Meriam El Mansour

Emmanuel Lepinette
email: emmanuel.lepinette@ceremade.dauphine.fr

Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty

Keywords: Super-hedging prices, Delayed information, Uncertainty, Conditional random sets, AIP condition Mathematics Subject Classification (2010): 49J53, 60D05, 91G20, 91G80. JEL Classification: C02, C61, G13

We solve the problem of super-hedging European or Asian options for discrete-time financial market models where executed prices are uncertain. The risky asset prices are not described by single-valued processes but measurable selections of random sets that allows to consider a large variety of models including bid-ask models with order books, but also models with a delay in the execution of the orders. We provide a dynamic programming principle under a weak no-arbitrage condition, the so-called AIP condition, under which the prices of the non negative European options are non negative. This condition is weaker than the existence of a risk-neutral martingale measure but it is sufficient to numerically solve completely the super-hedging problem. We illustrate our method by a numerical example.

Introduction

As observed in practice, the executed value of an asset may depend on the order sent by the trader and, also, on the quantities offered by the order book. Among the possible causes of the well-known slippage phenomenon, delays in the execution of the orders, liquidity disorders, market impacts, or transaction costs may influence the executed value. An approach to overcome this difficulty is to assume that we do not know by advance the traded prices. In that case, as proposed in the paper, the order that the trader sends is a mapping that associates to each possible price offered by the market a quantity to sell or buy. This is exactly what we generally observe in practice, in a presence of an order book for example, since there is no single prices.

On the contrary, it is traditional in mathematical finance to suppose that we first observe a (new) single market price and, then, we choose almost instantaneously the number of assets to sell or buy in order to revise the portfolio. This means that the last traded price keep constants long enough in the order book. Moreover, it coincides with a bid and ask price so that the buy and sell orders are executed at the same value.

In the real life, there may be delayed information, see the recent paper [START_REF] Agram | A financial market with singular drift and no arbitrage[END_REF] or [START_REF] Øksendal | Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations[END_REF], [START_REF] Saporito | Stochastic control with delayed information and related nonlinear master equation[END_REF] among others on stochastic control. The delayed information in the problem of pricing is sometimes modeled through incomplete or restricted information as in [START_REF] Ichiba | Option pricing with delayed information[END_REF], [START_REF] Kazmerchuk | The pricing of options for securities markets with delayed response[END_REF], [START_REF] Valière | No-arbitrage properties for financial markets with transaction costs and incomplete information[END_REF], [START_REF] Dahl | Pricing of claims in discrete time with partial information[END_REF] or using a two filtrations setting as in [START_REF] Cuchiero | A fundamental theorem of asset pricing for continuous time large financial markets in a two filtration setting[END_REF].

Another type of uncertainty is due the choice of the model supposed to approximate the real financial market [START_REF] Bertsimas | Data-driven robust optimization[END_REF]. Model risk may lead to price misevaluations that are studied in recent papers, in the growing field of robust finance. Since the seminal work of Knight [START_REF] Knight | Uncertainty, and Profit[END_REF], it is now broadly accepted that uncertainty may be described by a parametrized family of models, instead of considering only one model, if there is a lack of information on the parameters, see [START_REF] Pham | Portfolio diversification and model uncertainty: a robust dynamic mean-variance approach[END_REF] , [START_REF] Burzoni | Viability and arbitrage under Knightian uncertainty[END_REF], [START_REF] Neufeld | Superreplication under volatility uncertainty for measurable claims[END_REF], [START_REF] Becherer | Good deal hedging and valuation under combined uncertainty about drift and volatility[END_REF], [START_REF] Becherer | Hedging under generalized good-deal bounds and model uncertainty[END_REF], [START_REF] Fadina | Affine processes under parameter uncertainty[END_REF], [START_REF] Tevzaze | Robust utility maximization for a diffusion market model with misspecified coefficients[END_REF]. Other models consider that the market is driven by a family of probability measures in such a way that uncertainty stems from the existence of several possible reference probability measures determining which events are negligible, see [START_REF] Quenez | Optimal portfolio in a multiple-priors model[END_REF], [START_REF] Hobson | Robust hedging of the lookback option[END_REF], [START_REF] Cheredito | Duality formulas for robust pricing and hedging in discrete time[END_REF], [START_REF] Bouchard | Arbitrage in nondominated discrete-time models[END_REF], [START_REF] Biagini | Robust fundamental theorem for continuous processes[END_REF], [START_REF] Burzoni | Universal arbitrage aggregator in discrete-time markets under uncertainty[END_REF], [START_REF] Jan | Robust estimation of superhedging prices[END_REF], [START_REF] Carassus | The robust superreplication problem: a dynamic approach[END_REF].

In any case, uncertainty is taken into account in the literature by considering either several probabilistic structures, e.g. a family of reference probability measures and filtrations for the same price process or a family of price process models on the same stochastic basis. In the recent paper [START_REF] Rásonyi | On utility maximisation under model uncertainty in discrete-time markets[END_REF], the choice is made to fix only one filtered probability space on which a collection of stochastic processes describes the possible dynamics of the stock prices. We follow this alternative approach. Precisely, we consider a unique stochastic basis but we suppose that, in discrete time, the next stock prices at any time are not modeled by a unique vector-valued random variable as it is usual to do. Instead, we assume that the next stock prices belong to a collection of possible processes. The approach we adopt in our paper is slightly different from [START_REF] Rásonyi | On utility maximisation under model uncertainty in discrete-time markets[END_REF] in the sense that the collections of possible prices we consider are connected from time to time in such a way that it is possible to represent

Formulation of the problem

Let (Ω, (F t ) t∈{0,...,T } , F T , P ) be a filtered complete probability space where T is the time horizon. We do not suppose that F 0 is the trivial σ-algebra. We also consider a complete sub-filtration (G t ) t∈{0,...,T } where G t represents the market information available at time t. We suppose that G 0 is the trivial σ-algebra containing all the negligible sets and G t ⊆ F t for all t ∈ {0, . . . , T }. The typical case we shall consider is when F t = G t . The financial market we consider is composed of d risky assets and a bond S 0 . We assume without loss of generality that S 0 = 1.

Let us consider, for each t ≤ T , Λ t ⊆ L 0 (R d + , F t ) a collection of F tmeasurable random variables representing the possible executed prices for the risky assets at time t. We suppose that, at time t, the set Λ t may depend on the observed traded prices before time t, i.e. to each vector of prices (S u ) u≤t-1 , we associate a set Λ t = Λ t ((S u ) u≤t-1 ) representing the possible next prices at time t given that we have observed the executed prices (S u ) u≤t-1 at time t. We adopt the financial principle that the executed price S t is only known strictly after the order is sent at time t but before time t + 1.

Definition 2.1. A price process is an (F t ) t=0,••• ,T -adapted non-negative pro- cess (S t ) t=-1,••• ,T such that S t ∈ Λ t ((S u ) u≤t-1 ) for all t = 1, • • • , T and S -1 ∈ R is given .
Recall that S t represents the prices (S 1 t , • • • , S d t ) of d ≥ 1 risky assets proposed by the market to the portfolio manager when selling or buying. A typical case could be Λ t = L 0 (I t , F t ) with

I t = Π d j=1 [S bj t , S aj t ],
where (S bj ) j=1,••• ,d and (S aj ) j=1,••• ,d are respectively the bid and the ask price processes observed in the market at time t that may depend on (S u ) u≤t-1 . They are not necessary the best bid/ask prices as, in practice, the real transaction price may be a convex combination of bid and ask prices. Indeed, a transaction is generally the result of an agreement between sellers and buyers but it also depends on the traded volume. Clearly, the portfolio manager does not benefit in general from the last traded price observed in the market when sending an order. On the contrary, he should face an uncertain price S t which depends on the type of order (which may be not executed) but it also depends on some random events he does not control, e.g. slippage. A simple way to model this phenomenon is to suppose that the executed prices obtained by the manager belong to random intervals.

Another interesting case is when Λ t = {S θ t : θ ∈ Θ} is a parametrized family of random variables. For instance, consider fixed processes (ξ u ) u≤T and (m u ) u≤T adapted to (F t ) t=0,••• ,T and independent of F t-1 . Let C be a compact subset of R and suppose that S -1 is given. We define recursively

Λ t ((S u ) u≤t-1 ) = {S t-1 exp(σξ t + m t ) : S t-1 ∈ Λ t-1 , σ ∈ C} , t ≤ T.
In this model, there is an uncertainty on prices because of the unknown parameter (volatility) σ. This is a classical problem in robust finance, see for example [START_REF] Neufeld | Superreplication under volatility uncertainty for measurable claims[END_REF].

A portfolio strategy is an (F t ) t=0,••• ,T -adapted process θ = (θ 0 , θ) where, for all t = 0, • • • , T , θ t ∈ R d (resp. θ 0 t ∈ R) describes the quantities of risky assets (resp. the bond) held in the portfolio between time t and time t + 1. Since the strategies are not supposed to be adapted to (G t ) t=0,••• ,T , the manager is not supposed to control the quantity of assets he wants to sell or buy. This is what happens in practice because the orders are not necessarily executed, for instance in the case of limit stock market orders. For such a strategy θ, we define the portfolio process with initial endowment V 0 ∈ L 0 (R, F 0 ), as the liquidation value:

V V 0 , θ = θ 0 + θS = θ 0 + d i=1 θ i S i .
Recall that S t is observed strictly after the portfolio manager sends an order for θ t at time t. This is why F t is not the information available on the market at time t but the information the portfolio manager has strictly after t once he knows whether his order has been executed or not and once he knows the executed price as well. Nevertheless, the portfolio manager may send an order which depends on the uncertain price. For instance, such an order could be Buy at most 1000 units at a price less than or equal to 145 euros so that the strategies and the executed prices are linked.

In the following, we only consider self-financing portfolio processes V x, θ, i.e. they satisfy by definition:

∆V x, θ t := V x, θ t -V x, θ t-1 = θ t-1 ∆S t ,
where ∆S t := S t -S t-1 . Notice that this dynamics holds if and only if -(θ 0 t -θ 0 t-1 )S 0 t = (θ t -θ t-1 )S t . This means that the cost of the new portfolio allocation (θ 0 t , θ t ), i.e. buying or selling the quantities (|θ i t -θ i t-1 |) d i=0 , at the executed price S t is charged to the cash account. Therefore,

V V 0 , θ t = V 0 + t u=1 θ u-1 ∆S u . (2.1)
It is then natural by (2.1) to write V V 0 ,θ = V V 0 , θ. Our goal is to solve the following problem: Construct the minimal super hedging strategy of an Asian option whose payoff is g(S 0 , • • • , S T ) for some convex deterministic function on (R d ) T +1 . Because of price uncertainty, this means that we shall construct a self-financing strategy θ and we shall determine the minimal initial endowment V 0 such that V V 0 ,θ T ≥ g(S 0 , S 1 , • • • , S T ) whatever the executed prices S t ∈ Λ t ((S u ) u≤t-1 ) are for t ≤ T . As the filtration does not correspond to the current information (G t ) t=0,••• ,T of the market, contrarily to [START_REF] Baptiste | Pricing without martingale measures[END_REF], one more step is necessary to deduce the initial endowment P 0 . Indeed, the initial value of any portfolio process is F 0 -measurable contrarily to P 0 which has to be G 0 -measurable, i.e. P 0 is a constant or equivalently P 0 ≥ ess sup G 0 (V 0 ).

The super-hedging problem

The one time step resolution

We first introduce the basic tools and theoretical results we need in this section. A set Λ of measurable random variables is said F-decomposable if for any finite partition (F i ) i=1,••• ,n ⊆ F of Ω, and for every family

(γ i ) i=1,••• ,n of Λ, we have n i=1 γ i 1 F i ∈ Λ.
In the following, we denote by Σ(Λ) the F-decomposable envelop of Λ, i.e. the smallest F-decomposable family containing Λ. Notice that

Σ(Λ) = n i=1 γ i 1 F i : n ≥ 1, (γ i ) i=1,••• ,n ⊆ Λ, (F i ) i=1,••• ,n ⊆ F s.t. n i=1 F i = Ω . The closure Σ(Λ) in probability of Σ(Λ) is decomposable even if Λ is not decomposable. By [26, Theorem 2.4], there exists a F-measurable closed ran- dom set σ(Λ) such that Σ(Λ) = L 0 (σ(Λ), F) is the set of all F-measurable selectors of σ(Λ).
We now introduce the general one step problem between the dates t -1 and t for t ≥ 1. To do so, we suppose that after time t -1 but strictly before time t the portfolio manager observes the price S t-1 , as a consequence of her/his order, see Definition 2.1. More precisely, the portfolio manager knows (S u ) u≤t-2 at time t -1 and sends an order at time t -1 which is executed with a delay so that the executed price S t-1 ∈ Λ t-1 ((S u ) u≤t-2 ) is only observed strictly after t -1. In the following, we consider the σ-algebra

F t-1 = σ(S u : u ≤ t -1) for all t ≥ 1.
Let us consider a random function g t defined on (R d ) t+1 , t ≥ 1. We assume that the mapping (ω, z)

→ g t (S 0 (ω), • • • , S t-1 (ω), z) is F t-1 × B(R d )- measurable and z → g t (S 0 , S 1 , • • • , S t-1 , z) is lower-semicontinuous (l.s.c.) almost surely whatever the price process (S u ) u≤t-1 is. The first goal is to characterise the set P t-1 of all V t-1 ∈ L 0 (R, F t-1 ) such that V t-1 + θ t-1 ∆S t ≥ g t (S 0 , • • • , S t ), a.s. for all S t ∈ Λ t ((S u ) u≤t-1 ), (3.2) for some θ t-1 ∈ L 0 (R d , F t-1 ) 1 . Note that V t-1 depends on (S u ) u≤t-1 .
We observe by lower-semicontinuity that (3.2) holds if and only if

V t-1 + θ t-1 ∆S t ≥ g t (S 0 , • • • , S t ), for all S t ∈ Σ(Λ t ((S u ) u≤t-1 )). (3.3)
This means that we may suppose w.l.o.g. that Σ(Λ t ((S u ) u≤t-1 )) = Λ t ((S u ) u≤t-1 ).

In the following, we denote by I t ((S u ) u≤t-1 ) the F t -measurable closed random set such that Σ(Λ t ((S u ) u≤t-1 )) = L 0 (I t ((S u ) u≤t-1 ), F t ), see [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF]Theorem 2.4].

By [START_REF] Mansour | Conditional interior and conditional closure of a random set[END_REF]Theorem 3.4], we deduce that (3.2) is equivalent to V t-1 ≥ p t-1 where p t-1 = p t-1 ((S u ) u≤t-1 , θ t-1 ) is given by

p t-1 = θ t-1 S t-1 + sup z∈cl (It((Su) u≤t-1 )|F t-1 ) (g t (S 1 , • • • , S t-1 , z) -θ t-1 z) , = θ t-1 S t-1 + f * t-1 (-θ t-1 ).
In the formula above, cl (I t ((S u ) u≤t-1 )|F t-1 ) is the conditional closure of I t ((S u ) u≤t-1 ), i.e. the smallest F t-1 -measurable closed random set which contains I t ((S u ) u≤t-1 ) almost surely. We refer the readers to [START_REF] Mansour | Conditional interior and conditional closure of a random set[END_REF]Theorem 3.4] for the existence and uniqueness of such conditional random set. Moreover,

f * t-1 (y) = sup z∈R d (yz -f t-1 (z)) is the Fenchel-Legendre conjugate function of f t-1 defined as f t-1 (z) := -g t (S 0 , • • • , S t-1 , z) + δ cl (It((Su) u≤t-1 )|F t-1 ) (z),
where δ cl (It((Su) u≤t-1 )|F t-1 ) ∈ {0, ∞} is infinite on the complimentary of cl (I t ((S u ) u≤t-1 )|F t-1 ) and 0 otherwise. Notice that f * t-1 is convex and l.s.c. as a supremum (on cl (I t ((S u ) u≤t-1 )|F t-1 )) of convex and l.s.c. functions. Moreover, by [START_REF] Mansour | Conditional interior and conditional closure of a random set[END_REF]Theorem 3.4

], (ω, y) → f * t-1 (ω, y) is F t-1 × B(R d )-measurable. Therefore, Dom f * t-1 := {y : f * t-1 (ω, y) < ∞} is an F t-1
-measurable random set. We deduce that the F t-1 -measurable prices at time t -1 are given by

P t-1 ((S u ) u≤t-1 ) = θ t-1 S t-1 + f * t-1 (-θ t-1 ) : θ t-1 ∈ L 0 (R d , F t-1 ) +L 0 (R + , F t-1
). The second step is to determine the infimum super-hedging price as

p t-1 ((S u ) u≤t-1 ) = ess inf F t-1 P t-1 ((S u ) u≤t-1 ).
To do so, we use the arguments of [2, Theorem 2.8] and we obtain that:

p t-1 ((S u ) u≤t-1 ) = ess inf F t-1 θ t-1 S t-1 + f * t-1 (-θ t-1 ) : θ t-1 ∈ L 0 (R d , F t-1 ) , = ess inf F t-1 -θ t-1 S t-1 + f * t-1 (θ t-1 ) : θ t-1 ∈ L 0 (R d , F t-1 ) , = -ess sup F t-1 θ t-1 S t-1 -f * t-1 (θ t-1 ) : θ t-1 ∈ L 0 (R d , F t-1 ) , = -ess sup F t-1 θ t-1 S t-1 -f * t-1 (θ t-1 ) : θ t-1 ∈ L 0 (Dom f * t-1 , F t-1 ) , = -sup z∈Dom f * t-1 zS t-1 -f * t-1 (z) , = -sup z∈R d zS t-1 -f * t-1 (z) , = -f * * t-1 (S t-1 ).

Main properties satisfied by the one time step infimum super-hedging price

In the following, we suppose that, for all price process (S u ) u≤t-1 , there exists

α t-1 ∈ L 0 (R d , F t-1 ) and β t-1 ∈ L 0 (R, F t-1 ) that depend on (S u ) u≤t-1 such that g t (S 0 , • • • , S t-1 , x) ≤ α t-1 x + β t-1 , ∀x ∈ cl (I t ((S u ) u≤t-1 )|F t-1 ). (3.4)
This is the case for Asian options whose payoffs are for example of the form

k(S 0 + S 1 + • • • + S t -K) + , k ≥ 0. By [2, Theorem 2.8], we know that p t-1 ((S u ) u≤t-1 ) = (3.5) inf {αS t-1 + β : αx + β ≥ g t (S 0 , • • • , S t-1 , x), ∀x ∈ cl (I t ((S u ) u≤t-1 )|F t-1 )} .
We first establish the following result:2 Proposition 3.1. Let (S u ) u≤t-1 be a price process. Suppose that the mapping

(ω, z) → g t (S 0 (ω), • • • , S t-1 (ω), z) is F t-1 × B(R d )-measurable and the func- tion z → g t (S 0 , • • • , S t-1 , z) is l.s.c. almost surely. If S t-1 / ∈ conv cl (I t |F t-1 ), then p t-1 (((S u ) u≤t-1 )) = -∞. Moreover, if z → g t (S 0 , • • • , S t-1 , z) is a.s. convex, then p t-1 ((S u ) u≤t-1 ) ≥ g t (S 0 , • • • , S t-1 , S t-1 ) if S t-1 ∈ conv cl (I t |F t-1 ). At last, if g t (S 0 , • • • , S t-1 , •) is bounded from below by m t-1 ∈ L 0 (R, F t-1 ) on cl (I t |F t-1 ), then p t-1 (((S u ) u≤t-1 )) ≥ m t-1 if S t-1 ∈ conv cl (I t |F t-1 ). Proof. Suppose that S t-1 / ∈ conv cl (I t |F t-1 )
. By the Hahn-Banach separation theorem and a measurable selection argument, there exists

α * t-1 in L 0 (R d \ {0}, F t-1 ) and c 1 t-1 , c 2 t-1 ∈ L 0 (R d , F t-1
) such that we have the inequality

α * t-1 y < c 1 < c 2 < α * t-1 S t-1 for all y ∈ cl (I t |F t-1
). Multiplying the inequality by a sufficiently large positive multiplier, we may suppose that α * t-1 (S t-1 -y) ≥ n where n ∈ N is arbitrarily chosen. Let us introduce αt-1 = α t-1 -α * t-1 and βn

t-1 = β t-1 + α * t-1 S t-1 -n, n ≥ 1. By construc- tion, α t-1 x + β t-1 ≤ αt-1 x + βn t-1 for all x ∈ cl (I t |F t-1 ). It follows that αt-1 x + βn t-1 ≥ g t (S 1 , • • • , S t-1 , x), for all x ∈ cl (I t |F t-1 )
. By (3.5), we deduce that p

t-1 ≤ αt-1 S t-1 + βn t-1 = α t-1 + β t-1 -n. As n → ∞, we deduce that p t-1 = -∞. Suppose that z → g t (S 1 , • • • , S t-1 , z
) is a.s. convex and, furthermore, S t-1 ∈ conv cl (I t |F t-1 ). By Proposition 3.12,

p t-1 ((S u ) u≤t-1 ) ≥ g t (S 0 , • • • , S t-1 , S t-1 ). At last, suppose that z → g t (S 0 , S 1 , • • • , S t-1 , z) is bounded from below by m t-1 ∈ L 0 (R, F t-1 ) on cl (I t |F t-1
) and S t-1 ∈ conv cl (I t |F t-1 ). Then, S t-1 = lim n→∞ S n where S n ∈ conv cl (I t |F t-1 ), i.e. S n = Jn i=1 α i,n x i,n where α i,n ≥ 0 with Jn i=1 α i,n = 1 and x i,n ∈ cl (I t |F t-1 ) for all i, n. Consider (α, β) such that αx + β ≥ g t (S 0 , • • • , S t-1 , x) for all x ∈ cl (I t |F t-1 ). Then, αS t-1 + β = lim n→∞ (αS n + β) with

αS n + β = Jn i=1 α i,n (αx i,n + β) ≥ Jn i=1 α i,n g t (S 1 , • • • , S t-1 , x i,n ) ≥ m t-1 .
We deduce that αS t-1 + β ≥ m t-1 hence p t-1 ≥ m t-1 by (3.5). 2 Corollary 3.2. Let (S u ) u≤t-1 be a price process. Suppose that the mapping

(ω, z) → g t (S 0 (ω), • • • , S t-1 (ω), z) is F t-1 × B(R d )-measurable and the func- tion z → g t (S 0 , • • • , S t-1 , z
) is l.s.c. a.s. and convex or bounded from above by m t-1 ∈ L 0 (R, F t-1 ) on cl (I t ((S u ) u≤t-1 )|F t-1 ). Then, p t-1 ((S u ) u≤t-1 ) = -∞ if and only if S t-1 ∈ conv cl (I t |F t-1 ). In particular, the infimum superhedging price of any non negative payoff function is finite if and only if it is non negative or equivalently if S t-1 ∈ conv cl (I t (S u ) u≤t-1 |F t-1 ).

As studied in [START_REF] Baptiste | Pricing without martingale measures[END_REF], the non negativity of the prices for the zero claim or more generally for non negative European call options corresponds to a weak no arbitrage condition (AIP) which is naturally observed in practice. Adapted to our setting, we introduce the following definition: Definition 3.3. We say that condition AIP holds between t -1 and t if the prices at time t -1 of the time t zero claim is non negative for every price process (S u ) u≤t-1 . Moreover, we say that the condition AIP holds when AIP holds at any time step.

As observed in [START_REF] Baptiste | Pricing without martingale measures[END_REF] and above, when AIP fails, the infimum of the zero claim, and more generally of non negative payoffs, may be -∞. In that case, the dynamic programming principle we develop in this paper is still valid but unrealistic and non-implementable in practice. By Corollary 3.2, we have: Corollary 3.4. The condition AIP holds between t -1 and t if and only if S t-1 ∈ conv cl (I t ((S u ) u≤t-1 )|F t-1 ) for any price process (S u ) u≤t-1 , i.e.

I t-1 ((S u ) u≤t-2 ) ⊆ conv cl (I t (S u ) u≤t-1 |F t-1 ) if t ≥ 1.
In the following, if g is a function defined on R d and D is a subset of R d , we denote by conc(g, D) the (relative) concave envelope of g on D, i.e. the smallest concave function defined on R d which dominates g only on D. Observe that g ≤ h on D is equivalent to g -δ D ≤ h on R d . Therefore, conc(g, D) always exists as soon as g is dominated by an affine function on D.

Lemma 3.5. Let (S u ) u≤t-1 be a price process. Suppose that the mapping

(ω, z) → g t (S 0 (ω), • • • , S t-1 (ω), z) is F t-1 × B(R d )-measurable and the func- tion z → g t (S 0 , • • • , S t-1 , z) is l.s.c. almost surely. Consider the concave envelope h(x) = conc (g t (S 0 , • • • , S t-1 , •), cl (I t ((S u ) u≤t-1 )|F t-1 )) (x).
Then,

p t-1 ((S u ) u≤t-1 ) (3.6) = inf {αS t-1 + β : αx + β ≥ h(x), for all x ∈ cl (I t ((S u ) u≤t-1 )|F t-1 )} .
Proof. By definition, h is the smallest concave function which dominates g. We deduce that the set of all affine functions dominating g coincides with the set of all affine functions dominating h. By (3.5) we deduce that (3.6) holds.

Proposition 3.6. Suppose that AIP holds. Let (S u ) u≤t-1 be a price process. Suppose that the mapping (ω, z)

→ g t (S 0 (ω), • • • , S t-1 (ω), z) is F t-1 ×B(R d )- measurable and z → g t (S 0 , • • • , S t-1 , z) is l.s.c
. almost surely. Moreover, suppose that there exists α t-1 ∈ L 0 (R d , F t-1 ) and β t-1 ∈ L 0 (R, F t-1 ) such that g t (S 0 , • • • , S t-1 , z) ≤ α t-1 z + β t-1 for all z ∈ conv cl (I t ((S u ) u≤t-1 )|F t-1 ) and consider the concave envelope h(x) = conc (g t (S 0 , • • • , S t-1 , •), cl (I t ((S u ) u≤t-1 )|F t-1 )) (x).

(3.7)

We have

p t-1 ((S u ) u≤t-1 ) ∈ [g t (S 0 , • • • , S t-1 , S t-1 ), α t-1 S t-1 + β t-1 ]
. Moreover, if the super-differential ∂h(S t-1 ) = ∅ , then p t-1 ((S u ) u≤t-1 ) = h(S t-1 ) is a price, i.e. p t-1 ((S u ) u≤t-1 ) ∈ P t-1 ((S u ) u≤t-1 ) with the super-replicating strategies θ t-1 ∈ ∂h(S t-1 ).

Proof. It is clear by Lemma 3.5 that p t-1 ((S u ) u≤t-1 ) ≥ h(S t-1 ) when S t-1 belongs to cl (I t ((S u ) u≤t-1 )|F t-1 ). By definition, for all r t-1 ∈ ∂h(S t-1 ) = ∅, for all x ∈ conv cl (I t (S u ) u≤t-1 |F t-1 ), h(x) ≤ h(S t-1 ) + r t-1 (x -S t-1 ) =: δ(r t-1 , x).

(3.8)

Therefore, p t-1 ((S u ) u≤t-1 ) ≤ δ(r t-1 , S t-1 ) = h(S t-1 ), and finally

p t-1 ((S u ) u≤t-1 ) = h(S t-1 ).
At last, applying (3.8) with x = S t ∈ I t ((S u ) u≤t-1 ) ⊆ cl (I t ((S u ) u≤t-1 )|F t-1 ), we deduce that

p t-1 ((S u ) u≤t-1 ) + r t-1 ∆S t ≥ h(S t ) ≥ g t (S 0 , • • • , S t-1 , S t ).
Since x → g t (S 0 , • • • , S t-1 , x) is l.s.c., we consider the following random set:

G t-1 := {(ω, r t-1 ) : δ(r t-1 , x) ≥ g t (S 0 , • • • , S t-1 , x), ∀x ∈ conv cl (I t (S u ) u≤t-1 |F t-1 )}, = {(ω, r t-1 ) : δ(r t-1 , γ n t-1 ) ≥ g t (S 0 , • • • , S t-1 , γ n t-1 ), ∀n ∈ N},
where (γ n t-1 ) n≥1 is a Castaing representation of conv cl (I t (S u ) u≤t-1 |F t-1 ). Since G t-1 is F t-1 × B(R d )-measurable and G t-1 = ∅ a.s, it admits a measurable selection which is a measurable strategy θ t-1 for the price p t-1 ((S u ) u≤t-1 ). 2 Remark 3.7. As the function h in (3.7) is concave and finite a.s. on the conditional closure conv cl (I t (S u ) u≤t-1 |F t-1 ), see proof of Proposition 3.1, the super-differential ∂h(S t-1 ) of h at the point S t-1 is not empty when S t-1 belongs to the interior of conv cl (I t (S u ) u≤t-1 |F t-1 ).

The following result proves the measurability of the infimum super-hedging price p t-1 ((S u ) u≤t-1 ): Proposition 3.8. Suppose that cl (I t ((S u ) u≤t-1 )|F t-1 ) admits a Castaing representation (ξ m t-1 ) m≥1 where ξ m t-1 = x m ((S u ) u≤t-1 ), for all m ≥ 1, and x m are Borel functions on (R d ) t independent of (S u ) u≤t-1 . Then, there exist a Borel function φ t-1 on (R d ) t such that p t-1 ((S u ) u≤t-1 ) = φ t-1 ((S u ) u≤t-1 ).

Proof. Let (S u ) u≤t-1 be a price process. We denote by S (t-1) = (S u ) u≤t-1 and I t-1 = cl (I t (S (t-1) )|F t-1 ).

Recall that

p t-1 (S (t-1) ) = inf (α,β) αS t-1 + β : αx + β ≥ g t (S (t-1) , x), for all x ∈ I t-1 .
By assumption x m is a Borel function on (R d ) t independent of the price process (S u ) u≤t-1 . So:

p t-1 (S (t-1) ) = inf (α,β) αS t-1 + β : αx m (S (t-1) ) + β ≥ g t (S (t-1) , x m (S (t-1) )), ∀m = inf α αS t-1 + f * t-1 (-α, S (t-1) )
such that f * t-1 (-α, S (t-1) ) = sup m g t (S (t-1) , x m (S (t-1) )) -αx m (S (t-1) ) . 1) ) . We claim that p t-1 (S (t-1) ) = φ t-1 (S (t-1) ).

Let us denote

Q d = {α n = (α n 1 , ..., α n d ), n ≥ 1, α i ∈ Q} and define the real- valued mapping φ t-1 as φ t-1 (S (t-1) ) = inf n α n S t-1 + f * t-1 (-α n , S (t-
(3.9)

It is clear that p t-1 (S (t-1) ) ≤ φ t-1 (S (t-1) ). Conversely, let α ∈ R d , and α n ∈ Q d a sequence such that for arbitrary fixed ∈ int(R d + ), we have α n ≥ α and α > α n -componentwise. Then, by definition of f * t-1 , we have :

f * t-1 (-α, S (t-1) ) ≥ g t (S (t-1) , x m (S (t-1) )) -αx m (S (t-1) ), ∀m ≥ 1 ≥ g t (S (t-1) , x m (S (t-1) )) -α n x m (S (t-1) ) +(α n -α)x m (S (t-1) ), ∀m ≥ 1. Notice that x m (S (t-1) ) ∈ R d + because x m (S (t-1) ) ∈ I t-1 . So, f * t-1 (-α, S (t-1) ) ≥ g t (S (t-1) , x m (S (t-1) )) -α n x m (S (t-1) ), ∀m ≥ 1, ∀n ≥ 1 ≥ f * t-1 (-α n , S (t-1) ), ∀n ≥ 1.
Hence,

αS t-1 + f * t-1 (-α) ≥ αS t-1 + f * t-1 (-α n ), ∀n ≥ 1 ≥ α n S t-1 + f * t-1 (-α n ) -S t-1 , ∀n ≥ 1 ≥ α n S t-1 + f * t-1 (-α n ) -S t-1 , ∀n ≥ 1 ≥ φ t-1 (S (t-1) ) -S t-1 .
As → 0, we get αS t-1 + f * t-1 (-α) ≥ φ t-1 (S (t-1) ). Therefore, we deduce that p t-1 (S (t-1) ) ≥ φ t-1 (S (t-1) ). Hence, the equality (3.9) holds, which proves that the infimum superhedging price p t-1 ((S u ) u≤t-1 ) is measurable with respect to the argument (S u ) u≤t-1 .

The rest of this section aims to prove that, under some technical conditions, the mapping (S u ) u≤t-1 -→ p t-1 ((S u ) u≤t-1 ) is lower-semicontinuous. Definition 3.9. We say that the mapping

I t : (S u ) u≤t-1 -→ cl (I t ((S u ) u≤t-1 )|F t-1 ) z ∈ cl (I t ((S n u ) u≤t-1 )|F t-1
) in the ball B(z, ) of center z and radius , we have:

g t ((S u ) u≤t-1 , z) ≤ g t ((S n u ) u≤t-1 , z) + |α t-1 | × |((S u ) u≤t-1 , z) -((S n u ) u≤t-1 , z)|, g t ((S u ) u≤t-1 , z) ≤ g t ((S n u ) u≤t-1 , z) + |α t-1 | sup u≤t-1 |S n u -S u | + |α t-1 | , g t ((S u ) u≤t-1 , z) ≤ h (n) (z) + |α t-1 | sup u≤t-1 |S n u -S u | + |α t-1 | , (3.10)
where h (n) is an arbitrary affine function satisfying

h (n) ≥ g t ((S n u ) u≤t-1 , •) on cl (I t ((S n u ) u≤t-1 )|F t-1 ). Let us define h(n) (z) = inf z∈ B(z, )∩cl (It((S n u ) u≤t-1 )|F t-1 ) h (n) (z) + |α t-1 | sup u≤t-1 |S n u -S u | + |α t-1 | . By convention, we set inf ∅ = -∞. Let us show that h(n) is concave. To see it, observe that z ∈ B(z, ) ∩ cl (I t ((S n u ) u≤t-1 )|F t-1 ) if and only if z = z + u where u ∈ E n (z) = B(0, ) ∩ (cl (I t ((S n u ) u≤t-1 )|F t-1 ) -z) . Therefore, h(n) (z) = inf u∈E n (z) h (n) (z + u) + |α t-1 | sup u≤t-1 |S n u -S u | + |α t-1 | . Let z = λz 1 +(1-λ)z 2 .
We only need to consider the case where E n (z 1 ) = ∅ and E n (z 2 ) = ∅. We deduce that E n (z) = ∅. Moreover, by assumption, any u ∈ E n (z) may be written as u = αu 1 + (1 -α)u 2 where u i ∈ E n (z i ), i = 1, 2. Therefore,

h (n) (z + u) = αh (n) (z 1 + u 1 ) + (1 -α)h (n) (z 2 + u 2 ), ≥ α h(n) (z 1 ) + (1 -α) h(n) (z 2 ).
Taking the infimum in the right hand side of the inequality above, we deduce that h(n

) (λz 1 + (1 -λ)z 2 ) ≥ α h(n) (z 1 ) + (1 -α) h(n) (z 2 ), i.e. h(n) is concave. By (3.10), we deduce that p t-1 ((S u ) u≤t-1 ) ≤ h(n) (S t ) for all h (n) . As S n t-1 ∈ E n (S t-1
), for n large enough, under AIP, we deduce that

p t-1 ((S u ) u≤t-1 ) ≤ h (n) (S n t-1 ) + |α t-1 | sup u≤t-1 |S n u -S u | + |α t-1 | .
Taking the infimum over all affine functions h (n) , we get that for n large enough:

p t-1 ((S u ) u≤t-1 ) ≤ p t-1 ((S n u ) u≤t-1 ) + |α t-1 | sup u≤t-1 |S n u -S u | + |α t-1 | .
As is arbitrarily chosen, we may conclude that

p t-1 ((S u ) u≤t-1 ) ≤ lim inf n p t-1 ((S n u ) u≤t-1 ). 3.3. Case where x → g t (S 0 , • • • , S t-1 , x) is a convex function
We shall prove that p t-1 ((S u ) u≤t-1 ) is a convex function of the price process (S u ) u≤t-1 if so Λ t-1 is. In the following, we say that the mapping

Λ t-1 : (S u ) u≤t-1 -→ Λ t-1 ((S u ) u≤t-1 ) := conv (cl (I t ((S u ) u≤t-1 )|F t-1 )) is convex for the inclusion if, for λ ∈ [0, 1], Λ t-1 ((λ((S u ) u≤t-1 )+(1-λ)(( Su ) u≤t-1 ) ⊆ λΛ t-1 ((S u ) u≤t-1 )+(1-λ)Λ t-1 (( Su ) u≤t-1 ),
for all price process (S u ) u≤t-1 , ( Su ) u≤t-1 .

Proposition 3.12. Suppose that the mapping (ω, z) → g t (S 0 , S 1 (ω), ..., S t-1 (ω), z) is F t-1 ⊗ B(R d ) measurable, non negative and z → g t (S 0 , S 1 , ..., S t-1 , z) is lower semi-continuous and convex almost surely and suppose that the mapping Λ t-1 : (S u ) u≤t-1 -→ Λ t-1 ((S u ) u≤t-1 ) is convex. Then, the mapping (S u ) u≤t-1 → p t-1 ((S u ) u≤t-1 ) is convex .

Proof. Let (S u ) u≤t-1 , (S u ) u≤t-1 be two price processes. Let us define the following price process (S u

) u≤t-1 = λ(S u ) u≤t-1 + (1 -λ) (S u ) u≤t-1 for λ ∈ [0, 1].
We consider the following random sets:

Λ t-1 = conv (cl (I t ((S u ) u≤t-1 )|F t-1 )) , t ≥ 1, Λt-1 = conv cl (I t ( (S u ) u≤t-1 )|F t-1 ) , t ≥ 1, Λ t-1 = conv cl (I t ((S u ) u≤t-1 )|F t-1 ) , t ≥ 1.
By assumption, we have Λ t-1 ⊆ λΛ t-1 + (1 -λ) Λt-1 for λ ∈ [0, 1]. Let h and h be two affine functions such that:

h(x) ≥ g t ((S u ) u≤t-1 , x), ∀x ∈ Λ t-1 . h(x) ≥ g t (( Su ) u≤t-1 , x), ∀x ∈ Λt-1 .
Thus, for λ ∈]0, 1[, we have

λh(x) + (1 -λ) h(x) ≥ λg t ((S u ) u≤t-1 , x) + (1 -λ)g t (( Su ) u≤t-1 , x) ≥ g t (λ((S u ) u≤t-1 ) + (1 -λ)(( Su ) u≤t-1 ), λx + (1 -λ)x).
Let x ∈ Λ t-1 such that x = λx + (1 -λ)x. By above, we have:

λh(x) + (1 -λ) h(x) ≥ g t ((S u ) u≤t-1 ), x) =: ĝt (x).
Now, let us consider

E x = λ -1 λ Λt-1 + 1 λ x, λ ∈]0, 1[ ∩ Λ t-1 .
Observe that αE

x 1 + (1 -α)E x 2 = E αx 1 +(1-α)
x 2 for all α ∈ [0, 1], and x 1 , x 2 ∈ R d . Then, with x = αx 1 + (1 -α)x 1 , any x ∈ E x may be written as

x = αx 1 + (1 -α)x 2 , where x i ∈ E x i , i = 1, 2. As (x, x) → h( 1 1-λ (x -λx)) is affine, we deduce that λh(x) + (1 -λ) h( 1 1 -λ (x -λx)) ≥ α λh(x 1 ) + (1 -λ) h( 1 1 -λ (x 1 -λx 1 )) +(1 -α) λh(x 2 ) + (1 -λ) h( 1 1 -λ (x 2 -λx 2 )) , λh(x) + (1 -λ) h( 1 1 -λ (x -λx)) ≥ α ĥ(x 1 ) + (1 -α) ĥ(x 2 ),
where ĥ(x) = inf

x∈Ex {λh(x) + (1 -λ) h( 1 1-λ (x -λx))}.
Therefore, taking the infimum in the right side of the inequality above, we deduce that ĥ is a (non negative) concave function with finite values. So, it is continuous and we have ĥ(x) ≥ ĝt (x) for all x ∈ Λ t-1 . We deduce that p t-1 ((S u ) u≤t-1 ) ≤ ĥ(S t-1 )

≤ λh(S t-1 ) +

(1 -λ) h( St-1 ), ∀S t-1 ∈ Λ t-1 , St-1 ∈ Λt-1 .
Taking the infimum over all the affine functions h and h, we deduce that

p t-1 ((S u ) u≤t-1 ) ≤ λp t-1 ((S u ) u≤t-1 ) + (1 -λ)p t-1 (( Su ) u≤t-1 )
and the conclusion follows.

Remark 3.13. Suppose that the AIP condition holds and that (3.4) holds.

Consider φ t-1 (u) = inf n α n u t-1 + f * t-1 (-α n , u) , u = (u 0 , ..., u t-1 ) ∈ (R d ) t , where f * t-1 (-α, u) = sup m [g t (u, x m (u)) -αx m (u)].
, M t-1 ∈ L 0 ([0, ∞], F t-1 ) such that conv (cl (I t ((S u ) u≤t-1 )|F t-1 )) = [m t-1 , M t-1 ].
By Lemma 3.5, we deduce that under (AIP)

p t-1 ((S u ) u≤t-1 ) = g t (S 0 , • • • , S t-1 , m t-1 ) (3.11) + g t (S 0 , • • • , S t-1 , M t-1 ) -g t (S 0 , • • • , S t-1 , m t-1 ) M t-1 -m t-1 (S t-1 -m t-1 ).
If we suppose that m t-1 = k d t-1 S t-1 and M t-1 = k u t-1 S t-1 as in [START_REF] Baptiste | Pricing without martingale measures[END_REF], where k d t-1 and k u t-1 are deterministic coefficients, then p t-1 ((S u ) u≤t-1 ) = g t-1 ((S u ) u≤t-1 ) with

g t-1 (x 0 , • • • , x t-1 ) = λ t-1 g t (x 0 , • • • , x t-1 , k d t-1 x t-1 )+(1-λ t-1 )g t (x 0 , • • • , x t-1 , k u t-1 x t-1 )
,

where λ t-1 = k u t-1 -1 k u t-1 -k d t-1
and g T is the payoff function.

The multistep backward procedure

The main results of Section 3.2 for the one step model may be applied recursively, starting from time T , as the payoff function g T is known. Consider the case where the conditional support cl (I t ((S u ) u≤t-1 )|F t-1 ) admits a Castaing representation (ξ m ) m≥1 where ξ m = x m ((S u ) u≤t-1 ), for all m ≥ 1, and x m are Borel functions on (R d ) t . Then, by Proposition 3.8, we know that the infimum price at time T -1 is a Borel function g T -1 of the prices S 0 , • • • , S T -1 . Then, we may repeat the procedure if we are in position to verify that g T -1 is also l.s.c. This is the case by Proposition 3.12 and Remark 3.13, under convexity conditions.

Numerical illustration

Formulation of the problem with d = 1

In this section we consider the example of the European call option at time T = 2, i.e. with the payoff function g(S 2 ) = (S 2 -K) + , K > 0. Let (S t ) t=0,1,2 be the executed price process. Recall that S t belongs to the random set Λ t , for t = 0, 1, 2, respectively. We suppose that the risk-free asset is given by S 0 . Recall that there exist F t -measurable closed random sets I t = I t ((S u ) u≤t-1 ) such that:

Σ(Λ t ((S u ) u≤t-1 )) = L 0 (I t ((S u ) u≤t-1 ), F t ), t = 0, 1, 2.
We may suppose that Λ = Σ(Λ) so that S t ∈ I t a.s. for t = 0, 1, 2. At each step, we shall apply the procedure we have developed in the sections above. In particular, we seek for the strategy θ and we deduce the portfolio value V associated to the executed price process S. Then, we may estimate the error between the terminal value of V 2 and the payoff g 2 (S 2 ) that we denote by 2 = V 2 -g 2 (S 2 ). We start from a known price S -1 at time t = 0, which corresponds to the last traded price. We suppose that I t = [S t-1 m t , S t-1 M t ], t = 0, 1, 2, where the two random variables m t and M t are independent of S t-1 and are uniformly distributed as m t ∼ U[0.7, 1] and M t = m t + spr t such that spr t ∼ U[0, 0.4] is independent of m t . Observe that m - t = 0.7 and M + t = 1.4. At time t = 0, we suppose that there is a single asset price S 0 in L 0 (I 0 , F 0 ). We choose in our model to pick randomly S 0 in the interval I 0 . Precisely, S 0 = S -1 m 0 + k 0 S -1 (M 0 -m 0 ), where k 0 is a random variable such that k 0 ∼ U[0, 1]. We may interpret this choice as if the bid and ask prices of the market were the same and S 0 is the market price. The order we sent is of the form buy or sell the quantity θ 0 (S 0 ) at the price S 0 .

At time t = 1, we choose to model the bid and ask prices S bid 1 , S ask

Explicit computation of the strategy

We deduce the portfolio value and the strategy value at any time by dominating the payoff function by the smallest affine function on the conditional support of S, as mentioned in (3.5). We consider the terminal payoff function g(S T ) = (S T -K) + for several strikes.

1 respectively as: S bid 1 = S 0 m 1 and S ask 1 = S 0 M 1 .S ask 1 ≤ S * 1 (Fig 1 2 = S bid 2 ∈

 111111122 Fig 1

4. 2 . 1 . 1 =

 211 The strategy at time t = 1Recall that S 2 ∈ Λ 2 (S 1 ) ∼ I 2 = [S 1 * m 2 , S 1 * M 2 ]. In order to compute the strategy θ 1 = θ 1 (S 1 ) we first compute the function ϕ 1 given by (3.5) which dominates the the pay-off function g 2 on the conditional support cl (I 2 (S 1 )|F 1 ) = [S 1 m - 2 , S 1 M + 2 ]. So, V 0 (S 0 ) = ϕ 0 (S 0 ) = (S 0 M + 1 -K)(1 -m - : g 0 (S 0 ), θ 0 (S 0 ) = S 0 M + 1 -K S 0 (M + 1 -

Fig 11 :

 11 Fig 11: The empirical results.

Fig 12 Fig 13 :

 1213 Fig 12 Fig 13: K=100.

  Recall that, by Proposition 3.8, p t-1 ((S u ) u≤t-1 ) = φ t-1 ((S u ) u≤t-1 ). When g t is convex, then φ t-1 is convex by Proposition 3.12. Moreover, if g t ≥ 0, 0 ≤ φ t-1 < ∞ by Proposition 3.6. Then, dom φ t-1 = (R d ) t and we deduce that φ t-1 is continuous on (R d ) t . Remark 3.14. Let us denote by conv (cl (I t ((S u ) u≤t-1 )|F t-1 )) the closed convex envelop of cl (I t ((S u ) u≤t-1 )|F t-1 ). By a measurable selection argument, we may show that there exists m t-1

  m - 1 ) * 10 -6 1, 9 * 10 -7 5.975 * 10 -8 0 max( R )

	K	50	75	100	125	150
	E(S 0 )	95.002	94.983	95.006	94.98	95.001
	E(S 1 )	99.56	94.94	87.085	82.104	81.736
	E(S 2 )	94.56	90.180	82.716	78.01	77.664
	E(V 0 )	46.503	29.357	16.960	11.244	6.7
	max V 0	89.677	66.72	49.726	33.05	22.562
	E(V (S 0 )/S -1 )	0.465	0.294	0.170	0.112	0.067
	E(V (S 0 )/S 0 )	0.483	0.300	0.173	0.114	0.066
	min(V (S 0 )/S 0 )	0.359	0.163	0.098	0.032	0
	max(V (S 0 )/S 0 ) 0.642	0.479	0.358	0.237	0.162
	E( R )	0.017	0.077	0.076	0.064	0.039
	σ( R )	0.024	0.045	0.04	0.037	0.0317
	min R	0 2.23 0.18 0.19	0.195	0.187	0.187
	E(θ 0 S 0 /V 0 )	199%	255%	322%	333%	313%
	E(θ 1 S 1 /V 1 )	205%	230%	134%	32%	3%

Note that the condition V t-1 ∈ L 0 (R, F t-1 ) is not sufficient for the portfolio manager to observe it when t = 1 as V 0 is not G 0 -measurable.

The notation conv (A) designates the closed convex hull of A, i.e. the smallest convex closed set containing A.

is lower-semicontinous if the following property holds: For all sequence of price processes ((S n u ) u≤t-1 ) n≥1 converging a.s. to a process (S u ) u≤t-1 , and for all z ∈ cl (I t ((S u ) u≤t-1 )|F t-1 ), there exists a sequence (z n ) n≥1 such that lim n z n = z and z n ∈ cl (I t ((S n u ) u≤t-1 )|F t-1 ) for all n ≥ 1. Example 3.10. Suppose that d = 1 and

where m t-1 , M t-1 ∈ L 0 (R + , F t-1 ) and m t-1 ≤ M t-1 .

Consider z ∈ cl (I t ((S u ) u≤t-1 )|F t-1 ), i.e. z = α t m t-

and

In the following, we define the closed convex random sets

where B(z, ) is the closed ball of center z = 0 and radius > 0. We say that the mapping z → E t-1 ((S u ) u≤t-1 , z) is convex if, for all α ∈ [0, 1], and z 1 , z 2 ∈ R d , we have

Note that this convexity property above is automatically satisfied if d = 1.

Proposition 3.11. Consider a payoff function g t defined on (R d ) t+1 such that, there exists

. By assumption, we know that for all z ∈ cl (I t ((S u ) u≤t-1 )|F t-1 ), there exists a sequence z n ∈ cl (I t ((S n u ) u≤t-1 )|F t-1 ) such that lim n z n = z. We may suppose that |z -z n | ≤ where > 0 is arbitrarily fixed. By assumption, for all

The dominating affine function ϕ 1 , see Figure 2, is given by:

and

.

A simple computation shows that:

In this case, we have

Observe that the dominating affine function ϕ 1 coincides with the x-axis on the support [S 1 m - 2 , S 1 M + 2 ], see Figure 4. Therefore, V 1 (S 1 ) = g 1 (S 1 ) := 0 and we deduce that θ 1 (S 1 ) = 0.

We finally deduce that

The graph of the payoff function g 1 is represented in Figure 4.

The strategy at time t = 0

In order to determine the strategy θ 0 , we compute the smallest affine function ϕ 0 that dominates g 1 on the conditional support cl(I 1 (S 0 )|F 0 ).

We have V 0 (S 0 ) = g 0 (S 0 ) = 0 and θ 0 (S 0 ) = 0, see Figure 5.

We find that (see Figure 6):

=: g 0 (S 0 ), and

.

We have, see Figure 7: We obtain that (see Figure 9):

x

. Then,

.

We have V 0 (S 0 ) = (S 0 -K) + =: g 0 (S 0 ) and θ 0 (S 0 ) = 1, see Figure 10.

Empirical results

For an observed price S -1 at time t = 0 (which corresponds to the last traded price), and for different strike values K, we test the infimum super hedging strategy by computing the relative error R from a data set of 10 6 simulated prices S t for t ∈ 0, 1, 2. To do so, we wrote a script in Python. The relative error is given by

In the following table 11, empirical results are presented for different values of the strike K and a sample of 10 6 scenarios.

We observe that the executed prices depend on the strike K > 0. Indeed, as expected, the orders we send depend on the payoff function. As K increases, the payoff decreases and, as expected, the option price V 0 decreases. The distribution of S 1 admits two regimes as seen in Figure 13 that correspond to the bid and ask prices.

Notice that the proportion of the portfolio value invested in the risky assets at time t = 1 decreases as the payoff decreases. We also observe that this proportion decreases (resp. increases) when the price S decreases (resp. increases) between time t = 0 and t = 1, i.e. when ∆S 1 < 0 (resp. ∆S 1 ≥ 0). At last, the empirical results obtained for the relative error confirm the efficiency of the super-hedging strategy, see Figure 15.


Laetitia Van Schoors
email: laetitia.van-schoors@univ-eiffel.fr

Nicolas Beauzieres

Thomas Cadu

Olivier Sicot

Emmanuel Keita

Relationship between physicochemical evolution and the failure process of flax fibers aged in water

Keywords: Flax fiber, Ageing, Failure, Griffith, microstructure

Increasing environmental concern has put forward the use of flax fibers instead of glass fibers in composite materials. However, durability performances of these bio-fibers remain one of their main issues. This study focuses on the hydrothermal ageing of flax fibers. Flax tows were immersed in distilled water at a temperature of 80°C for different durations. The effect of the hydrothermal ageing on mechanical properties of flax tows was evaluated. Results showed a strong decrease in the maximal strength and the stiffness by 31% and 49% respectively until one week of ageing. Multi-scale analyses were realized to explain these evolutions. Morphological characterization highlighted a washing of fiber surfaces during the hydrothermal ageing, extracting amorphous components as pectins, lignins and amorphous hemicelluloses from cortical tissues. We showed that this morphological evolution impacted the fiber crystallinity. Based on a mechanical analysis, we showed that amorphous components extraction may be at the origin of the material softening. Moreover, the amorphous phase in particular the natural binder pectin would play a major role in the fiber stiffness but does not modify the flaws at the origin of failure.

Introduction

Renewable resources like flax fibers have regained interest due to an increasing need for environmentalfriendly materials. Flax fibers offer many advantages. They are CO2 neutral, thermal and acoustic insulators and most importantly, they have high specific mechanical properties.

Flax fibers consist of cellulose, hemicellulose, lignin, pectin, waxes and water (cf. Table 1). They are at microscopic scale a complex multi-layer arrangement with a polygonal shape (5 to 7 sides), which is characteristic of bio-fibers. They have a diameter between 10 and 25 µm and length between [START_REF] Moothoo | A study of the tensile behaviour of flax tows and their potential for composite processing[END_REF] Their multi-layer arrangements could be separated into three schematized parts (cf. Fig. 1). The primary cell wall (CWI), 0.2 µm thick, is mainly made up of a little amount of cellulose and hemicelluloses, and an important quantity of pectins [START_REF] Lefeuvre | Analysis of the role of the main constitutive polysaccharides in the flax fibre mechanical behaviour[END_REF][START_REF] Gorshkova | Plant Fiber Formation: State of the Art, Recent and Expected Progress, and Open Questions[END_REF]. The secondary cell wall (CWII) is the thickest (10 µm). This wall is composed of three layers (S1, S2 and S3). The thickest layer of the CWII is S2 and consists of highly crystalline cellulose micro-fibrils embedded in a matrix of amorphous hemicellulose, pectin and lignin [START_REF] Lefeuvre | Analysis of the role of the main constitutive polysaccharides in the flax fibre mechanical behaviour[END_REF][START_REF] Lefeuvre | Elementary flax fibre tensile properties: Correlation between stress-strain behaviour and fibre composition[END_REF][START_REF] Alix | Pressure impact of autoclave treatment on water sorption and pectin composition of flax cellulosic-fibres[END_REF]. The third part of the elementary fiber is called lumen. It is a small cavity, generally contributing to the water uptake [START_REF] Baley | Analysis of the flax fibres tensile behaviour and analysis of the tensile stiffness increase[END_REF]. At the mesoscopic scale, elementary fibers are bound together through a pectinbased interface called the middle lamella [START_REF] Lefeuvre | Analysis of the role of the main constitutive polysaccharides in the flax fibre mechanical behaviour[END_REF][START_REF] Lamon | Investigation of statistical distributions of fracture strengths for flax fibre using the tow-based approach[END_REF].

Fig. 1 Schematic representation of three sections of flax fiber

The chemical composition and the multi-layer arrangement confer to flax fibers very good tensile properties. Baley and Bourmaud [START_REF] Baley | Average tensile properties of French elementary flax fibers[END_REF] studied the mechanical properties of 12 varieties of flax fibers and found that their specific stiffness was higher than the one of glass fiber.

For flax tows (cf. Fig. 2), scale of fibers specimens widely used in multi-layer composites, the overall tensile properties could be slightly different. At the microscopic scale, the elementary flax fibers, which tensile properties have been discussed above, are widely studied. At the mesoscopic scale, the fiber bundle or the technical fiber (50 -100 µm diameter) consists of multiple elementary fibers (10 -40 elementary fibers) tied together mainly by pectin through the middle lamella [START_REF] Lefeuvre | Analysis of the role of the main constitutive polysaccharides in the flax fibre mechanical behaviour[END_REF][START_REF] Lamon | Investigation of statistical distributions of fracture strengths for flax fibre using the tow-based approach[END_REF]. Finally, at the macroscopic scale, the flax tows are made up of an assembly of flax technical fibers aligned longitudinally so the overall cohesion is ensured by the pectin binders and some entanglement between fibers. This architecture gives strength to the tows as a whole [START_REF] Stamboulis | Effects of environmental conditions on mechanical and physical properties of flax fibers[END_REF]. Tensile properties of flax technical fibers and tows are significantly lower than tensile properties of elementary fibers. For identical lengths, the tensile strength of a technical fiber could be much lower than the one of an elementary fiber. Bos describes this phenomenon as bundle effects [START_REF] Bos | The potential of flax fibres as reinforcement for composite materials[END_REF]. Beakou and Charlet. [17] suggest this lower strength results from the competition between the tensile breakage of elementary fibers and the shear fracture of the pectin, which bonds the fibers together. For Moothoo et al. [START_REF] Moothoo | A study of the tensile behaviour of flax tows and their potential for composite processing[END_REF], the mechanical properties of flax tows are essentially influenced by the fiber/fiber interface, i.e.

the weak pectin interface between technical fibers. Charlet and beakou [START_REF] Charlet | Mechanical properties of interfaces within a flax bundle -Part I: Experimental analysis[END_REF] have investigated the mechanical properties of the pectin interface within flax bundles. They found that mechanical properties of the interface were dramatically low. At least 90% of the tested samples had lower strength than the weakest fiber/matrix interface.

A summary of the average tensile properties of flax fibers at different scales is presented in As stress is applied, the material stored the energy by elastic deformation. Above a critical strength, it is favorable to broaden preexisting flaws leading to failure [START_REF] Griffith | VI. The phenomena of rupture and flow in solids[END_REF]. This general energetic approach, develop by Griffith [START_REF] Griffith | VI. The phenomena of rupture and flow in solids[END_REF][START_REF] Griffith | The theory of rupture[END_REF] has been applied to explain failure at the microscopic scale in various materials [START_REF] Shah | Griffith fracture criterion and concrete[END_REF][START_REF] Xiao | Stress analysis for a Zener-Stroh crack interacting with a coated inclusion[END_REF][START_REF] Lazarus | Fracture spacing in tensile brittle layers adhering to a rigid substrate[END_REF][START_REF] Rice | A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks[END_REF][START_REF] Rice | Mathematical analysis in the mechanics of fracture[END_REF][START_REF] Dubois | Viscoelastic crack growth process in wood timbers: An approach by the finite element method for mode I fracture[END_REF]. In this study, we refer to this theory to understand the relationship between macroscopic behavior and modification of the micro-structure.

The mechanical properties of the flax fibers could be influenced by other parameters and could be strongly dependent on the water content because of their hydrophilic nature. The main components involved in the water uptake of the fiber are cellulose, hemicellulose and pectin, due to their large number of hydroxyl groups [START_REF] Mokhothu | Review on hygroscopic aging of cellulose fibres and their biocomposites[END_REF][START_REF] Célino | The hygroscopic behavior of plant fibers: a review[END_REF][START_REF] Célino | Qualitative and quantitative assessment of water sorption in natural fibres using ATR-FTIR spectroscopy[END_REF]. Thuault et al. [START_REF] Thuault | Effects of the hygrothermal environment on the mechanical properties of flax fibres[END_REF] have studied the influence of water ageing of flax fibers on the mechanical characteristics. Fibers were immersed up to two months at 20°C and 70°C. Both temperatures seemed to have the same effect on mechanical properties. Thus, after two months of ageing, the failure strength decreased by more than 60%, whereas the stiffness and the failure strain decreased by 55% and 26% respectively. Stamboulis et al. [START_REF] Stamboulis | Effects of environmental conditions on mechanical and physical properties of flax fibers[END_REF] have suggested that high-water content within the fiber might lead to the disorganization of the micro-fibrils network, resulting in lower mechanical properties without highlighting it.

Le Duigou et al. [START_REF] Duigou | Etude des mécanismes d'adhérence entre une fibre de lin et le PLLA-Influence d'un traitement faiblement impactant à l'eau Study of adherence mechanism between flax fibre and PLLA matrix-Influence of low environmental impact water treatment[END_REF] have shown that the immersion (or washing) for 72h at 23°C of flax fibers leads to a decrease of about 15% of failure strength and about 18% of the failure strain, but stiffness remained stable. According to these authors, in these conditions, those diminutions are non-relevant and they concluded that the washing at 23°C for 72h did not significantly affect the mechanical properties of the fibers. Bourmaud et al. [START_REF] Bourmaud | Importance of fiber preparation to optimize the surface and mechanical properties of unitary flax fiber[END_REF] analyzed the components extracted by immersion of flax fibers at 23°C for 72h, then 100°C for 1h. They found that immersion at room temperature extracted mainly oligomers from cortical tissues, while boiling water extracted polysaccharides from pectins. The same result was found by Alix et al. [START_REF] Alix | Pressure impact of autoclave treatment on water sorption and pectin composition of flax cellulosic-fibres[END_REF] using autoclave treatments of flax fibers. However, the influence of hydrothermal ageing was studied only at the elementary fiber scale.

The present paper aims at studying and understanding the effect of hydrothermal ageing on mechanical properties and embrittlement at the scale of the flax tows. We immersed flax tows for weeks in hot water at 80°C and measured mechanical and physicochemical properties. Comparing macroscopic behavior and molecular information, we show that extraction of amorphous phase plays a minor role on the macroscopic behavior but degradation of the amorphous compounds especially pectin is responsible for the softening of the material after ageing.

Material and methods

Material and hydrothermal ageing

The flax fibers used for this study were cultivated in Normandy and supplied by the company LINEO ©.

The manufacturing process begins with shelling and extracting the bundles from the flax stems. They are then scutched to clean them and to remove the shive residue. The fibers are aligned, stretched, then moistened with a water mist and continuously dried by passing through an infrared oven. The reinforcement is in the form of long fibers arranged unidirectionally without twisting. Then, the flax bundles are assembled in the form of a purely unidirectional reinforcing tape 40 cm wide, with a basis weight of 110 gr / m² and branded under the name FlaxTape ™ 110. No weft thread is used to maintain the alignment of the fibers between them. Cohesion is guaranteed by the pectic binder on the surface of the fibers. Two layers (40 cm width and 35 cm length) were stored at 23°C / 50% relative humidity (RH) for one week without any drying or chemical pre-treatments, in order to keep the fibers in the same conditions before every test. Hydrothermal ageing was performed in an oven (MEMMERT) with a temperature set at 80°C. Flax tow specimens were randomly cut from a layer, then immersed in distilled water at 80°C. The ageing durations were 1 week and 3 weeks. After ageing, the tows were conditioned one week at 23°C / 50% RH prior to tensile testing and microstructure analysis to limit the fluctuations in water content of the fiber, which are responsible for the variability in the mechanical properties of the flax fibers. So, the potential difference of water content will be in this case related to the fiber hydrophilic characteristic and not at the humidity rate in which the fibers are stored.

Tow specimens

A gauge length of 150 mm (5 mm width, 210 ± 10 µm thick) with a linear density de 0.55kg/km was used for tensile tests in this study, as recommended by Moothoo et al. [START_REF] Moothoo | A study of the tensile behaviour of flax tows and their potential for composite processing[END_REF] (Fig. 3). This gauge length was chosen because, according to Bos [START_REF] Bos | The potential of flax fibres as reinforcement for composite materials[END_REF], for a gauge length shorter than 25 mm, the behavior of the technical fiber was influenced by elementary fibers, resulting in high strengths. Besides, for gauge length longer than 100 mm, the tensile properties are stable. So, for this author, when the gauge length is higher than the average elementary fiber length (15 to 36 mm), the failure presumably occurs through the pectin interface of the fibers. Moothoo et al. [START_REF] Moothoo | A study of the tensile behaviour of flax tows and their potential for composite processing[END_REF] have observed the same behaviors and have pointed out a stabilization of the tensile properties as the gauge length increases.

After ageing, the specimens were dried at ambient air over one day, and then stuck on a thin aluminum sheet. Afterwards, as already mentioned, they were conditioned at 23°C / 50% RH for 1 week before tensile tests. with a 2 kN load cell at room temperature. Force-strain curves were then recorded at a constant crosshead displacement rate of 1 mm.min -1 . 25 specimens were tested for raw fibers, the same number of specimens were tested after 1 week and 3 weeks of immersion.

Physicochemical analysis of the fiber

A Scanning Electron Microscope (SEM FEI QUANTA 400) was used in the secondary electrons mode, at 7.5kV voltage and at a working distance of 7.5 ± 0.1mm. The samples were bonded on adhesive carbon taps in order to inspect the surface of the fiber and the effect of water on these fibers. In order to avoid electronic charging, the specimens have been coated with a thin carbon layer.

The fiber crystalline microstructure was determined by X-ray diffraction (XRD) analysis in reflection The two well-defined peaks observed around 17° are in accordance with Tserki et al. [START_REF] Tserki | A study of the effect of acetylation and propionylation surface treatments on natural fibres[END_REF].

The chemical composition of the first few µm of the flax fibers surface was analyzed using Attenuated Total Reflectance Fourier Transform Infrared (ATR-FTIR) method. Spectra were recorded using a Nicolet impact 380 spectrometer and Durascope Diamond ATR equipment. 32 scans were collected on raw and aged fibers over the 4000-650 cm -1 spectral range, with a resolution of 4 cm -1 .

Dynamic Vapor Sorption (SMS DVS-ADVANTAGE) was used to investigate the water uptake of flax fibers after hydrothermal ageing. The DVS is able to point out highly accurate mass changes in humid environments. Two ramps were recorded: 36 hours at 25°C and 90% RH, 36 hours at 25°C and 40% RH.

3. Results

Tensile behavior of flax tows

The tensile behaviors of flax tows were characterized on one-week conditioned and aged specimens.

Tensile tests were performed afterwards and their typical force-strain curve is provided Figure 5. The tensile behavior of flax tows displays four periods. The first period (0 -0.05% strain) is non-linear whereas the second (0.05 -0.3% strain) is linear and was considered to calculate the stiffness of the tow. In the third period (0.3 -0.7% strain), the stress increase is no longer linear and reaches a maximum (c) for a critical strain of c. Unlike elementary fibers, a second modulus E2 cannot be determined between 0.25 and 0.5% strain because the slope is not linear between 0.3 and 0.5% strain for flax tows. At higher strain, the force decreases until the complete failure of the tow. This general behavior is similar for aged samples with smaller stiffness, maximal strengths and higher critical strain.

For our results to be statistically significant, each experiment was repeated on 25 samples. The evolution of the mechanical properties of flax tows as a function of the ageing time is shown in Fig. 6. During the first week of ageing, water immersion affects the mechanical properties of the tows.

Mechanical properties slightly evolve between one and three weeks. The average maximal strength and the average elastic modulus decreased by 25% and 44% respectively after one week of ageing; after three weeks these decreases reach 37% and 57%. Besides, the failure strain increased by 40% in the same period. The standard deviation on tensile tests are high between 10 and 30% of the absolute values, nevertheless the evolution of elastic modulus and critical strain after one week is significant.

However, the slight modification of the properties between one and three weeks is of the order of the uncertainty of the protocol.

To understand the mechanical evolutions, multi-scale analyses have been performed.

Physicochemical analysis

Firstly, the effect of one week of ageing on the hydrophilic behavior of the flax fiber has been evaluated by Dynamic Vapor Sorption (DVS). Fig. 7 presents the moisture adsorption behavior of raw fibers (black curve) and fibers after 1 week of hydrothermal ageing (red curve). A first isotherm was carried out at 25°C and 50% RH for 60 min. Two ramps were then processed: the first one at 25°C / 90% RH during 36h and the second one at 25°C / 40% RH during 36h. It appears that the moisture uptake of raw fibers is slightly higher (15%) than for 1-week-aged fibers up to 36h at 90% RH. Also, at 40% RH, the release of water is more important for raw fibers. These results

mean that fibers after one week of ageing are less sensitive to humidity than non-aged fibers. This difference could be associated to an extraction of hydrophilic components present in the fibers, such as pectin, hemicellulose and cellulose. Concerning lignin, which is a hydrophobic component, Hill et al. [START_REF] Hill | The water vapor sorption behavior of natural fibers[END_REF] showed by dynamic vapor sorption that moisture nevertheless increases with the lignin fraction. They explain that it is due to the ability of the lignin network to deform and accommodate water within the cell walls.

In order to assess if the decrease of hydrophilic behavior could be related to the fiber surface evolution, Scanning Electron Microscopy (SEM) observations were performed before and after hydrothermal ageing. At the technical fiber scale, fiber surfaces appear cleaner after one week of ageing in water at 80°C (Fig. 8). So, components present on the surface of the fibers have been extracted during ageing.

However, it seems that elementary fibers are still held together, probably by a part of the middle lamella [START_REF] Lefeuvre | Analysis of the role of the main constitutive polysaccharides in the flax fibre mechanical behaviour[END_REF]. Hydrothermal ageing would therefore lead to the extraction of hydrophilic compounds present especially at the surface of the fibers.

Fig. 8 SEM Scan of technical fiber surfaces before and after 3 weeks of ageing in water at 80°C

In order to characterize the microstructure of extracted components on the fibers by hot water, X-ray diffraction (XRD) analyses have been carried out.

Fig. 9 shows the evolution of the crystallinity of the fiber as a function of the ageing time. The crystallinity ratio slightly increases with ageing time (cf. Fig. 9). This slight increase of crystallinity could be attributed to a part of the amorphous phase which would have been crystallized or have been extracted. With regards to the morphological analysis, extraction by water seems to be favored. Thus, the amorphous phase decreases from 27 to 24%, extraction represents around 10% of the amorphous phase.

To confirm this interpretation, chemical analysis of fiber was performed. Fig. 10 shows IRTF spectra of fibers before and after water ageing. Spectra were baseline-corrected and normalized at 1160 cm -1 This peak is attributed to the C-O-C stretching asymmetric bridge of the cellulose and does not evolve during ageing [START_REF] Alix | Pressure impact of autoclave treatment on water sorption and pectin composition of flax cellulosic-fibres[END_REF][START_REF] Célino | The hygroscopic behavior of plant fibers: a review[END_REF][START_REF] Célino | Qualitative and quantitative assessment of water sorption in natural fibres using ATR-FTIR spectroscopy[END_REF]. As shown in Fig. 10, water seems to affect bands located at 1247cm -1 and 1726cm -1 . After one week of ageing in water and even more after three weeks, a decrease of these peaks amplitude has been observed. Peaks at 1726cm -1 are attributed to C=O stretching vibration of hemicellulose ester groups and carboxylic acid groups of pectins [START_REF] Célino | The hygroscopic behavior of plant fibers: a review[END_REF][START_REF] Célino | Qualitative and quantitative assessment of water sorption in natural fibres using ATR-FTIR spectroscopy[END_REF]. The 1247cm -1 peak corresponds to C-O functions of these groups.

At the end of each ageing time, the liquid of the bath was dried using a rotary evaporator at 70°C and a constant pressure of 90 mbar. The dry extract was retrieved, then analyzed. Fig. 11 shows the infrared spectra of dry extracts after 1 week and 3 weeks of immersion in distilled water at 80°C. Several welldefined peaks and shoulders can be noticed. The assignment of bands and shoulders are shown in Table 3. Firstly, the peak at 1587cm -1 reveals the presence of aromatic rings found in lignins. Secondly, peaks at 1030cm -1, 1230cm -1 , 1400cm -1, and shoulders at 1635cm -1 and 1726cm -1 confirm the extraction of hemicellulose and pectin during the immersion of flax fibers in water at 80°C. So, this hydrothermal ageing at 80°C until 3 weeks leads to a decrease of tows mechanical properties.

The physicochemical analyses have highlighted that this ageing leads to extraction of hydrophilic amorphous components, especially present on the surface of the fibers. The FTIR analyses revealed ester acid and ether functions present in the pectins and hemicellulose. But, as some fiber compounds have similar chemical functions, it was not possible to discriminate the component or components extracted by ageing. This study also reveals the extraction of lignins, consequently, the decrease of tows mechanical properties induced by ageing results from the extraction of amorphous surface compounds such as pectins, hemicelluloses and lignins.

In this section, we observed the evolution of flax fibers submitted to ageing in hot water. At macroscopic scale, flax fibers become softer and their strength decreases. We also investigated their hydric response and evolution of chemical composition. In the next section, the relationship between the microscale evolution and the macroscale behavior will be discussed.

Discussion

Based on the mechanical, physical and chemical properties measured while ageing, we performed a multi-scale analysis to identify the degraded components and their effects at macroscopic scale. At the macroscopic scale, the flax tows show a clear weakening after ageing. The failure of the structure occurs at lower stress. Moreover, the decrease of the elastic modulus and the increase of the critical strain shows that the material is softer than before ageing (see Fig. 6).

Griffith's model is a classic theory to study the mechanical failure, [START_REF] Griffith | VI. The phenomena of rupture and flow in solids[END_REF][START_REF] Griffith | The theory of rupture[END_REF][START_REF] Rice | A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks[END_REF][START_REF] Rice | Mathematical analysis in the mechanics of fracture[END_REF][START_REF] Irwin | Analysis of stresses and strains near the end of a crack transversing a plate[END_REF]. This theory compares the energy elastically stored to the one needed to propagate a crack from a preexisting microcrack (length a ). It links the strength (𝜎 𝑐 ) to the elastic modulus (E) as following:

𝜎 𝑐 = √ 𝑊•𝐸 𝑎 ( 1
)
where W is the crack surface energy. Thus, 𝜎 𝑐 2 𝐸 ⁄ , representing the elastic stored energy as the failure starts, depends mostly on the microcrack length.

Here, the maximal strength and the modulus (cf. Fig. 6) decrease with hydrothermal ageing but the elastic stored energy 𝜎 𝑐 2 𝐸 ⁄ is constant (cf. Fig. 12). Thus, based on equation 1, the characteristic size of microcracks leading to failure is unchanged during ageing. Flaws at the origin of failure are not modified. It suggests that the failure process remains the same. It appears from this analysis that the organization of the fibers e.g. the stress localization and the microcracks are not modified by the ageing conditions.

Although the microcracks did not evolved, the macroscopic behavior is strongly affected by ageing.

Indeed, the elastic modulus halves during the hydrothermal ageing, thus the material should be affected and becomes homogenously softer. Some authors [START_REF] Baley | Analysis of the flax fibres tensile behaviour and analysis of the tensile stiffness increase[END_REF][START_REF] Lefeuvre | Analysis of the role of the main constitutive polysaccharides in the flax fibre mechanical behaviour[END_REF] explain these drops in elastic modulus by an increase in diameter due to the cell-wall relaxation. In the case of our study, no increase in diameter was observed However, based on XRD results (cf. Fig. 9), the extracted components represent less than 10% of the amorphous phases (25% of the total mass). Thus, crystalline stiff phase increases its relative fraction during ageing, the general softening of the material is not straightforward. Based on FTIR, we observed an extraction of different amorphous compounds i.e. pectins, hemicelluloses and lignins (see Fig. 10).

Amorphous compounds extraction appears to be at the origin of the general softening of the material, suggesting that this phase, in particular pectin which is the natural binder that ensures cohesion between fibers [START_REF] Moothoo | A study of the tensile behaviour of flax tows and their potential for composite processing[END_REF]17], plays a major role in the stiffness of the fibers tows but does not modify the flaws at the origin of failure.

Conclusion

The paper studied the effect of hydrothermal ageing on mechanical properties and embrittlement at the scale of the flax tows. Fibers were immersed in distilled water at a temperature of 80°C for different durations. Mechanical properties of tows were studied after weeks of immersion. Tensile tests showed that the maximal force and the stiffness decreased by 31% and 49% respectively up to one week and stabilized after three weeks. The failure strain increased by 47% in the same period, and also stabilized after three weeks. SEM micrographs showed that fibers surfaces appeared cleaner after the ageing. In order to explain these tensile evolutions, chemical and microstructural analyses were performed. Thus, X-ray diffraction and IRTF analysis showed that the crystallinity of fibers increased during the hydrothermal ageing, due to the extraction of amorphous components. By compilation of macroscopic behavior and physicochemical evolutions, it appears that the fiber structure is not modified and extraction of the amorphous components, including pectin, is responsible for the softening of the material.

Fig. 2

 2 Fig. 2 Schematic representation of the flax tow adapted from Moothoo et al. [15] (a) cross-sectional view, (b) longitudinal view

Fig. 3

 3 Fig. 3 Flax tow specimen

  mode. A PHILIPS PW 1830 diffractometer with CoKα1 (λ=1,79 Å) radiation set has been used. The 2θ angle range was 4-76° and samples were laid on a silicon disc. One method was considered to assess the fiber crystallinity rate: to use a symmetrical Gaussian function and through Origin TM software, calculate the crystallinity rate by deconvolution. The apparent crystallinity is determined from the ratio of the area of the crystalline peaks to the total area including the amorphous halo with the following equation 1:𝑪 = 𝟏𝟎𝟎.𝑰 𝒄𝒓 𝒑𝒆𝒂𝒌𝟏 +𝑰 𝒄𝒓 𝒑𝒆𝒂𝒌𝟐 +𝑰 𝒄𝒓 𝒑𝒆𝒂𝒌𝟑 𝑰 𝒄𝒓 𝒑𝒆𝒂𝒌𝟏 +𝑰 𝒄𝒓 𝒑𝒆𝒂𝒌𝟐 +𝑰 𝒏𝒐𝒏-𝒄𝒓 +𝑰 𝒄𝒓 𝒑𝒆𝒂𝒌𝟑 (1)C: apparent crystallinity (%), Icr peakx is the area under the crystalline peaks at 16.8°, 18.8° and 25.9° Inon-cr is the area of the amorphous halo.

Fig. 4 X

 4 Fig. 4 X-Ray diffractogram of flax fibers (dark), the deconvolution's elementary peaks (green) and the sum of the peaks (red)

Fig. 5 Fig. 6

 56 Fig. 5 Typical tensile behavior of flax tow unaged ( ___ ), aged 1 week (----) and aged 3 weeks ( ….. )

Fig. 7

 7 Fig. 7 Thermogravimetric behavior of raw fibers (black line), after 1-week ageing (red line) and Relative Humidity (blue dash line)

Fig. 9

 9 Fig.[START_REF] Gorshkova | Plant Fiber Formation: State of the Art, Recent and Expected Progress, and Open Questions[END_REF] Crystallinity ratio of the fibers at each ageing time obtained from XRD

Fig. 10

 10 Fig. 10 Infrared spectra of unaged fibers (black), 1 week-aged (red) and 3 week-aged (blue) (a).Focus on wavenumber between 1780 and 1180 cm -1 (b)

Fig. 11

 11 Fig. 11 Infrared spectra of dry extracts after 1 week (red) and 3 weeks (blue) of immersion

Fig. 12

 12 Fig. 12 Evolution of the volumic stored elastic energy  2 /E as a function of the number of weeks

Table 1 :

 1 Compositions of flax fibers by different authors.

	and 36

Table 2 .

 2

		Strength (MPa)	Stiffness (GPa)	Strain (%)	Ref
	Elementary fiber
	(Single fiber 33 mm length -19mm width)	945 ± 200	52.5 ± 8.6	2.07 ± 0.45	[13]
	Technical fiber
	(10-40 elementary fibers held together)	264 ± 127	-	-	[19]
	Flax bundle (Bunch of technical	166 ± 25	22 ± 2	1.06 ± 0.06	[15]
	fiber held together)
	Fiber/Fiber interface	2.9 ± 2.4	18.7 10 -6 ± 10.1 10 -6	-	[18]

Table 2 :

 2 Average tensile properties of flax fibers at different scales.

Table 3 : Absorption bands characteristic of the dry extracts ([30, 36

 3

, 37])

Acknowledgments

The authors declare that there is no conflict of interest.

Appendix

The flax fibers used in these studies had the following composition