculture...).

Discussion

The Neoclassical model gene concept (A stretch of DNA sequence that codes for a particular protein) (55.3%) was quite popular as was the Mendelian model (unit determining a character) (33%). But, it is also found that, hybrid models consisting of features from several of the historical models. 26. 8% of students knew the chemical nature of gene (DNA) and defined the gene by its relationship to a phenotype regardless of the specific molecular sequence and the whole developmental mechanisms involved.

Hence, it was noticed that the understanding of gene with modern concept is poor in majority of the students. Only one student mentioned that genes code for products other than proteins and enzyme such as RNA-molecules (RNAs and RNAt). Although the students had a course on molecular genetics, they were unable to mobilize this knowledge to define the gene and its function. Much of students had difficulties in distinguishing structure of gene and their functions. Several students indicated that genes are responsible for traits, but could not give any explanation of the mechanisms for this. Even the students have studied the biosynthesis of proteins in the course of Biochemistry and Molecular Genetics, they can not explain with the appropriate biochemical terms the process by which the gene controls a character. Similar results were found by others researchers (Lewis, et al., 2000a).

Few students have cited the location of genes (chromosome, nucleus), this shows that students found it difficult to make connections between the different organizational levels (molecular and cellular levels) and were unable to mobilize knowledge learned in other disciplines (cell biology,...) when they faced situations that required it. Students know and often hear words such as DNA, chromosomes, mRNA, genetic information, genes, ... , but they were unable to link with related gene concepts (chromosomes as organizers of genetic information; the physical entity of the gene; interrelationship between replication of the chromosome and genetic information; distinction between genes and genetic information; regulation of genes, interactions between gene and environment; … ).

These difficulties have been investigated by many researchers who presumed that time gaps between the teaching of related topics is important for understanding genetic relationships, and the compartmentalization between these genetics concepts and processes are the main obstacles to students' understanding and the development of a holistic concept of genetics (Lewis, et al., 2000a(Lewis, et al., , 2000b(Lewis, et al., , 2000c;;[START_REF] Marbach-Ad | Attempting to break the code in students' comprehension of genetic concepts[END_REF]. We have also identified several erroneous conceptions such as: "The gene is a sequence of amino acids which is in the form of a chain and is expressed as a phenotype", "DNA is a sequence of several genes=nucleotides", "Codon is responsible for the formation of proteins", "The gene is the whole of hereditary characteristic", "The gene is the unit that constitutes the DNA, each gene contains a genetic character". Similar conceptions were identified in students by other researchers [START_REF] Johnstone | Isolating topics of high perceived difficulty in school biology[END_REF][START_REF] Kindfield | Understanding a basic biological process: Expert and novice models of meiosis[END_REF]Bahar, et al., 1999a;Lewis, et al., 2000b). One out of five of students have difficulties in separating the concept of gene and the concept of alleles, one of these students has indicated that the gene is composed of only 2 or 3 alleles. These difficulties were found by other researchers [START_REF] Wood-Robinson | Young peoples' ideas about inheritance and evolution[END_REF]Lewis, et al., 2000a).

No students have referred to the fact that there may be interaction between genes and environment in the expression of different phenotypes. In another side, the 915 citations attached to the term "gene" seem to reflect a latent genetic determinism (linear causality "a gene → protein, trait, …") because there is no citation that shows the action of the environment on cellular gene expression. The remarkable absence of the interaction between genes and environment in the responses confirms the anchoring of the deterministic ideology among these students This was also found among upper secondary school students [START_REF] Agorram | Analysis of students' conceptions among genetics[END_REF], teachers and future teachers [START_REF] Castera | Conceptions of primary and secondary teachers on biological determinism of human personality in 12 countries in Europe, Africa and Middle East[END_REF][START_REF] Agorram | Analysis of students' conceptions among genetics[END_REF][START_REF] Agorram | Moroccan primary and secondary teachers and futures teachers on the genetic determinism[END_REF].

Conclusion and recommendations

This study shows that, even after receiving instruction in genetics in upper secondary school and university, students (objects of this study) find it difficult to define correctly the word "gene" (one of the fundamental concepts of genetics), to link concepts in genetics and have conceptions about genetics which differ from the current scientific model of heredity. This corroborates other researches conducted in other countries [START_REF] Clough | How secondary students interpret instances of biological adaptation[END_REF]. One of the causes of these difficulties is that the teaching of genetics is fragmented and is given on several levels.

For a better understanding of this central concept of genetics by students, teachers must use approaches other than traditional ones such as historical approach. The historical approach does not mean that the teacher cites only the chronological record of various discoveries, but it should insist on the epistemological and methodological obstacles and related them with the science of the time of these discoveries. Thus, students will acquire the competence to take a critical view on scientific concepts and how they were built. Teachers should encourage students to use concept maps to generate their ideas on the gene concept and to compare it with historical models cited in the literature. Thus, students can identify and characterize the internal and external weaknesses of each model and discover the epistemological and methodological obstacles encountered by researchers when developing these models. Teachers can use the gene models described by [START_REF] Gericke | Definition of historical models of gene function and their relation to students' understanding of genetics[END_REF] and updated by other researchers [START_REF] Smith | Updating the model definition of the gene in the modern genomic era with implications for instruction[END_REF] as a basis for teaching the gene concept. The teachers must also use strategies that will facilitate the connection of concepts about genes and about genetics in general.

Table 2 Distribution of concepts related to gene cited by students

 2

	Categories	Number of citations and %
	Materiel entity (nucleotides, DNA, chromatin, promoter, ... )	234 (25.6%)
	Information-transmission (translation, transcription, meiosis, codon, ... )	324 (35.4%)
	Products of gene expression (protein, enzyme, ... )	64 (7.0%)
	Cellular environment where the gene is expressed (nucleus, ribosome, cytoplasm, cell, ... )	70 (7.6%)
	Pathological manifestations (colour blindness, down syndrome, ... )	67 (7.3%)
	Applications (GMO, cloning, restriction enzymes, gene therapy, ... )	42 (4.6%)
	Others (biodiversity, genetics, Mendel, Morgan, molecular biology, … )	114 (12.4%)
	Total of citations	915 (100.0%)
	No answer	2/94 (2.1%)


Pierre Clément

Marie-Pierre Quessada

A recent News of the Week story ("Creationist beliefs persist in Europe," A. Curry, 27 February, p. 1159) Citizenship, 2004Citizenship, -2008)). This project involved data collection in 19 countries, 13 of which were in Europe (1).

In each country, the sample was a balanced set of primary and secondary school teachers who taught biology or the national language. This study differentiated between antievolutionist creationist teachers, t e a c h e r s w h o a r e creationist and evolutionist, and teachers who are evolutionists. There was a large contrast across countries: from 2% antievolutionist creationists in Estonia or France to more than 80% in Morocco or Algeria. In Europe, results revealed 47% antievolutionist creationists in Romania, 30% in Poland, and more than 25% in Cyprus and Malta. Creationist beliefs were more likely in those with greater belief in God or greater religious observance, regardless of religion. Biology teachers were more evolutionist than their colleagues i n o n l y h a l f o f t h e c o u n t r i e s surveyed. The longer a teacher trained at a university, the greater the acceptance of evolutionist ideas.

Pierre Clement 1 * and MariePierre

Quessada 2


Mathieu Acher
email: mathieu.acher@irisa.fr

Roberto E Lopez-Herrejon
email: roberto.lopez@jku.at

Rick Rabiser
email: rick.rabiser@jku.at

Cdl Mevss

SPLTea 2014: First International Workshop on Software Product Line Teaching

Keywords: K.3.2 [Computer and Information Science Education]: Computer Science Education; D.2.9 [Software Engineering]: Management-Life cycle Education, Software Engineering Software Product Lines, Software Engineering Teaching 1

Education has a key role to play for disseminating the constantly growing body of Software Product Line (SPL) knowledge. Teaching SPLs is challenging and it is unclear how SPLs can be taught, what are the possible benefits, or what is the material available. This workshop aims to explore and explain the current status and ongoing work on teaching SPLs at universities, colleges, and in industry (e.g., by consultants). Participants will discuss gaps and difficulties faced when teaching SPLs, benefits to research and industry, different ways to teach SPL knowledge, common threads, interests, and problems. The overall goal is to strengthen the important aspect of teaching in the SPL community.

Currently, it is unclear how Software Product Lines (SPLs) are taught, what are the possible difficulties faced, what are the benefits, or what is the material available. To address this gap, we conducted a survey [START_REF] Acher | A survey on teaching of software product lines[END_REF] with the purpose of capturing a snapshot of the state of teaching in our community. Our goal was to identify common threads, interests, and problems and build upon them to further understand and hopefully strengthen this important need in our community.

This workshop aims to further explore the current status and ongoing work on teaching SPLs and variability at universities, colleges, and in industry (e.g., by consultants). Participants will discuss gaps and difficulties faced when teaching SPLs, benefits to research and industry, different ways to teach SPL knowledge, common threads, interests, and problems. The overall goal is to strengthen the important aspect of teaching in the SPL community and outline future directions for research and practice.

As a first result we expect to consolidate a network of academics and practitioners interested and committed with SPL education and consulting. With this group we plan to create a virtual community in the form of a group in a social network or a dedicated Wiki. The virtual community aims to fill an educational vacuum: at least 30+ instructors have already taught SPLs [START_REF] Acher | A survey on teaching of software product lines[END_REF], but very few materials or experiences are shared among them. This will be the focal point to disseminate news, contribute courses, exercises, case studies, tools, and other coordination activities.

TOPICS

Topics include, but are not limited to:

• Experiences with teaching SPLs to students or practitioners

•

•

  Best practices for teaching SPLs • Innovative curricula or course formats • Impact of the online education movement (MOOCs) on teaching SPLs • Innovative methods for teaching SPLs in online courses • Integration of SPL research into teaching and training • Continuing SPL education in the face of rapid technological change The influence of new paradigms, such as cloud computing or global software development, on SPL teaching • Ensuring graduated students meet industry needs through the understanding of SPL techniques • Innovative use of social media for knowledge management in SPL teaching


Existing results for the Frenkel-Kontorova model

i i -2 i -1 i + 1 i + 2 m d 2 U j dt 2 + γ dU j dt = (U j+1 -U j ) + (U j-1 -U j ) + sin(2πU j ) + f N.
Forcadel Homogenization for traffic flow models

Existing results for the Frenkel-Kontorova model

i i -2 i -1 i + 1 i + 2 m d 2 U j dt 2 + γ dU j dt = F j (U j
u ε (t, x) = εu t ε , x ε ε u ε (t, x) 0 x u ε →? N.
Forcadel Homogenization for traffic flow models

u ε (t, x) = εu t ε , x ε u 0 (t, x) 0 x u ε → u 0 N. Forcadel
Homogenization for traffic flow models

For ε > 0, we define

u ε (t, x) = εu t ε , x ε , u ε (0, x) = u 0 (x) Theorem (F., Imbert, Monneau)
Under certain assumptions on u 0 , we have u ε → u 0 with

u 0 t = F (u 0 x ), u 0 (0, x) = u 0 (x)
The density of particles ρ = 1 u 0 x satisfies formally a conservation law

ρ t = ( H(ρ)) x with H(ρ) = -ρ F (1/ρ) N. Forcadel
Homogenization for traffic flow models

Ingredients of the proof

Main idea: inject the system of particles into a (system of) PDE.

If m is small, the system of PDE satisfies a comparison principle Order of the particles (link between the F j ).

Notion of hull functions

N. Forcadel Homogenization for traffic flow models

Üj (t) = a j V j (U j+1 (t) -U j (t)) -Uj (t) .
We set

Ξ j (t) = U j (t) + 1 α Uj (t) with α = 1 2 min j∈{1,...,n 0 } (a j ).
Then

Uj (t) = α (Ξ j (t) -U j (t)) Ξj (t) = (a j -α)(U j (t) -Ξ j (t)) + a j α V j (U j+1 (t) -U j (t)) .
N. Forcadel Homogenization for traffic flow models

Change of variables

Üj (t) = a j V j (U j+1 (t) -U j (t)) -Uj (t) .
We set

Ξ j (t) = U j (t) + 1 α Uj (t) with α = 1 2 min j∈{1,...,n 0 } (a j ).
Then

Uj (t) = α (Ξ j (t) -U j (t)) Ξj (t) = (a j -α)(U j (t) -Ξ j (t)) + a j α V j (U j+1 (t) -U j (t)) .
This system is monotone if

a j ≥ 4 V ′ ∞ N. Forcadel
Homogenization for traffic flow models Injection in a system of PDE

For j ∈ Z, pour tout (t, x) ∈ (0, +∞) × R, u j (t, x) = U j+n 0 ⌊x⌋ (t) and ξ j (t, x) = Ξ j+n 0 ⌊x⌋ (t). Then                      ∂u j ∂t = α(ξ j (t, x) -u j (t, x)) ∂ξ j ∂t = (a j -α)(u j -ξ j ) + a j α V j (u j+1 -u j ) u j+n 0 (t, x) = u j (t, x + 1) ξ j+n 0 (t, x) = ξ j (t, x + 1).
(2)

N. Forcadel Homogenization for traffic flow models

Ordering the vehicles

We assume that V j (h) = 0 for h ≤ 2h 0 and we consider the worst case

Using the solution of this ode as barrier, we get U j+1 ≥ U j + h 0 for a good choice of h 0

This implies that Ξ j+1 ≥ Ξ j N. Forcadel Homogenization for traffic flow models

Notion of hull function

N. Forcadel Homogenization for traffic flow models

Hull function h

         ∂u j ∂t = α(ξ j (t, x) -u j (t, x)) ∂ξ j ∂t = (a j -α)(u j -ξ j ) + a j α V j (u j+1 -u j )
We search particular solutions of the form

u j (t, x) = h j (λt + px), ξ j (t, x) = g j (λt + px) (h j , g j ) = hull functions λ = mean velocity n 0 p = mean density U i+n 0 l -U i l → p as l → +∞ N. Forcadel
Homogenization for traffic flow models

Equation of the hull functions

λ = α(g j (z) -h j (z)) λ = (a j -α)(h j (z) -g j (z)) + a j α V j (h j+1 (z) -h j (z))
N. Forcadel Homogenization for traffic flow models Theorem (F., Salazar)

For every p ∈ (0, +∞), there exists a unique λ := F (p) for which there exists hull functions (h j , g j ). Moreover, (h j , g j ) can be constructed such that h j (z) = z + h j (0), g j (z) = z + g j (0).

N. Forcadel Homogenization for traffic flow models

Construction of the hull functions N. Forcadel Homogenization for traffic flow models

Existence of hull functions

Theorem (F., Salazar)

For every p > 0, there exists some (ū ∞ j , ξ∞ j ) such that there exists a unique λ =: F (q) such that

u ∞ j (t, x) j , ξ ∞ j (t, x) j = px + λt + ū∞ j j , px + λt + ξ∞ j j ,
is a solution of (2). Moreover, if we define (h j , g j ) such that u ∞ j (t, x) = h j (λt + qx), and ξ ∞ j (t, x) = g j (λt + qx) then (h j , g j ) is a hull function and satisfies h j (z) = z + h j (0) and g j (z) = z + g j (0).

N. Forcadel

Homogenization for traffic flow models 1 (Initial data) u(0, x) = ξ(0, x) = px N. Forcadel Homogenization for traffic flow models 1 (Initial data) u(0, x) = ξ(0, x) = px 2 (Particular form of the solutions)

u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0) 3 (Long time behavior) u j (t, x) t , ξ j (t, x) t → λ as t → +∞ N. Forcadel
Homogenization for traffic flow models 1 (Initial data) u(0, x) = ξ(0, x) = px 2 (Particular form of the solutions)

u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0) 3 (Long time behavior) u j (t, x) t , ξ j (t, x) t → λ as t → +∞ 4 (Global control of the solution) |u j (t, 0) -λt| ≤ C and |ξ j (t, 0) -λt| ≤ C N. Forcadel
Homogenization for traffic flow models 1 (Initial data) u(0, x) = ξ(0, x) = px 2 (Particular form of the solutions) u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0)

3 (Long time behavior)

u j (t, x) t , ξ j (t, x) t → λ as t → +∞

Homogenization results

N. Forcadel Homogenization for traffic flow models

u j (t, x) = U j+n 0 ⌊x⌋ (t) 4 0 x U1 u1(t, x) U3n 0+1 U2n 0+1 Un 0+1 1 2 3
N. Forcadel Homogenization for traffic flow models

u ε j (t, x) = εu j t ε , x ε ε u ε j (t, x) 0 x u ε j →?
N. Forcadel Homogenization for traffic flow models

u ε j (t, x) = εu j t ε , x ε u 0 (t, x) 0 x u ε j → u 0 N. Forcadel
Homogenization for traffic flow models

Formal idea of the proof v ε j (t, x) := u 0 (t, x) + εh j (0), w ε j (t, x) := u 0 (t, x) + εg j (0)

N. Forcadel Homogenization for traffic flow models v ε j (t, x) := u 0 (t, x) + εh j (0), w ε j (t, x) := u 0 (t, x) + εg j (0) We then have very formally

(v ε j ) t (t, x) =u 0 t = F (u 0 x ) = α(g j (0) -h j (0)) = α ε (v ε j -w ε j ) (w ε j ) t (t, x) =u 0 t = F (u 0 x ) =(a j -α)(h j (0) -g j (0)) + a j α V j (h j+1 (0) -h j (0)) =(a j -α) v ε j -w ε j ε + a j α V j v ε j+1 -v ε j ε N.
Forcadel Homogenization for traffic flow models v ε j (t, x) := u 0 (t, x) + εh j (0), w ε j (t, x) := u 0 (t, x) + εg j (0) We then have very formally (v ε j ) t (t, x) =u 0 t = F (u 0 x ) = α(g j (0) -h j (0)) = α ε (v ε j -w ε j ) (w ε j ) t (t, x) =u 0 t = F (u 0 x ) =(a j -α)(h j (0) -g j (0)) + a j α V j (h j+1 (0) -h j (0))

=(a j -α) v ε j -w ε j ε + a j α V j v ε j+1 -v ε j ε (v ε j , w ε j
) and (u ε j , ξ ε j ) satisfy (almost) the same equation, so u ε j ≃ v ε j and ξ ε j ≃ w ε j .

N. Forcadel Homogenization for traffic flow models

  -m , . . . , U j+m )

(Initial data) u(0, x) = ξ(0, x) = px

(Particular form of the solutions)u j (t, x) = px + u j (t, 0) and ξ j (t, x) = px + ξ j (t, 0) N. ForcadelHomogenization for traffic flow models

(Global control of the solution)|u j (t, 0) -λt| ≤ C and |ξ j (t, 0) -λt| ≤ C

(Translation at infinity) By considering (u j (t + n, y) -λn, ξ j (t + n, y) -λn) as n -→ +∞, one can construct global in time solution (u ∞ j , ξ ∞ j )

An example of computation of F (W. Salazar)

N. Forcadel

Homogenization for traffic flow models

Properties of λ

N. Forcadel

Homogenization for traffic flow models

Homogenization results

For ε > 0, we define

Under certain assumptions on u 0 , we have u ε j → u 0 and ξ ε j → u 0 with

Ansatz for the proof :

Homogenization for traffic flow models Motivations Homogenization of traffic flows models

Homogenization results

For ε > 0, we define

Under certain assumptions on u 0 , we have u ε j → u 0 and ξ ε j → u 0 with

Ansatz for the proof :

N. Forcadel Homogenization for traffic flow models