In this thesis we introduce two new types of conditional random sets taking values in a Banach space: the conditional interior and the conditional closure. The conditional interior is a version of the conditional core, as introduced by A. Truffert and recently developed by Lépinette and Molchanov, and may be seen as a measurable version of the topological interior. The conditional closure is a measurable version of the topological closure. These concepts are useful for applications in mathematical finance and conditional optimization as we shall see in the thesis.

In the second part of the thesis we apply the theoretical results established in the first part to solve the problem of super-hedging European or Asian options for discrete-time financial market models where executable 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 numerical procedure to compute the infimum price under a weak no-arbitrage condition, the so-called AIP condition, under which the prices of any 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 the super-hedging problem. We illustrate our method by a numerical example.

In the last part we show that, in discrete-time, it is possible to evaluate the minimal super-hedging price when we restrict ourselves to integer-valued strategies. In fact, usual theory of asset pricing in finance assumes that the financial strategies, i.e. the quantity of risky assets to invest, are real-valued so that they are not integervalued in general, as we can see in the Black and Scholes model for instance. This is clearly contrary to what it is possible to do in the real world. Surprisingly, it seems that there are few contributions in that direction in the literature (see the reference in the last version). To do so, we only consider terminal claims that are continuous piecewise affine functions of the underlying asset. We formulate a dynamic programming principle that can be directly implemented on an historical data, and which also provides the optimal integer-valued strategy.

Résumé

Dans cette thèse nous introduisons deux nouveaux types d'ensembles aléatoires conditionnels à valeurs dans un espace de Banach : l'intérieur conditionnel et la clôture conditionnelle des ensembles aléatoires. L'intérieur conditionnel est une version mesurable des ouverts topologiques, tel qu'introduit par A. Truffert et récemment développé par Lépinette et Molchanov, et peut être vu comme une version mesurable de l'intérieur topologique. La clôture conditionnelle est une généralisation de la notion de support conditionnel d'une variable aléatoire. Ces concepts sont utiles pour des applications en mathématique financière et en optimisation conditionnelle.

Dans la deuxième partie, nous appliquons les résultats théoriques établis dans la première partie pour résoudre le problème de sur-réplication des options européennes ou asiatiques pour les modèles de marchés financiers en temps discret où les prix exécutables sont incertains. Les prix des actifs risqués ne sont pas décrits par des processus à valeur unique mais des sélections mesurables d'ensembles aléatoires qui permettent de considérer une grande variété de modèles, y compris des modèles bidask avec carnets d'ordres, mais aussi des modèles avec un retard dans l'exécution des ordres. Nous proposons un procédé numérique pour calculer le prix minimal sous une condition faible de non-arbitrage, dite condition AIP, sous laquelle les prix positifs des options européennes sont positifs. Cette condition est plus faible que l'existence d'une probabilité risque neutre mais elle est suffisante pour résoudre numériquement le problème de super-couverture. Nous illustrons notre méthode par un exemple numérique.

Dans la dernière partie nous montrons que, dans un modèle de marché en temps discret, il est possible d'évaluer le prix minimal de sur-réplication lorsqu'on se limite aux stratégies à valeurs entières. En fait, la théorie habituelle de l'évaluation des options en finance suppose que les stratégies financières, c'est-à-dire la quantité d'actifs risqués à investir, sont à valeurs réelles de sorte qu'elles ne sont pas à valeurs entières en général, comme on peut le voir dans le modèle de Black et Scholes par exemple. Ceci est clairement contraire à ce qu'il est possible de faire dans le monde réel. Il semble qu'il y'ait peu de contribution en ce sens dans la littérature. Pour ce faire, nous ne considérons que des payoffs qui sont des fonctions continues affines par morceaux de l'actif sous-jacent. Nous formulons un principe de programmation dynamique qui peut être directement implémenté sur une donnée historique et qui fournit également la stratégie optimale à valeurs entières. I want to thank my supervisor Pr. Emmanuel Lépinette who during these years of thesis, have managed this work with all the skills and human qualities that can be expected from a supervisor. He was very patient and I learned a lot from him. I would like to tell him how much his advices were beneficial to me. I am also grateful to Pr. Dorra Bourguiba for her help during this journey and her support. I also like to thank Pr. Yuri Kabanov and Pr. Serguei Pergamenchtchikov for reporting this work. I am grateful to all the members of my thesis committee for accepting to participate in my thesis defense. I want to take a moment to express my sincerest gratitude to Mohamed Amine Ben Amor for his unwavering support throughout my academic journey. His guidance, mentorship, and encouragement have been invaluable to me. Also I want to thank Pr Youssef Azouzi for sharing his knowledge and expertise with me and for his advices. I am very grateful to my family, my friends for their support. Finally, i would like to express my gratitude to my mother, who has been my pillar of strength, my constant support and my source of inspiration.

Chapter 1

General Introduction 1.1 Motivation

In mathematical finance, options plays an important role. They are a powerful tool for investors to manage and hedge any potential risk. In fact, options are contracts that give the holder the right (but not the obligation) to buy or sell the underlying asset at a fixed price on a specified date. As a result, finding a fair price for both the buyer and the seller has been a challenging problem that several mathematicians tried to answer: The seller of the option will deliver a random amount at a specific time to the buyer. In exchange the seller receives from the buyer some amount of capital the we call price.

So, we introduce first the classical theory of arbitrage and pricing for financial markets without frictions. We recall the definition of arbitrage opportunity and then we present the the fundamental theory of asset pricing based on convex duality and the dual characterization of super-hedging prices. We refer to [START_REF] Kabanov | Markets with transaction costs[END_REF] and [START_REF] Shiryaev | Essentials of stochastic finance[END_REF] and the references therein for a general overview.

Framework

In order to determine the price of an asset, we consider a discrete financial market model defined by a complete filtered probability space (Ω, F, F, P) where F := (F t ) 0≤t≤T is a filtration that represents the information that the investor holds at time t and such that F T = F. Let M(R d , F t ) be the set of R d -valued F t -measurable random variables. The random variables are defined up to neglugible set, which means that we work directly with quotient spaces. We consider a price process S := (S 0 t , S t ) 0≤t≤T such that S t ∈ M(R d + , F t ). A strategy θ = (θ 0 t , θ t ) 0≤t≤T is a stochastic process adapted to the filtration F.

The liquidation value at time t using the strategy θ is given by:

V θ t := θ 0 t S 0 t + θ t .S t = θ 0 t S 0 t + d i=1 θ i t S i t Definition 1.2.
1. An European option is a financial contract between a buyer and a seller that gives the buyer the right to execute the contract at a fixed maturity time T and to get a terminal wealth ξ T also known as the payoff. In return he should pay the price of the contract to the seller at time t = 0. For example the payoff ξ T of a Call option is (S T -K) + where K ≥ 0 is a constant called strike.

One of the interesting problems in mathematical finance is to find the price of an option. So we need to recall some definitions and theorems that will be useful. Definition 1.2.2. A portfolio process (V θ t ) t∈{0,...,T } is said to be self-financing if:

θ 0 t-1 S 0 t + θ t-1
.S t = θ 0 t S 0 t + θ t .S t , ∀t = 1, ..., T.

We have the following equivalences:

1. (V θ t ) t∈{0,...,T } is said to be self-financing.

2. Let ∆V θ t = V θ t -V θ t-1 , and ∆S t = S t -S t-1 , then ∆V θ t = θ 0 t-1 ∆S 0 t + θ t-1 .∆S t ∀t = 1, ..., T. In the sequel, we denote by D T 0 the set of all the discounted terminal values Ṽ θ T with initial value Ṽ θ 0 = V 0 = 0 that is:

Let Ṽ

D T 0 = T t=1 θ t-1 .∆ St : θ t ∈ M(R, F t ), t = 0, ..., T -1 .
We denote also by SH T 0 = D T 0 -M(R + , F T ) the set of super-hedgeable claims, i.e ξ T ∈ SH T 0 if and only if there exists V θ T ∈ D T 0 such that V θ T ≥ ξ T a.s. A portfolio process with terminal value V θ T super-replicates ξ T at time T if V θ T ≥ ξ T , a.s. If V θ T = ξ T a.s then V θ T replicates ξ T . Now, let's give the definition of a price: Definition 1.2.3. Let ξ T be a payoff and V θ be a self-financing portfolio process. A price is any initial endowment V θ 0 such that V θ ≥ ξ T . We denote by P(ξ T ) the set of all prices for the payoff ξ T .

We can define now the super hedging price. Let x be the initial capital of a self-financing startegy denoted by V x, θ. If ξ T is a contingent claim and Θ is the set of all strategies, then the (infimum) super-hedging price is denoted by p(ξ T ), defined by: p(ξ)

:= inf p ∈ R, ∃ θ ∈ Θ s.t V p, θ T ≥ ξ T a.s .
Note that it is an infimum price and we are not sure that it is a minimal price. The classical approaches to solve the super-hedging problem is assuming the absence of arbitrage (NA). To do so, let us recall the definition of an arbitrage opportunity. Definition 1.2.4. An arbitrage opportunity is a financial strategy θ starting from a zero initial endowment such that:

• V θ T ≥ 0, P a.s .

• P(V θ T > 0) > 0.

Definition 1.2.5. The (NA) condition holds if there is no arbitrage condition.

This means that it is impossible to make a profit without taking some risk. The (NA) condition can be expressed in the set setting as follow:

• NA holds if D T 0 ∩ M(R + , F T ) = {0}. • NA holds if SH T 0 ∩ M(R + , F T ) = {0}.
In the sequel we state the first fundamental theorem of asset pricing (FTAP) that gives a characterisation of the absence of arbitrage using the martingale criterion first proven by Harrison and Krep in [START_REF] Harrison | Martingales and arbitrage in multiperiod securities markets[END_REF] and Harrison and Pliska in [START_REF] Harrison | Martingales and stochastic integrals in the theory of continuous trading[END_REF] in the case of finite universe Ω while Dalang, Morton and Willinger in [START_REF] Dalang | Equivalent martingale measures and no-arbitrage in stochastic securities market models[END_REF] gave the proof for an arbitrary Ω. There exist several proofs of this theorem based on the Hahn Banach theorem and some results of functional analysis. One can find in [START_REF] Shiryaev | Essentials of stochastic finance[END_REF] another proof where the martingale measures are explicitly constructed using the Esscher conditional transformation. In their seminal work [START_REF] Jouini | Martingales and arbitrage in securities markets with transaction costs[END_REF], Jouini.E and Kallel.H worked on arbitrage theory in presence of transaction costs and in [START_REF] Kabanov | Hedging and liquidation under transaction costs in currency markets[END_REF] extend the result to the case of multi-asset models especially the case of the currency models by considering martingale density and solvency cones rather then measures.

Theorem 1.2.6. The financial market is arbitrage free if and only if there exists a probability measure Q equivalent to P such that the discounted price process S is a Q-martingale measure 1 .

In fact FTAP states that: NA holds if and only if EMM(S) = ∅, where EMM(S) is the set of all equivalent martingale measures. The dual elements of the previous theorem allow to minimal super-replicating price.

Theorem 1.2.7. If NA holds then:

p(ξ) = sup Q∈EMM(S) E Q (ξ).
The second theorem of asset pricing is given by the following theorem:

Theorem 1.2.8. If EMM(S) = ∅, then the following holds: The market is complete if and only if EMM(S) is a singleton.

The completeness of financial markets is an important condition. In fact, if the market is complete then we can compute the super-hedging price as the expectation of the contingent claim under the unique martingale measure which can be determined numerically using for example Monte Carlo methods. But in many cases, this condition doesn't hold true which complicates the task of determining the superhedging price using the dual characterisation. In [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF], the authors introduced a weak no-arbitrage condition called the absence of immediate profit (AIP) that ensures the possibility to price a European claim in a financial market. Definition 1.2.9. We say that a financial market satisfies the AIP condition if the super-hedging price of the zero claim is zero.

Some contributions

In this thesis we introduce in a first place theoretical results that will be used to solve the super-hedging problem. Actually we define two new types of conditional random sets: Conditional interior and conditional closure sets and then we solve the super-hedging problem without any no-arbitrage condition, by considering the price process as measurable selections of random sets, contrarily to what it is usual to do. We then provide a numerical procedure to compute the infimum of the superhedging price under AIP condition. We shall follow the idea of L.Carassus and E.Lépinette [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF] by considering the infimum value of the set of super-hedging prices and then computing the super-hedging price by using the convex duality without any assumptions on the market. To do so, we use the notions of Fenchel-Lengendre conjugate and biconjugate, the conditional essential supremum and the conditional support of random variables. Then we provide the optimal integer-valued strategy and we formulate a dynamic programming principle implemented on an historical data base. Several important notions are defined in the Appendix that will be used throughout the thesis. For further references see [START_REF] Kabanov | Markets with transaction costs[END_REF], [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF], [START_REF] Rockafellar | Variational analysis, volume 317 of Grundlehren der mathematischen Wissenschaften[END_REF] and [START_REF] Aliprantis | Infinite dimensional analysis[END_REF].

Chapter 2

Conditional interior and conditional closure of random sets

Introduction

The conditional essential supremum and infimum of a real-valued random variable have been introduced in [START_REF] Barron | Conditional essential suprema with applications[END_REF]. A generalization is then proposed in [START_REF] Kabanov | Essential supremum with respect to a random partial order[END_REF] for vectorvalued random variables with respect to random preference relations. Actually, these two concepts are related to the notion of conditional core as first introduced in [START_REF] Truffert | Conditional expectation of integrands and random sets[END_REF] and developped in [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF] for random sets in separable Banach spaces, with respect to a complete σ-algebra H. A conditional core of a set-valued mapping Γ(ω), ω ∈ Ω, is defined as the largest H-graph measurable random set Γ (ω) such that Γ (ω) ⊂ Γ(ω). This concept provides a natural conditional risk measure, that generalizes the concept of essential infimum for multi-asset portfolios in mathematical finance. Applications are deduced for geometrical market models with transaction costs: see [START_REF] Lépinette | Risk arbitrage and hedging to acceptability under transaction costs[END_REF] and [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF] and the theory with transaction costs developed in [START_REF] Kabanov | Markets with transaction costs[END_REF].

In this chapter, we first introduce the open version of the conditional core as proposed in [START_REF] Truffert | Conditional expectation of integrands and random sets[END_REF], [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF]. Precisely, if H is a complete sub-σ-algebra on a probability space, the conditional interior (or open conditional core)of a set-valued mapping Γ(ω), ω ∈ Ω, is defined as the largest H-measurable random open set Γ (ω) such that Γ (ω) ⊂ Γ(ω) P-almost every ω ∈ Ω. It may be seen as a measurable version of the classical interior in topology. One of our main contribution is to show the existence and uniqueness of such a conditional interior for an arbitrary random set in a separable Banach space. Then, the dual concept, the conditional closure, is introduced as a generalization of the conditional support of a real-valued random variable to a family of vector-valued random variables. A numerical application of the latter is deduced in conditional random optimization: We show that an essential supremum is a pointwise supremum on a conditional closure.

The chapter is organized as follows. In Section 2.2, we recall the definition and usual properties of measurable random sets in Banach spaces. Then, we introduce the notion of conditional interior and show the existence of such sets in Banach spaces. In Section 2.3, the conditional closure is introduced and an application in conditional optimization is formulated in the next section. At last, we present an application in mathematical finance. The definitions of the conditional support,the conditional essential supremum and the conditional essential infimumum are postponed to the appendix.

Conditional Interior

We consider a complete probability space (Ω, F, P) and a complete sub-σ-algebra H of F. Set-valued mappings we consider take their values in the family of all subsets of a separable Banach space X equipped with its Borel σ-algebra B(X ). In the following, we first recall the concept of graph measurable random set. We then introduce the concept of conditional interior of a random set. Existence is deduced from the existence of the conditional core of a closed random set, see [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF], even if it is not always non-empty.

Recall that a random set Γ(ω), ω ∈ Ω, is a set-valued mapping that assigns to each ω ∈ Ω a subset Γ(ω) of X . We say that Γ is

H-measurable if, for any open set O ⊆ X , {ω ∈ Ω, Γ(ω) ∩ O = ∅} ∈ H.
We say that Γ is H-graph measurable, if We say that a F-measurable random variable ξ : Ω → X is an F-measurable selection of Γ if ξ(ω) ∈ Γ(ω) for almost all ω ∈ Ω. The set of such selections is denoted by L 0 (Γ, F) ⊆ M(X , F) where the set M(X , F) of all X -valued F-measurable random variables is equipped with the metric of convergence in probability.

graph Γ := {(ω, x) ∈ Ω × X : x ∈ Γ(ω)} ∈ H ⊗ B(X ).
The following result may be found in [START_REF] Hess | Set-valued integration and set-valued probability theory: an overview[END_REF]Th. 4.4] and, when applied, it is usually mentioned as measurable selection argument, see also [START_REF] Molchanov | Theory of random sets[END_REF]Theorem 2.3].

Theorem 2.2.2 (Measurable selection argument).

If Γ is an F-graph measurable random set which is non-empty and closed a.s., then L 0 (Γ, F) = ∅.

The following result is proven in [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF]Proposition 2.7] whenever the random set we consider is closed or not. We denote by cl (X) the closure of any subset X ⊆ X with respect to the norm topology. The closure of any subset of M(X , F) is taken with respect to the metric topology of L 0 (X , F). Proposition 2.2.3. Suppose that F is complete. Suppose that Γ is an F-graph measurable random set which is non empty a.s.. Then, cl (Γ) is F-graph measurable and admits a Castaing representation, i.e., there exits a countable family (ξ n ) n≥1 of measurable selections of Γ such that

cl (Γ(ω)) = cl (ξ n (ω) : n ≥ 1), ω ∈ Ω. Moreover, we have cl (L 0 (Γ, F)) = L 0 (cl (Γ), F).
Corollary 2.2.4. If Γ is an F-graph measurable random set, then its topological interior int (Γ(ω)), defined for each ω ∈ Ω, and its boundary ∂Γ(ω) are also Fgraph measurable.

Proof. Indeed, consider Y (ω) = X \ Γ(ω). It is clear that Y is F-graph measur- able. Moreover, we have int (Γ(ω)) = int (X \ Y (ω)) = X \ cl (Y (ω)),
where cl (Y (ω)) is F-graph measurable by the proposition above, hence the complement set int (Γ(ω)) is F-graph measurable. We also deduce that the boundary

∂Γ(ω) = cl (X(ω)) \ int (X(ω)) is F-graph measurable.
For x ∈ X and r ≥ 0, B(x, r) denotes the open ball in X of center x and radius r and B(x, r) is its closure. For any A ⊆ X and λ ∈ R, we use the convention that λ × A = {λa : a ∈ A}. In particular, 0 × A = {0}. Recall the following definition, see the paper by A. Truffert [START_REF] Truffert | Conditional expectation of integrands and random sets[END_REF] and [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF]: Definition 2.2.5. The H-conditional core m(Γ|H), of a set-valued mapping Γ, is the largest H-graph measurable random set Γ such that Γ (ω) ⊆ Γ(ω) a.s.. Note that m(Γ|H) is the largest subset in the sense that, if Γ is another H-graph measurable random set contained in Γ a.s., then Γ(ω) ⊆ m(Γ|H)(ω) P-almost all ω ∈ Ω. The following result is proved in [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF] and is an extension of the primal result of [START_REF] Truffert | Conditional expectation of integrands and random sets[END_REF]. Recall that, if Γ is a random set, L 0 (Γ, H) ⊆ M(X , H) is the set of all H-measurable random variables γ with values γ(ω) ∈ Γ(ω) a.s.. Proposition 2.2.6. Suppose that Γ is a closed F-graph measurable set that may be empty. Then, the conditional core m(Γ|H) exists and we have

L 0 (m(Γ|H), H) = L 0 (Γ, H). If γ ∈ M(X , H) and r ∈ M([0, ∞), F), m( B(γ, r)|H) = B(γ, ess inf H r).
The conditional core plays a role in mathematical finance as it naturally appears when considering the dynamics of a self-financing discret-time portfolio process (V t ) T t=0 of the form V t-1 ∈ V t + G t , t ≥ 1, where G t is the solvency set, see [START_REF] Kabanov | Markets with transaction costs[END_REF], i.e., V t-1 ∈ m(V t +G t |F t-1 ), see [START_REF] Lépinette | Risk arbitrage and hedging to acceptability under transaction costs[END_REF]. When the σ-algebra is trivial, the conditional core becomes the set of fixed points of X; it is also related to the essential intersection considered in [START_REF] Hess | Essential intersection and approximation results for robust optimization[END_REF]. Notice that the H-measurable conditional core is mainly independent of F, i.e., instead of completing the underlying measurable space with respect to a single (or a dominated family of) probability measure(s), one may pass to the universal completion, and hence avoid postulating the existence of any reference measure.

We now introduce an open version of the conditional core: When ∂O(ω) = ∅, we define d(x, ∂O(ω)) = ∞ for all x ∈ X . Otherwise, ∂O admits a Castaing representation on {∂O = ∅} by Proposition 2.2.3. We deduce that the random mapping (ω, x) → d(x, ∂O(ω)) is F ⊗ B(X )-measurable. Therefore, the random sets

F n := {x : d(x, ∂O) ≥ 1/n} ∩ cl O, n ≥ 1, are closed F-
⊆ n m(F n |H) hence O H ⊆ o(O|H).
We then deduce the general case:

Theorem 2.2.9. For any F-graph measurable random set Γ, the H-graph measurable interior of Γ exists and o(Γ|H) = o(int Γ|H) is unique.

The following lemma is recalled for the sake of completeness. It is used in the proof above.

z n ∈ B(x, r * + n -1 ) such that z n / ∈ O hence r * ≤ z n -x ≤ r * + n -1 .
As the sequence (z n ) n is bounded, using the completness of the Banach space we deduce the existence of a subsequence, that we denote (z n ) n converges to some z in the Banach space. Then, z -x = r * hence z ∈ cl O. In the case where z ∈ O, z n ∈ O for n large enough since O is open, which yields a contradiction. Therefore, z ∈ ∂O. This implies that r * = d(x, z) ≥ d(x, ∂O) and finally r * = d(x, ∂O).

Conditional Closure

We now introduce the concept of conditional closure. The existence is proved in the following theorem which is the second main contribution of this chapter. As previously, F and H are supposed to be complete with respect to some probability measure P.

Definition 2.3.1. The H-graph measurable conditional closure cl (Γ|H) of a set- valued mapping Γ is the smallest H-graph measurable random closed set Γ such that Γ(ω) ⊆ Γ (ω) a.s.. Theorem 2.3.2.
For any F-graph measurable random set Γ, the H-graph measurable conditional closure cl (Γ|H) of Γ exists and is unique. We have

cl (Γ|H) = X \ o(X \ Γ|H).
Moreover, for all measurable selection γ of cl (Γ|H) and ε ∈ M(]0, ∞[, H), for all H ∈ H such that P(H) > 0, we have P({Γ ∩ B(γ, ε) = ∅} ∩ H) > 0.

Proof. The first part is a direct consequence of Corollary 2.2.9. Indeed, we first observe that Γ ⊆ X \ o(X \ Γ|H). Moreover, if Γ H is a closed-valued H-graph measurable set containing Γ, then X \ Γ H ⊆ X \ Γ. We deduce that X \ Γ H ⊆ o(X \ Γ|H) and, finally, X \ o(X \ Γ|H) ⊆ Γ H . Suppose that P({Γ ∩ B(γ, ε) = ∅} ∩ H) = 0 for some H ∈ H and ε ∈ M((0, ∞), H). Therefore, by definition of the conditional closure as a smallest set, we have

cl (Γ|H) = cl (Γ|H)1 Ω\H + cl (Γ|H) ∩ (X \ B(γ, ε)) 1 H .
Indeed, the H-graph measurable set in the r.h.s. above contains Γ by assumption hence it contains cl (Γ|H). We get a contradiction since γ ∈ L 0 (cl (Γ|H), H) a.s. by assumption. Uniqueness is clear as the conditional closure of Γ is the smallest closed set, up to a negligible set, in the sense that it is included in any other H-measurable set containing Γ.

Remark 2.3.3. We may deduce the conditional support of a random variable X ∈ L 0 (X , F). Precisely, there exists a smallest H-graph measurable random closed set denoted by supp H (X) such that P(X ∈ supp H (X)) = 1. It is given by supp H (X) = cl ({X}|H) and is called the H-conditional support of X. Moreover, for all γ ∈ L 0 (supp H (X), H) and ε ∈ M(]0, ∞[, H), for all H ∈ H such that P(H) > 0, we have P({X ∈ B(γ, ε)} ∩ H) > 0.

Notice that the conditional support is necessarily non empty a.s. hence it admits a measurable selection. In the following, we adopt the notation kA = {ka : a ∈ A} for any subset A ⊆ X and k ∈ A.

Ω \ H. Example 2.3.5. Let X = R and γ 1 , γ 2 ∈ M(X , F) be such that γ 1 ≤ γ 2 a.s.. Consider Γ = [γ 1 , γ 2 ]. Then, cl (Γ|H) = [ess inf H γ 1 , ess sup H γ 2 ] ∩ R. Indeed, first observe that Γ ⊆ [ess inf H γ 1 , ess sup H γ 2 ] ∩ R a.s..
= {ξ / ∈ Γ H } ∈ H, d(ξ, Γ H ) > 0. Note that d(ξ, Γ H ) < ∞.
There is a non-null F-measurable subset of A H on which ess sup H r -d(ξ, Γ H ) < r by definition of the essential supremum. Let us define ξ = γ + α(ξ -γ) where α = r/(r + d(ξ, Γ H )). As we have ξ -γ ≤ r + d(ξ, Γ H ) when ess sup H r -d(ξ, Γ H ) < r, we deduce that ξ ∈ Γ and ξ is only F-measurable. Therefore, ξ ∈ Γ H and d(ξ, Γ H ) ≤ ξ -ξ where we may show that ξ -ξ < d(ξ, Γ H ) hence a contradiction. Consider now the case where ess sup H r = ∞. We need to show that Γ H = X . In the contrary case, on a non-null set, we may construct a H-measurable selection ζ of X \ Γ H . Moreover, on a smaller non-null set, r ≥ ζ -γ as ess sup H r = ∞. It follows that ζ ∈ Γ hence ζ ∈ Γ H , i.e., a contradiction.

Applications

Conditional Optimization

The second main contribution is the following. It allows one to compute numerically an essential supremum as a pointwise supremum on the conditional closure. This result is a generalization of [3, Proposition 2.7] and is useful in robust finance dynamic programming, see our example below. In this section we suppose that X = R d . We recall that an integrand h(ω, x), (ω, x) ∈ Ω × X , is a jointly measurable function, which is lower semi-continuous in x and takes values in the extended real line R ∪ {+∞}, see [START_REF] Rockafellar | Variational analysis, volume 317 of Grundlehren der mathematischen Wissenschaften[END_REF]Corollary 14.34].

Lemma 2.4.1. Let h(ω, x), (ω, x) ∈ Ω × X , be an H ⊗ B(X )-measurable integrand. Let Γ be a non-empty closed F-graph measurable set of X . Then, sup x∈cl (Γ|H)(ω) h(x) = sup n h(γ n ) where (γ n ) n∈N is a Castaing representation of cl (Γ|H).

Proof. As cl (Γ|H) is H-graph measurable and closed-valued, it admits a Castaing representation cl (Γ|H)(ω) = cl{γ n (ω) : n ∈ N} a.s. where, for all n, γ n ∈ L 0 (cl (Γ|H), H), by Proposition 2.2.3. Notice that we may adjust the values on a set of measure zero, and therefore assume that the equality holds everywhere on Ω, see the proof of [START_REF] Kabanov | Markets with transaction costs[END_REF]Proposition 5.4.4].

As

(γ n ) n ⊂ cl{γ n : n ∈ N} = cl (Γ|H), h(γ n ) ≤ sup x∈cl (Γ|H) h(x). Therefore, sup n h(γ n ) ≤ sup x∈cl (Γ|H) h(x). Moreover, if x ∈ cl (Γ|H)(ω)
, we may write almost surely x = lim n γ n (ω) by the Castaing representation (γ n ) n of cl (Γ|H). By the lower semicontinuity of h, we get that h(x) ≤ lim inf n h(γ n ). Thus, h(x) ≤ sup n h(γ n ) and sup x∈cl (Γ|H)(ω) h(x) ≤ sup n h(γ n ) on Ω. The equality is then deduced. Theorem 2.4.2. Let h(ω, x), (ω, x) ∈ Ω×X , be an H⊗B(X )-measurable integrand. Let Γ be a non-empty closed F-graph measurable set of X . Then, with the notation h(γ)(ω) = h(ω, γ(ω)), we have

ess sup H {h(γ) : γ ∈ L 0 (Γ, F)} = sup x∈cl (Γ|H)(ω)
h(x) = sup n h(γ n ) ≤ ess sup H {h(γ), γ ∈ L 0 (cl (Γ|H), H)}.
If γ ∈ L 0 (cl (Γ|H), H), then γ ∈ cl{γ n : n ∈ N} i.e., γ = lim n γ n for a subsequence and, by lower semi-continuity,

h(γ) = lim inf n h(γ n ) ≤ sup n h(γ n ) = sup x∈cl (Γ|H) h(x).
Since the family {h(γ), γ ∈ L 0 (cl (Γ|H)} is directed upward, we also deduce that sup x∈cl (Γ|H) h(x) = lim n ↑ h(γ n ) where γ n ∈ L 0 (cl (Γ|H), H).

Step 2. We show that ess sup H {h(γ)

: γ ∈ L 0 (Γ, F)} = sup x∈cl (Γ|H) h(x) a.s.. First notice that sup x∈cl (Γ|H) h(x) = sup n h(γ n ) is H-measurable, since h is H ⊗ B(X )-measurable. Moreover, as Γ(ω) ⊆ cl (Γ|H)(ω) a.s., we deduce that h(γ) ≤ sup x∈cl (Γ|H) h(x), for every γ ∈ L 0 (Γ, F). We deduce that ess sup H {h(γ) : γ ∈ L 0 (Γ, F)} ≤ sup x∈cl (Γ|H) h(x) a.s..
To show the reverse inequality, consider any H-measurable selection γ of cl (Γ|H) and a deterministic sequence ε n > 0 with lim n ε n = 0. Let us define

Λ n = {Γ ∩ B(γ, ε n ) = ∅} ∈ F.
By Theorem 2.3.2, P(Λ n |H) > 0 a.s.. Indeed, in the contrary case, on a non-null H-measurable set Λn , we have P( Λn ∩ Λ n |H) = 0 hence P( Λn ∩ Λ n ) = 0, i.e., a contradiction with Theorem 2.3.2.

By a measurable selection argument, we consider γn ∈ L 0 (Γ, F) such that γn ∈ B(γ, ε n ) on Λ n . We define Λn = {γ n ∈ B(γ, ε n )}. Since Λ n ⊆ Λn , we have P( Λn |H) > 0 a.s.. Moreover, using the conditional expectation, we have

ess sup H {h(γ) : γ ∈ L 0 (Γ, F)}1 Λn ≥ h(γ n )1 Λn , ess sup H {h(γ) : γ ∈ L 0 (Γ, F)}P( Λn |H) ≥ E(h(γ n )1 Λn |H), a.s.. (2.4.1)
Moreover, we have

E(h(γ n )1 Λn |H) ≥ E( inf z∈ B(γ,εn) h(z)1 Λn |H).
Notice that inf z∈ B(γ,εn) h(z) = inf z∈X h(z) where h = h on B(γ, ε n ) and h = +∞ otherwise. As h is also an integrand, we deduce by [START_REF] Rockafellar | Variational analysis, volume 317 of Grundlehren der mathematischen Wissenschaften[END_REF]Theorem 14.37

h(z) = h(z n ) where z n ∈ B(γ, ε n ) converges pointwise to γ as n → ∞.
We finally deduce by lower semi-continuity that

ess sup H {h(γ) : γ ∈ L 0 (Γ, F)} ≥ lim inf n h(z n ) ≥ h(γ).
This inequality holds for any selection γ of cl(Γ|H). Therefore, we get that

ess sup H {h(γ) : γ ∈ L 0 (Γ, F)} ≥ ess sup H {h(γ), γ ∈ L 0 (cl(Γ|H), H)}.
The conclusion of the lemma follows.

Recall that a set Λ of measurable random variables is said F-decomposable if for any finite partition (F i ) n i=1 ⊆ F of Ω, and for every family (γ i ) n i=1 of Λ, we have n i=1 γ i 1 F i ∈ Λ. Decomposability was initially introduced by Rockafellar: see also [START_REF] Hiai | Integrals, conditional expectations, and martingales of multivalued functions[END_REF]. In the following, we denote by Σ(Λ) the F-decomposable envelope 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 [START_REF] Kabanov | Markets with transaction costs[END_REF]Proposition 5.4.3], there exists an F-graph measurable closed random set σ(Λ) such that Σ(Λ) coincides with L 0 (σ(Λ), F), the set of all measurable selectors of σ(Λ).

Theorem 2.4.3. Let h(ω, x), x ∈ X , be an H ⊗ B(X )-measurable integrand. Let us consider a family Λ of measurable random variables so that

Σ(Λ) = L 0 (σ(Λ), F)
is the set of all measurable selectors of some F-graph measurable random closed set σ(Λ). Then,

ess sup H {h(γ) : γ ∈ Λ} = ess sup H {h(γ) : γ ∈ Σ(Λ)} = sup x∈cl (σ(Λ)|H) h(x).
Proof. Notice that for any finite partition (F i ) n i=1 ⊆ F of Ω, n ≥ 1, and for every family (γ i ) n i=1 of Λ, we have

h n i=1 γ i 1 F i = n i=1 h(γ i )1 F i .
Therefore, as ess sup H {h(γ) : γ ∈ Λ} ≥ h(γ) a.s. for any γ ∈ Λ, we deduce that ess sup H {h(γ) : γ ∈ Λ} ≥ h(γ) a.s. for any γ ∈ Σ(Λ). Since h is l.s.c. and any γ ∈ Σ(Λ) is a limit of elements of Σ(Λ), we get that the inequality also holds for any γ ∈ Σ(Λ). Taking the essential supremum over all γ ∈ Σ(Λ), we deduce that

ess sup H {h(γ) : γ ∈ Λ} ≥ ess sup H {h(γ) : γ ∈ Σ(Λ)}
and, finally, the equality holds since Λ ⊆ Σ(Λ). The last equality of the corollary is deduced from Theorem 2.4.1.

Application in Finance: Robust Super-Hedging of an European or Asian Option

We consider a financial market in discrete time defined by a complete stochastic basis (Ω, (F t ) T t=0 , P). We suppose that there is a non-risky asset whose price is S 0 = 1, without loss of generality. The (discounted) prices are modeled by a vector-valued stochastic process (S t ) T t=0 adapted to the filtration (F t ) T t=0 with values in R d , d ≥ 1. We consider the one step super-hedging problem between two dates t -1 and t with t ≥ 1. 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. We denote by S t 0 the price process (S u ) u≤t , for all 0 ≤ t ≤ T. More precisely, the portfolio manager knows S t-2 0 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 is only observed strictly after t -1.

Let us consider, for each t ≤ T , Λ t ⊆ L 0 (R d + , F t ) an F t -measurable random set representing the possible prices for the risky assets at time t. We suppose that, at time t, the set Λ t may depend on the observed prices before time t -1, i.e. to each vector of prices S t-1 0 , we associate a set Λ t = Λ t (S t-1 0 ) representing the possible next prices at time t given that we have observed the executed prices S t-1 0 . Therefore: Definition 2.4.4. A price process is an

(F t ) T t=0 -adapted non-negative process (S t ) T t=-1 such that S t ∈ Λ t (S t-1 0 ) for all t = 1, • • • , T and S -1 ∈ R d is given . Note that S t represents the prices (S 1 t , • • • , S d t )
of the 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 ) d j=1 and (S aj ) d j=1 are respectively the bid and the ask price processes observed in the market at time t that may depend on S t-1 0 . 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 the result of an agreement between sellers and buyers but it also depends on the traded volume. Clearly, the portfolio manager does not benefit from the last 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 could be when Λ t coincides with a parametrized family {S θ t : θ ∈ Θ} of random variables. For instance, consider fixed processes (ξ u ) u≤T and (m u ) u≤T adapted to (F t ) T t=0 and independent of F t-1 . Let C be a compact set and suppose that S -1 is given. We define recursively

Λ t (S t-1 0 ) = {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 (e.g. volatility) σ.

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 t , 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)-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 t-1 0 . Our 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 1 , • • • , S t ), a.s. for all S t ∈ Λ t (S t-1 0 ), (2.4.2)
for some θ t-1 ∈ L 0 (R d , F t-1 ). We observe that, by lower-semicontinuity, (3.3.2) holds if and only if

V t-1 + θ t-1 ∆S t ≥ g t (S 1 , • • • , S t ), for all S t ∈ Σ(Λ t (S t-1 0 ). (2.4.3)
Recall that Σ(Λ t (S t-1 0 ) is defined in the previous section. This means that we may suppose w.l.o.g. that Σ(Λ t (S t-1 0 )) = Λ t (S t-1 0 ). In the following, we denote by

I t (S t-1 0 ) the F t -measurable closed random set such that Σ(Λ t (S t-1 0 )) = L 0 (I t (S t-1 0 ), F t ): see [45, Theorem 2.4]. By Theorem 2.4.2, we deduce that (3.3.3) is equivalent to V t-1 ≥ p t-1 where p t-1 = p t-1 (S t-1 0 , θ t-1 ) is given by p t-1 = θ t-1 S t-1 + sup z∈cl (It(S t-1 0 )|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, 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 1 , • • • , S t-1 , z) + δ cl (It(S t-1 0 )|F t-1 ) (z), where δ cl (It(S t-1 0 )|F t-1 ) ∈ {0, ∞} is infinite on the complimentary of cl (I t (S t-1 0 )|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 t-1 0 )|F t-1 )) of convex and l.s.c. functions. Moreover, by Theorem 2.4.2, (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 t-1 0 ) = 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 t-1 0 ) = ess inf F t-1 P t-1 (S t-1 0 ).
To do so, we use the arguments of [3, Theorem 2.8] and we obtain that:

p t-1 (S t-1 0 ) = 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 ∈ Dom 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 ).
In the following, we suppose that, for all price process S t-1 0 , 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 1 , • • • , S t-1 , x) ≤ α t-1 x + β t-1 , ∀x ∈ cl (I t (S t-1 0 )|F t-1 ).
This is the case for instance for Asian options whose payoff is of the form k(S 0 +

S 1 + • • • + S t -K) + , k > 0. By [3, Theorem 2.8],
we then deduce that

p t-1 (S t-1 0 ) = inf αS t-1 + β : αx + β ≥ g t (S 0 , • • • , S t-1 , x), ∀x ∈ cl (I t (S t-1 0 )|F t-1 ) .
In this section, we have solved the super-hedging problem without any noarbitrage condition, contrarily to what it is usual to do.

Conclusions

In section 2.4.2, we have solved the general super-hedging problem in one step for an Asian option. The next step is to repeat the whole procedure to deduce backwardly the infimum prices and the associated super-hedging strategy from the maturity date T to the starting date t = 0. It will show the relevance of the conditional closure.

The conditional closure could be also useful more generally for robust finance dynamic programming as in the paper [START_REF] Elkaraoui | Capacities, measurable selection and dynamic programming part i: abstract framework[END_REF]. We conjecture that it is possible to solve a discrete-time stochastic control problem through random set conditioning.

At last, some interesting problems leave open in the direction of conditional topologies: see [START_REF] Drapeau | The algebra of conditional sets and the concepts of conditional topology and compactness[END_REF]. A deeper study of the basic properties of the conditional closure and interior of random sets may be interesting with a comparison to the classical results of topology but also with the paper by Truffert [START_REF] Truffert | Conditional expectation of integrands and random sets[END_REF]. This also allows to consider new types of martingales, see [START_REF] Hiai | Integrals, conditional expectations, and martingales of multivalued functions[END_REF], and, in continuous time, new problems should arise.

Chapter 3

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

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 available in 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 in advance the traded prices. In that case, as proposed in the chapter, the order that the trader sends is a mapping that associates to each possible price available in 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 price.

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 is kept constant 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 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] Oksendal | 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] Devalière | 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] or in [START_REF] Devallière | No-arbitrage criteria for financial markets with transaction costs and incomplete information[END_REF].

Another type of uncertainty is due to 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 | Risk, uncertainty, and profits[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] Bechererc | 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] Oblój | 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 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 them through measurable random sets.

Moreover, a less common type of uncertainty is introduced in this chapter. Recall that it is usual in the literature, even in the recent papers on robust finance, to suppose that the transactions are executed at a price which is known in advance. For example, in the Black and Scholes model, the delta-hedging strategy for the European Call option at time t is a function Φ(t, S t ) of the single price S t observed at time t. In practice, the strategy is discretized at some dates (t i ) i=0,••• ,n with n → +∞ so that the number of stocks to trade at time t i is ∆Φ t i = Φ(t i , S t i ) -Φ(t i-1 , S t i-1 ). In the case where ∆Φ t i < 0, the executed price at time t i should be a bid price in the order book and an ask price otherwise, i.e. there should be at least two possible prices.

We take into account this ambiguity or uncertainty in our chapter by assuming that there may be several possible executable prices at the next instant. This means in particular that we do not know in advance the price when we send an order to be executed. Precisely, an executed price S t at time t is only F t+1 -measurable where F t describes the market information available at time t. This is illustrated in our numerical example where the stock price is modeled by a pair of bid and ask prices. This chapter addresses the super-hedging problem of European or Asian options under uncertainty and may be easily adapted to American options in discrete time.

Here the uncertainty mainly refers to the uncertainty in executed prices due to the delay, which is modeled by random sets, and there is one single physical probability measure. Moreover, uncertainty may also refers to the presence of an order book so that several prices may exist and depend on the traded volumes. The advantage of the approach we consider is its flexibility, including a large variety of possible models, e.g. with transaction costs or limit order books. Contrarily to the classical approach, we do not suppose the existence of a risk-neutral probability measure but we work under the AIP condition of [START_REF] Baptiste | Pricing without martingale measures[END_REF], [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF], i.e. we suppose that the super-hedging prices of the non-negative European claims are non-negative, as it is easily observed in the real financial market. We recall that the AIP condition is weaker than the usual NA condition but it is sufficient to deduce numerically tractable pricing estimations, as illustrated in our numerical example.

The chapter first focuses on the one-period case, see Section 3.3.1, and the multiperiod case is automatically obtained by (measurably) paste all periods together. The one-period hedging problem can be described as:

V t-1 + θ t-1 ∆S t ≥ g t (S 0 , • • • , S t ), a.s. for all S t ∈ Λ t (S t-1 0 )
. Here S t is a possible executed price which is F t+1 -measurable, θ t-1 is a trading strategy which is made at time t -1 and its outcome is revealed at the same time t as S t-1 due to execution delay and, thus, V t-1 , which models the portfolio value at time t -1, is also F t -measurable; g is an Asian option to be hedged while Λ t (S t-1 0 ) represents the set of all possible prices S t that can be traded strictly after time t. We recall that S t 0 denotes the price process (S u ) u≤t , for all 0 ≤ t ≤ T. The problem is essentially converted to the one without delay by taking supremum conditioned on F t in the above equation, and the (minimal) super-hedging price is provided in Theorem 3.3.1 in terms of the concave envelope of some related function restricted on the conditional closure of Λ t (S t-1 0 ), see [START_REF] Elmansour | Conditional interior and conditional closure of random sets[END_REF]. Properties of the hedging price, including continuity, convexity, and measurability are analyzed in Section 3.3.2. These properties are important to deduce backwardly the multi-period case which involves a measurable pasting. The benefit of our approach is its easy implementation as illustrated in Section 3.4. Indeed, roughly speaking, our main results state that we only need to know the range of the future price values in terms of the observed prices to deduce the strategy θ t to be followed. This can be achieved from a historical data. The strategy depends at time t on the price S t , i.e. θ t = θ t (S t ) where S t is only revealed at time t + 1 so that the order a time t is the F t -measurable mapping z → θ t (z) and not θ t (S t ). Note that the executed price S t will depend on the model, e.g. S t may be one of the several bid and ask prices, and the delayed observation of S t at time t + 1 allows to deduce the quantity θ t (S t ) to hold in the portfolio.

Formulation of the problem

Let (Ω, (F t ) t∈{0,...,T +1} , F T , P ) be a filtered complete probability space where T is the time horizon. We suppose that F 0 is the trivial σ-algebra and the σ-algebra F t represents the information available on the market at time 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.

In the following, we shall consider random subsets A of R d , i.e. A = A(ω) may depend on ω ∈ Ω. We then denote by L 0 (A, F t ) the set of all random variables X t which are F t -measurable and satisfies

X t (ω) ∈ A(ω) a.s.. At last, R d + is the set of all x = (x i ) d i=1 ∈ R d such that x i ≥ 0 for all i = 1, • • • , d. Let us consider, for each t ≤ T + 1, Λ t ⊆ L 0 (R d + , F t+1
) a collection of F t+1measurable random variables representing the possible executable prices for the risky assets between time t and time t + 1. 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 t-1 0 , we associate a set Λ t = Λ t (S t-1 0 ) representing the possible next prices S t after time t given that we have observed the executed prices S t-1 0 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 3.2.1. A price process is an (F t+1 ) t=-1,••• ,T -1 -measurable non-negative process (S t ) t=-1,••• ,T such that S t ∈ Λ t (S t-1 0 ) is F t+1 measurable for all t = 0, • • • , T and S -1 ∈ R is given. Example 3.2.2. 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+1 ) with F t+1 = σ(S u , u ≤ t) for all t ≥ 1 and

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 between time t and t + 1 that may depend on S t-1 0 .

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 that depends on the type of order (and 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 executable prices obtained by the manager belong to random intervals.

Example 3.2.3. 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+1 ) t=0,••• ,T and independent of F t . Let C be a compact subset of R and suppose that S -1 is given. We define recursively

Λ t (S t-1 0 ) = {S t-1 exp(σξ t + m t ) : σ ∈ C} , S t-1 ∈ Λ t-1 , t ≤ T,
where Θ = {σ ∈ C}. 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+1 ) t=-1,••• ,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 (F t ) t=0,••• ,T but only adapted to (F t+1 ) 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. Precisely, the portfolio manager may send an F t -measurable order at time t that depends on the uncertain price S t which is only F t+1 measurable. 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 example, the executed quantity should be deduced from an order book as the minimum between 1000 and the number of assets we may obtain for a price less than 145. Then, the executed price is a weighted average of all prices available for less than 145 in the order book.

For such a strategy θ = (θ 0 , θ), we define the portfolio process with initial endowment V 0 ∈ L 0 (R, F 1 ), as the liquidation value

V 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. In the super-hedging problem we solve, we expect orders which are mapping x → ∆θ t (x) = θ t (x) -θ t-1 (S

V θ t = V 0 + t u=1 θ u-1 ∆S u . (3.2.1)
It is then natural by (3.2.1) to write

V θ = V θ.
The aim of the chapter 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 g 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 = V θ 0 such that we have V θ T ≥ g(S 0 , S 1 , • • • , S T
) independently of the value of the executable prices S t ∈ Λ t (S t-1 0 ) are for t ≤ T . Note that V t is F t+1 -measurable hence one more step is necessary to deduce the initial endowment P 0 at time t = 0 we need for initiating a super-hedging portfolio process V , i.e. P 0 ≥ V 0 . Indeed, P 0 should be F 0 -measurable, i.e. a constant, or equivalently P 0 ≥ esssup F 0 (V 0 ). We refer to [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF] for the definitions of conditional essential supremum and infimum.

The super-hedging problem 3.3.1 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 envelope 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 [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF]Theorem 2.4], there exists a F-measurable closed random 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 3.2.1. More precisely, the portfolio manager knows S t-2 0 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 t-2 0 ) is only observed strictly after t -1, i.e. S t-1 is F t -measurable. In the following, we consider the σ-algebra F t = 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 ⊗ B(R d )-measurable and z → g t (S 0 , S 1 , • • • , S t-1 , z
) is lower-semicontinuous (l.s.c.) almost surely independently the price process S t-1 0 is. The first goal is to characterise the set P t-1 of all V t-1 ∈ L 0 (R, F t ) that depend on S t-1 0 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 t-1 0 ), (3.3.2)
for some θ t-1 ∈ L 0 (R d , F t ) 1 . As θ t-1 is only F t -measurable, we also expect a dependence between θ t-1 and S t-1 0 as we shall see later. Nevertheless, we do not suppose an explicit dependence of Λ t (S t-1 0 ) with respect to θ t-1 , which is an open problem. We observe by lower-semicontinuity that (3.3.2) holds if and only if

V t-1 + θ t-1 ∆S t ≥ g t (S 0 , • • • , S t ), a.s. for all S t ∈ Σ(Λ t (S t-1 0 )). (3.3.3)
This means that we may suppose w.l.o.g. that Σ(Λ t (S t-1 0 )) = Λ t (S t-1 0 ). In the following, we denote by I t (S t-1 0 ) the F t+1 -measurable closed random set such that Σ(Λ t (S t-1 0 )) = L 0 (I t (S t-1 0 ), F t+1 ), see [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF]Theorem 2.4]. By [29, Theorem 4.2], we deduce that (3.3.2) is equivalent to V t-1 ≥ p t-1 where p t-1 = p t-1 (S t-1 0 , θ t-1 ) is given by

p t-1 = θ t-1 S t-1 + sup z∈cl (It(S t-1 0 )|Ft) (g t (S 0 , • • • , 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 t-1 0 )|F t ) is the conditional closure of I t (S t-1 0 ), i.e. the smallest F t -measurable closed random set which contains I t (S t-1 0 ) almost surely. We refer the readers to [START_REF] Elmansour | Conditional interior and conditional closure of random sets[END_REF]Theorem 3.1] 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(S t-1 0 )|Ft) (z), (3.3.4)
where δ cl (It(S t-1

0

)|Ft) ∈ {0, ∞} is infinite on the complimentary of cl (I t (S t-1 0 )|F t ) and 0 otherwise. Notice that f * t-1 is convex and l.s.c. as a supremum (on cl (I t (S t-1 0 )|F t )) of convex and l.s.c. functions. Moreover, by [29, Theorem 4.2], (ω, y)

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

P t-1 (S t-1 0 ) = θ t-1 S t-1 + f * t-1 (-θ t-1 ) : θ t-1 ∈ L 0 (R d , F t ) +L 0 (R + , F t ). (3.3.5)
The second step is to determine the infimum super-hedging price as

p t-1 (S t-1 0 ) = essinf Ft P t-1 (S t-1 0 ). (3.3.6)
To do so, we use the arguments of [16, Theorem 2.8] and we obtain our first main result:

Theorem 3.3.1. Suppose that the mapping (ω, z) → g t (S 0 (ω), • • • , S t-1 (ω), z) is F t ⊗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 t-1 0 is. Let us consider the function f t defined by (3.3.4) and the set of all prices given by (3.3.5). Then, the infimum price given by (3.3.6), satisfies p t-1

(S t-1 0 ) = -f * * t-1 (S t-1 ).
Proof. This is a consequence of the following chain of equalities:

p t-1 (S t-1 0 ) = essinf Ft θ t-1 S t-1 + f * t-1 (-θ t-1 ) : θ t-1 ∈ L 0 (R d , F t ) , = essinf Ft -θ t-1 S t-1 + f * t-1 (θ t-1 ) : θ t-1 ∈ L 0 (R d , F t ) , = -esssup Ft θ t-1 S t-1 -f * t-1 (θ t-1 ) : θ t-1 ∈ L 0 (R d , F t ) , = -esssup Ft θ t-1 S t-1 -f * t-1 (θ t-1 ) : θ t-1 ∈ L 0 (Dom f * t-1 , F t ) , = -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 ). (3.3.7)
Note that we do not need to suppose no-arbitrage conditions to establish the very general pricing formula above. It is only based on the lower-semicontinuity and measurability assumptions satisfied by the payoff g.

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

In this section we introduce the main contribution. They are needed to propagate the one time step pricing procedure of Section 3.3.1 to the multi-period case. In the following, we suppose that, for all price process S t-1 0 , there exists

α t-1 ∈ L 0 (R d , F t ) and β t-1 ∈ L 0 (R, F t ) that may depend on S t-1 g t (S 0 , • • • , S t-1 , x) ≤ α t-1 x + β t-1 , ∀x ∈ cl (I t (S t-1 0 )|F t ). (3.3.8)
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 [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF][Theorem 2.8], we know that

p t-1 (S t-1 0 ) = (3.3.9) inf αS t-1 + β : αx + β ≥ g t (S 0 , • • • , S t-1 , x), ∀x ∈ cl (I t (S t-1 0 )|F t ) .
We first establish the following result:2 Proposition 3.3.2. Let S t-1 0 be a price process. Suppose that the mapping (ω, z)

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

α * t-1 in L 0 (R d \ {0}, F t ) and c 1 t-1 , c 2 t-1 ∈ L 0 (R d , F t ) such that we have the inequality α * t-1 y < c 1 t-1 < c 2 t-1 < α * t-1 S t
p t-1 ≤ αt-1 S t-1 + βn t-1 , i.e. p 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 ). By (3.3.9),

p t-1 (S t-1 0 ) ≥ 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 ) on cl (I t |F t ) and S t-1 ∈ conv cl (I t |F t ). Then, S t-1 = lim n→∞ S n where S n ∈ conv cl (I t |F t ), i.e. S n = Jn i=1 λ i,n x i,n where λ i,n ≥ 0 with Jn i=1 λ i,n = 1 and x i,n ∈ cl (I t |F t ) 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 ). Then, αS t-1 + β = lim n→∞ (αS n + β) with αS n + β = Jn i=1 λ i,n (αx i,n + β) ≥ Jn i=1 λ 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 (3.3.9). Corollary 3.3.3. Let S t-1 0 be a price process. Suppose that the mapping (ω, z)

→ g t (S 0 (ω), • • • , S t-1 (ω), z) is F t ⊗B(R d )-measurable and the function z → g t (S 0 , • • • , S t-1 , z) is l.s.c. a.

s. and convex or bounded from below by m

t-1 ∈ L 0 (R, F t ) on cl (I t (S t-1 0 )|F t ). Then, p t-1 (S t-1 0 ) = -∞ if and only if S t-1 ∈ conv cl (I t (S t-1 0 )|F t ).
In particular, the infimum super-hedging 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 t-1 0 )|F t ). As studied in [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[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 3.3.4. 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 of the model. Moreover, we say that the condition AIP holds when AIP holds at any time step.

As observed in [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[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 numerical procedure we develop in this chapter is still valid but unrealistic and non-implementable in practice. By Corollary 3.3.3, we have: Corollary 3.3.5. The condition AIP holds between t -1 and t if and only if S t-1 ∈ conv cl (I t (S t-1 0 )|F t ) for any price process S t-1 0 , i.e.

I t-1 (S t-2 0 ) ⊆ conv cl (I t (S t-1 0 )|F t ), 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.

The following result allows us to compute the infimum price rather easily.

Lemma 3.3.6. Let S t-1 0 be a price process. Suppose that the mapping (ω, z)

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

p t-1 (S t-1 0 ) (3.3.10) = inf αS t-1 + β : αx + β ≥ h t-1 (x), for all x ∈ cl (I t (S t-1 0 )|F t ) .
Proof. By definition, h t-1 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 t-1 . By (3.3.9) we deduce that (3.3.10) holds.

The following result provides a criterion under which the infimum price is a price: Proposition 3.3.7. Suppose that AIP holds. Let S t-1 0 be a price process. Suppose that the mapping (ω, z)

→ g t (S 0 (ω), • • • , S t-1 (ω), z) is F t ⊗ 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 ) and β t-1 ∈ L 0 (R, F t ) 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 t-1 0 )|F t )
and consider the concave envelope

h t-1 (x) = conc g t (S 0 , • • • , S t-1 , •), cl (I t (S t-1 0 )|F t ) (x). (3.3.11)
We have p t-1

(S t-1 0 ) ∈ [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 t-1 (S t-1 ) = ∅ , then p t-1 (S t-1 0 ) = h t-1 (S t-1
) is a price, i.e. p t-1 (S t-1 0 ) ∈ P t-1 (S t-1 0 ) with the super-replicating strategies θ t-1 ∈ ∂h t-1 (S t-1 ). Proof. It is clear by Lemma 3.3.6 that p t-1 (S t-1 0 ) ≥ h(S t-1 ) when S t-1 belongs to cl (I t (S t-1 0 )|F t ). By definition, for all r t-1 ∈ ∂h(S t-1 ) = ∅, for all x ∈ conv cl (I t (S t-1 0 )|F t ), h(x) ≤ h(S t-1 ) + r t-1 (x -S t-1 ) =: δ(r t-1 , x).

(3.3.12)

Therefore, p t-1 (S t-1 0 ) ≤ δ(r t-1 , S t-1 ) = h(S t-1 ), and finally p t-1 (S t-1 0 ) = h(S t-1 ).

At last, applying (3.3.12) with x = S t ∈ I t (S t-1 0 ) ⊆ cl (I t (S t-1 0 )|F t ), we deduce that

p t-1 (S t-1 0 ) + 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 := {(ω, r t-1 ) : δ(r t-1 , x) ≥ g t (S 0 , • • • , S t-1 , x), ∀x ∈ conv cl (I t (S t-1 0 )|F t )}, = {(ω, r t-1 ) : δ(r t-1 , γ n t ) ≥ g t (S 0 , • • • , S t-1 , γ n t ), ∀n ∈ N},
where (γ n t ) n≥1 is a Castaing representation of conv cl (I t (S t-1 0 )|F t ). Since G t is F t ⊗ B(R d )-measurable and G t = ∅ a.s, it admits a measurable selection which is a measurable strategy θ t for the price p t-1 (S t-1 0 ). Remark 3.3.8. As the function h t-1 in (3.3.11) is concave and finite a.s. on the conditional closure conv cl (I t (S t-1 0 )|F t ), see proof of Proposition 3.3.2, the superdifferential ∂h(S t-1 ) of h t-1 at the point S t-1 is not empty when S t-1 belongs to the interior of conv cl (I t (S t-1 0 )|F t ).

The following result proves the measurability of the infimum super-hedging price p t-1 (S t-1 0 ) with respect to S t-1 0 . To do so, we suppose the existence of a Castaing representation, see [START_REF] Rockafellar | Variational analysis, volume 317 of Grundlehren der mathematischen Wissenschaften[END_REF], [START_REF] Lépinette | Conditional cores and conditional convex hulls of random sets[END_REF]. Proposition 3.3.9. Suppose that cl (I t (S t-1 0 )|F t ) admits a Castaing representation (ξ m t ) m≥1 where ξ m t = x m (S t-1 0 ), for all m ≥ 1, and x m are Borel functions on (R d ) t independent of S t-1 0 . Then, there exist a Borel function φ t-1 on (R d ) t such that p t-1 (S t-1 0 ) = φ t-1 (S t-1 0 ).

Proof. Let S t-1 0 be a price process. We denote by

S (t-1) = S t-1 0 and I t-1 = cl (I t (S (t-1) )|F t ).
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 t-1 0 . 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) ) .

Let us denote Q d = {α n = (α n 1 , ..., α n d ), n ≥ 1, α n 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-1) ) . We claim that p t-1 (S (t-1) ) = φ t-1 (S (t-1) ).

(3.3.13)

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 ncomponentwise. 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 (3.3.13) holds, which proves that the infimum super-hedging price p t-1 (S t-1 0 ) is measurable with respect to the argument S t-1 0 . The rest of this section aims to prove that, under some technical conditions, the mapping S t-1 0 -→ p t-1 (S t-1 0 ) is lower-semicontinuous, which is needed to propagate backwardly the numerical procedure of Theorem 3.3.6 in the multi-step model. Definition 3.3.10. We say that the mapping

I t : S t-1 0 -→ cl (I t (S t-1 0 )|F t )
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 t-1 0 , and for all z ∈ cl (I t (S t-1 0 )|F t ), 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 ) for all n ≥ 1.

(I t (S t-1 0 )|F t ) = [m t-1 S t-1 , M t-1 S t-1 ]
where

m t-1 , M t-1 ∈ L 0 (R + , F t ) and m t-1 ≤ M t-1 . Consider z ∈ cl (I t (S t-1 0 )|F t ), i.e. z = α t m t-1 S t-1 + (1 -α t )M t-1 S t-1 where α t ∈ L 0 ([0, 1], F t ). Let us define z n = α t m t-1 S n t-1 + (1 -α t )M t-1 S n t-1 for all n ≥ 1. Then, z n ∈ cl (I t ((S n u ) u≤t-1 )|F t ) and |z n -z| ≤ 2M t-1 |S n t-1 -S t-1 | hence lim n z n = z.
In the following, we define the closed convex random sets

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

E t-1 (S t-1 0 , αz 1 + (1 -α)z 2 ) ⊆ αE t-1 (S t-1 0 , z 1 ) + (1 -α)E t-1 (S t-1 0 , z 2 ).
Note that this convexity property above is automatically satisfied if d = 1.

Proposition 3.3.12. Consider a payoff function g t defined on (R d ) t+1 such that, there exists α t-1 ∈ L 0 ((R d ) t+1 , F t ) such that g t (x) -g t (y) ≥ α t-1 (x -y), x, y ∈ (R d ) t+1 . Suppose that I t : S t-1 0 -→ cl (I t (S t-1 0 )|F t ) is lower-semicontinous and that z → E t-1 (S t-1 0 , z) is convex for all S t-1 0 . Then, S t-1 0 -→ p t-1 (S t-1 0 ) is lowersemicontinuous, i.e. p t-1 (S t-1 0 ) ≤ lim inf n p t-1 ((S n u ) u≤t-1 ) if ((S n u ) u≤t-1 ) n≥1 converges a.s. to S t-1 0 .

Proof. Suppose that ((S n u ) u≤t-1 ) n≥1 converges a.s. to S t-1 0 . By assumption, we know that for all z ∈ cl (I t (S t-1 0 )|F t ), there exists a sequence z n ∈ cl (I t ((S n u ) u≤t-1 )|F t ) such that lim n z n = z. We may suppose that |z -z n | ≤ where > 0 is arbitrarily fixed. By assumption, for all z ∈ cl (I t ((S n u ) u≤t-1 )|F t ) in the ball B(z, ) of center z and radius , we have:

g t (S t-1 0 , z) ≤ g t ((S n u ) u≤t-1 , z) + |α t-1 | × |(S t-1 0 , z) -((S n u ) u≤t-1 , z)|, g t (S t-1 0 , 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 t-1 0 , z) ≤ h (n) (z) + |α t-1 | sup u≤t-1 |S n u -S u | + |α t-1 | , (3.3.14)
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 ). 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 ) 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 ) -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 left 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 (3.3.14), we deduce that p t-1 (S t-1 0 ) ≤ 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 t-1 0 ) ≤ 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 t-1 0 ) ≤ 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 t-1 0 ) ≤ lim inf n p t-1 ((S n u ) u≤t-1 ).

Case of a convex payoff function

We shall prove that p t-1 (S t-1 0 ) is a convex function of the price process S t-1 0 if so Λ t-1 is. In the following, we say that the mapping

Λ t-1 : S t-1 0 -→ Λ t-1 (S t-1 0 ) := conv cl (I t (S t-1 0 )|F t ) is convex for the inclusion if, for λ ∈ [0, 1], Λ t-1 ((λ(S t-1 0 ) + (1 -λ)(( Su ) u≤t-1 ) ⊆ λΛ t-1 (S t-1 0 ) + (1 -λ)Λ t-1 (( Su ) u≤t-1 ),
for all price process S t-1 0 , ( Su ) u≤t-1 . Proposition 3.3.13. Suppose that the mapping (ω, z) → g t (S 0 , S 1 (ω), ..., S t-1 (ω), z) is F t ⊗ 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

t-1 0 -→ Λ t-1 (S t-1 0 ) is convex. Then, the mapping S t-1 0 → p t-1 (S t-1 0 ) is convex . Proof. Let St-1 0 , S t-1 0
be two price processes. Let us define the following price process

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

Λ t-1 = conv cl (I t (S t-1 0 )|F t ) , t ≥ 1, Λt-1 = conv cl (I t ( St-1 0 )|F t ) , t ≥ 1, Λ t-1 = conv cl (I t (S t-1 0 )|F t ) , 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 t-1 0 , x), ∀x ∈ Λ t-1 . h(x) ≥ g t ( St-1 0 , x), ∀x ∈ Λt-1 . Thus, for λ ∈]0, 1[, we have λh(x) + (1 -λ) h(x) ≥ λg t (S t-1 0 , x) + (1 -λ)g t ( St-1 0 , x) ≥ g t (λ(S t-1 0 ) + (1 -λ)( St-1 0 ), 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 t-1 0 , 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 2 , 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 t-1 0 ) ≤ 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 t-1 0 ) ≤ λp t-1 (S t-1 0 ) + (1 -λ)p t-1 ( St-1 0
) and the conclusion follows.

Remark 3.3.14. Suppose that the AIP condition holds and that (3.3.8) 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)].
Recall that, by Proposition 3.3.9, p t-1 (S t-1 0 ) = φ t-1 (S t-1 0 ). When g t is convex, then φ t-1 is convex by Proposition 3.3.13. Moreover, if g t ≥ 0, 0 ≤ φ t-1 < ∞ by Proposition 3.3.7. Then, dom φ t-1 = (R d ) t and we deduce that φ t-1 is continuous on (R d ) t . Remark 3.3.15. Consider the case d = 1. By a measurable selection argument, we may show that there exists m

t-1 , M t-1 ∈ L 0 ([0, ∞], F t ) such that conv cl (I t (S t-1 0 )|F t ) = [m t-1 , M t-1 ]
. By Lemma 3.3.6, we deduce that under (AIP)

p t-1 (S t-1 0 ) = g t (S 0 , • • • , S t-1 , m t-1 ) (3.3.15) + 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 ).
Moreover, the strategy is given by

θ t-1 = 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 .
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 t-1 0 ) = g t-1 (S t-1 0 ) 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. At last, the order to be sent at time t is given by the deterministic mapping defined on R t by

θ t-1 (s 0 , • • • , s t-1 ) = g t (s 0 , • • • , s t-1 , k u t-1 s t-1 ) -g t (s 0 , • • • , s t-1 , k d t-1 s t-1 ) (k u t-1 -k d t-1 )s t-1
.

Remark 3.3.16 (Market impact). It is possible in our model to include a market impact. Indeed, it suffices to make the order (demand) mapping D t (x) = θ t (x)θ t-1 (S t-1 ) coincided at time t with the supply mapping O t (x), i.e. the available quantity we may buy or sell at price x in the order book. By convention, O t is negative for bid prices and positive for ask prices. It is an increasing function on R + starting from O t (0+) = -∞ at price 0 (we can sell as many assets as we want to the market at price 0) and ending up with O t (+∞) = +∞, i.e. we can buy as many assets as we want to the market at price +∞. As soon as D t is bounded, there exists executable bid prices S b t in the order book such that D t (S b t ) ≥ O t (S b t ) when D t (S b t ) ≤ 0, i.e. the order may be executed at price S b t as the quantity

|D t (S b t )| ≤ |O t (S b t )|.
The executed bid price is naturally the best one among all possible. Similarly, there exists executable ask prices S a t in the order book such that D t (S a t ) ≤ O t (S a t ) when D t (S a t ) ≥ 0 and the order may be executed at price S a t for the quantity D t (S a t ) ≤ O t (S a t ). Note that the executed bid price may be closed to 0 while the executed ask price may be very large. This liquidity phenomenon is then taken into account in the model through the conditional supports allowing to compute the strategy in our approach.

The multistep backward procedure

The main results of Section 3.3.2 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 t-1 0 )|F t ) admits a Castaing representation (ξ m ) m≥1 where ξ m = x m (S t-1 0 ), for all m ≥ 1, and x m are Borel functions on (R d ) t . Then, by Proposition 3.3.9, 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 3.3.13 and Remark 3.3.14, under convexity conditions.

Many questions could be investigated for future research, e.g. sensitivity to modeling assumptions, but also how to calibrate such a model from statistical estimations. Mainly, we need to estimate conditional supports. This is illustrated in the numerical example that we propose in the next section. A technical question is also to consider discontinuous payoff functions even if this is less usual in finance where g is generally a convex function. Actually, by Lemma 3.3.6, we may replace the payoff function by its concave envelope. Note that our analysis is general enough to consider a lot of models, e.g. with order books.

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 = 1. Recall that there exist F t+1 -measurable closed random sets I t = I t (S t-1 0 ) such that:

Σ(Λ t (S t-1 0 )) = L 0 (I t (S t-1 0 ), F t+1 ), 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 = I t (S t-1 0 ) = [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 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 make this choice for simplicity and that corresponds to the case where the bid and ask prices of the market coincide with the mid price S 0 . The order we sent is of the form buy or sell the quantity θ 0 (z) at the price z.

At time t = 1, we choose to model bid and ask prices S bid 1 , S ask 1 respectively as: S bid 1 = S 0 m 1 and S ask 1 = S 0 M 1 where S 0 is the last executed price. Notice that the order of buying or selling depends on the bid-ask values, see Figure 3 Note that the mapping s 1 → ∆θ 1 (s 1 ) is the F 1 -measurable order we send at time t = 1, see Figure 3.4. The later depends on S 0 , which is F 1 -measurable.

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 (3.3.9). We consider the terminal payoff function g(S T ) = (S T -K) + for several strikes.

The strategy at time

t = 1 Recall 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 (3.3.9) which dominates the the pay-off function g 2 on the conditional support cl (

I 2 (S 1 )|F 2 ) = [S 1 m - 2 , S 1 M + 2 ]. 1st case: K ∈ [S 1 m - 2 , S 1 M + 2 ] ⇔ S 1 ∈ [ K M + 2 , K m - 2 ] .
The dominating affine function ϕ 1 , see Figure 3.5, is given by:

ϕ 1 (x) = (S 1 M + 2 -K)(x -S 1 m - 2 ) S 1 (M + 2 -m - 2 ) . So, V 1 (S 1 ) = p 1 (S 1 ) = ϕ 1 (S 1 ) = (S 1 M + 2 -K)(1 -m - 2 ) M + 2 -m - 2 =: g 1 (S 1 ),
and

θ 1 (S 1 ) = S 1 M + 2 -K S 1 (M + 2 -m - 2 )
.

A simple computation shows that:

V 2 = V 1 (S 1 ) + θ 1 (S 1 )(S 2 -S 1 ) = ϕ 1 (S 2 ) ≥ g 2 (S 2 ).
2nd case: K ≤ S 1 m - 2 ⇔ S 1 ≥ K m - 2 .
In this case, we have

ϕ 1 (x) = (x -K) + for all x ∈ [S 1 m - 2 , S 1 M + 2 ], see Figure 3.6. Hence, V 1 (S 1 ) = (S 1 -K) + =: g 1 (S 1 ) and θ 1 (S 1 ) = 1. 3rd case: K ≥ S 1 M + 2 ⇔ S 1 ≤ K M + 2 .
Observe that the dominating affine function ϕ 1 coincides with the x-axis on the support

[S 1 m - 2 , S 1 M + Figure 3.7
We finally deduce that

g 1 (x) = (xM + 2 -K)(1 -m - 2 ) M + 2 -m - 2 1 K M + 2 , K m - 2 (x) + (x -K) + 1 K m - 2 ,∞ (x).
The graph of the payoff function g 1 is represented in Figure 3.7.

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 ).

1st case:

S 0 M + 1 ≤ K M + 2 , i.e. S 0 ≤ K M + 1 M + 2 .
We have V 0 (S 0 ) = g 0 (S 0 ) = 0 and θ 0 (S 0 ) = 0, see Figure 3.8.

S 0 m - 1 ≤ K M + 2 and S 0 M + 1 ∈ [ K M + 2 , K m - 2 ], i.e. S 0 ∈ [ K M + 1 M + 2 , K m - 1 M + 2 ∧ K m - 2 M + 1 ].
We find that (see Figure 3.9):

ϕ 0 (x) = (S 0 M + 1 M + 2 -K)(1 -m - 2 ) S 0 (M + 1 -m - 1 )(M + 2 -m - 2 ) (x -S 0 m - 1 ). So, V 0 (S 0 ) = ϕ 0 (S 0 ) = (S 0 M + 1 M + 2 -K)(1 -m - 2 )(1 -m - 1 ) (M + 2 -m - 2 )(M + 1 -m - 1 )
=: g 0 (S 0 ), and

θ 0 (S 0 ) = (S 0 M + 1 M + 2 -K)(1 -m - 2 ) S 0 (M + 2 -m - 2 )(M + 1 -m - 1 )
.

3rd case: S 0 m - 1 ≤ K M + 2 and S 0 M + 1 ≥ K m - 2 , i.e. S 0 ∈ [ K m - 2 M + 1 , K m - 1 M + 2 ].
We have, see Figure 3.10:

ϕ 0 (x) = S 0 M + 1 -K S 0 (M + 1 -m - 1 ) (x -S 0 m - 1 ).
V 0 (S 0 ) = ϕ 0 (S 0 ) = (S 0 M + 1 -K)(1 -m - 1 ) M + 1 -m - 1 =: g 0 (S 0 ), θ 0 (S 0 ) = S 0 M + 1 -K S 0 (M + 1 -m - 1 )
.

4th case: S 0 m - 1 ∈ [ K M + 2 , K m - 2 ] and S 0 M + 1 ∈ [ K M + 2 , K m - 2 ], i.e. S 0 ∈ [ K m - 1 M + 2 , K m - 2 M + 1 ].
We have ϕ 0 (x) = g 1 (x), for all x ∈ cl(I 1 (S 0 )|F 0 ), see Figure 3.11. Therefore,

V 0 (S 0 ) = ϕ 0 (S 0 ) = (S 0 M + 2 -K)(1 -m - 2 ) M + 2 -m - 2 =: g 0 (S 0 ), θ 0 (S 0 ) = M + 2 (1 -m - 2 ) M + 2 -m - 2 .
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

ε R = V 2 -(S 2 -K) + S 2 .
In the following table 3

E(θ 0 S 0 /V 0 ) 199% 255% 322% 333% 313% E(θ 1 S 1 /V 1 ) 205% 230% 134% 32% 3%
Figure 3.14: The empirical results.

We observe that the executed prices depend on the strike K > 0, i.e. there is a market impact of the orders on the prices. 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 3.16 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 3.18.

Chapter 4

Super-hedging an arbitrary number of European options with integer-valued strategies

Introduction

The problem of super-hedging a European claim, such as a call option, is very classical in mathematical finance but has only been solved for real-valued strategies so that the optimal strategy, corresponding to the minimal hedging or super-hedging price, is not integer-valued contrary to what it is allowed to do in the real life. In this joint work with D.Cherif and E.Lepinette we propose to solve the problem of super-hedging a European claim with integer-valued financial strategies. Actually, the main contribution in the literature for integer-valued financial strategies is the paper [START_REF] Gerhold | Dynamic trading under integer constraints[END_REF] where a finite set of states Ω is considered. In this setting, the authors adapt the usual theory for the real-valued strategies to the integer-valued ones, i.e. they introduce a no-arbitrage condition which is equivalent to the existence of a risk-neutral probability measure and show that the minimal super-hedging price may be characterized through the martingale measures similarly to the usual case with real-valued strategies. See also [START_REF] Xiaotie | Computational complexity of arbitrage in frictional security market[END_REF] where Ω is finite. The general case with an arbitrary state space Ω is still an open problem. Also, portfolio optimization problems of Markowitz type are considered in [START_REF] Bienstock | Computational study of a family of mixed-integer quadratic programming problems[END_REF], [START_REF] Bonami | An exact solution approach for portfolio optimization problems under stochastic and integer constraints[END_REF], [START_REF] Baumann | Portfolio-optimization models for small investors[END_REF].

Let us recall that the usual approach of pricing assumes that the financial market model satisfies a no-arbitrage condition NA, which is equivalent to the existence of a risk-neutral probability measure Q under which the discounted asset prices are martingales, see the Dalang-Morton-Willinger theorem [START_REF] Dalang | Equivalent martingale measures and no-arbitrage in stochastic securities market models[END_REF]. Under NA, we may show that there exists a minimal super-hedging price P * 0 (ξ T ) for the European claim ξ T ≥ 0 given by

P * 0 (ξ T ) = sup Q∈M(P ) E Q (ξ T ), (4.1.1)
where M(P ) is the set of all risk-neutral probability measures equivalent to the initial probability measure P of the model. Here, we suppose that the risk-free interest rate of the model is r = 0. Recall that the formula above holds in discrete time but also in continuous time with extra-conditions on the model. Indeed, the no-arbitrage condition needs to be strenghtened and it is only equivalent to the existence of Q ∼ P under which discounted asset prices are local martingales, see [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF], [START_REF] Delbaen | The fundamental theorem of asset pricing for unbounded stochastic processes[END_REF], [START_REF] Delbaen | The mathematics of arbitrage[END_REF].

In any case, the optimal strategy that achieves the minimal super-hedging price (4.1.1) is not, in general, integer-valued. The typical example is the continuoustime Black and Scholes model where the so-called delta-hedging strategy for the European call is explicit and lies in the set [0, 1] \ {0, 1}, see [START_REF] Black | The pricing of options and corporate liabilities[END_REF].

Clearly, a new approach is necessary to compute the super-hedging prices for only integer-valued financial strategies. We follow the ideas developed in [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF] where the problem is initially solved without any no-arbitrage conditions. Then, a no-arbitrage condition AIP naturally appears and means that the infimum price of the zero claim ( non negative claims more generally) is not -∞. This condition is clearly necessary for numerical purposes. Actually, it is shown that AIP is equivalent to the property that the infimum super-hedging price of any non negative claim is non negative, as observed in the real markets. In this chapter, we do not explicitly suppose such a no-arbitrage condition but the form of the conditional supports of the asset price we assume implies this condition for the model with integer-valued strategies.

Our aim is to compute super-hedging prices with only integer-valued financial strategies. We restrict ourselves to payoffs which are piecewise affine functions of the underlying asset and we assume specific conditional supports for the asset prices. Problems such as characterizations of the no-arbitrage condition NA with only integer-valued strategies or generalization of our work to arbitrary conditional supports of the asset prices remains open if Ω is not finite.

Notations If A ⊂ R is a random subset of R and F is a σ-algebra, we denote by L 0 (A, F) the family of all F-measurable random variables X on the probability space (Ω, F, P ) such that X ∈ A a.s..

Formulation of the problem

We consider n ≥ 1 options that we want to super-replicate in discrete time t = {0, ...T }. Let (Ω, (F t ) 0≤t≤T ) be a stochastic basis where (F t ) 0≤t≤T is supposed to be complete. We consider a financial market model composed of two assets. We suppose, without loss of generality, that the risk-free asset is S 0 t = 1 for all t ∈ {0, ...T }, while the risky asset price is described by a stochastic process S = (S t ) 0≤t≤T . Recall that a self-financing portfolio process (V t ) 0≤t≤T satisfies by definition:

∆V t = V t -V t-1 = θ t-1 ∆S t , t = 1, • • • , T,
where θ t-1 is F t-1 -measurable and represents the number of risky assets of the portfolio.

In this chapter, we consider European options whose payoffs are of the form ξ T = g(S T ) ∈ L 0 (R, F T ), where g is a continuous piecewise affine function. The typical example is the European call option, i.e. g(x) = (x -K) + , K > 0. Our goal is to compute the set of all super-hedging prices of ξ T , i.e the set of all V 0 , initial values of self-financing portfolio processes (V t ) 0≤t≤T , such that V T ≥ ξ T almost surely.

Contrarily to what it is usual to do in the literature, we restrict ourselves to the case of integer-valued strategies, i.e θ t ∈ Z almost surely, for all t ∈ {0, ...T }, where Z = N ∪ (-N) and N is the set of all non negative integers. In the case of superhedging an arbitrary number of options n ≥ 1, the problem reads as V T ≥ nξ T , a.s. and it is clearly interesting to analyse the impact of n on the strategies and the infimum prices, as linearity is not necessarily preserved with respect to the quantity n of claims.

To solve this problem, we follow the approach of [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF], [START_REF] Baptiste | Pricing without martingale measures[END_REF] that we adapt to integer-valued strategies. To do so, we first solve backwardly the super-hedging problem between two dates t -1 and t, and we show that the procedure may be propagated backwardly as the minimal super-hedging price we obtain at time t -1 is still a continuous piecewise affine payoff function of the underlying asset. It is then possible to deduce the minimal super-hedging price at time t = 0 by induction.

The super-hedging problem

Let t ≤ T and g t be a continuous piecewise affine function, i.e. there exists a subdivision 0 = a 0 < a 1 < ... < a n-1 < a n = ∞ of [0, ∞] such that g t is an affine function for all x ∈ [a i , a i+1 ), ∀i ∈ {0, ..., n -1}. As the asset prices are non negative, we suppose without loss of generality that a 0 = 0. We first solve the one step problem: find V t-1 and the strategy θ t-1 ∈ L 0 (R, F t-1 ) such that:

V t-1 + θ t-1 ∆S t ≥ g t (S t ), a.s.
This is equivalent to:

V t-1 ≥ g t (S t ) -θ t-1 ∆S t , ⇔ V t-1 ≥ g t (S t ) -θ t-1 S t + θ t-1 S t-1 , ⇔ V t-1 ≥ esssup F t-1 (g t (S t ) -θ t-1 S t ) + θ t-1 S t-1 .
Equivalently, we have:

V t-1 ≥ V t-1 (θ t-1 ) := sup x∈supp F t-1 (St) (g t (x) -θ t-1 x) + θ t-1 S t-1 , (4.3.2)
where supp F t-1 (S t ) is the conditional support of S t knowing F t-1 , see [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF] and [START_REF] Elmansour | Conditional interior and conditional closure of random sets[END_REF] for the definition and the proof of the inequality above.

In the following, we suppose that there exist two deterministic numbers k d t-1 ∈ (0, 1) and

k u t-1 ∈ (1, ∞) such that supp F t-1 (S t ) = [k d t-1 S t-1 , k u t-1 S t-1 ]
. This model may be seen as a generalization of the Binomial model and the conditions imposed on the coefficients k d t-1 and k u t-1 are equivalent to a no-arbitrage condition, see [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF]. In particular, we have:

V t-1 (θ t-1 ) = sup x∈[k d t-1 S t-1 ,k u t-1 S t-1 ] (g t (x) -θ t-1 x) + θ t-1 S t-1 ∈ L 0 (R, F t-1 ).
We define V * t-1 as the infimum of all the superhedging prices at time t -1 over all integer-valued strategies in Z, i.e.

V * t-1 := ess inf H θ t-1 ∈L 0 (Z,F t-1 )
V t-1 (θ t-1 ).

Lemma 4.3.1. We have

V * t-1 = inf θ∈Z V t-1 (θ). (4.3.3) Proof. Let us define γ = inf θ∈Z V t-1 (θ) ∈ L 0 (R, F t-1 )
, see [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF]. As

V * t-1 ≤ V t-1 (θ), for all θ ∈ Z, we get that V * t-1 ≤ inf θ∈Z V t-1 (θ) = γ. On the other hand, if θ t-1 ∈ L 0 (Z, F t-1 ),
then:

θ t-1 = θ∈Z θ1 {θ t-1 =θ} , V t-1 (θ t-1 ) = θ∈Z V t-1 (θ)1 {θ t-1 =θ} ≥ θ∈Z γ1 {θ t-1 =θ} = γ.
We deduce that V * t-1 ≥ γ and the conclusion follows.

Theorem 4.3.2 (One step problem). Let us consider t ∈ {1, ..., T } and suppose that ξ t = g t (S t ) where g t is a continuous piecewise affine function. Moreover, we assume that there exists two deterministic numbers k d t-1 ∈ (0, 1) and

k u t-1 ∈ (1, ∞) such that supp F t-1 (S t ) = [k d t-1 S t-1 , k u t-1 S t-1 ]. Then, V * t-1 = g t-1 (S t-1
) where g t-1 is a continuous piecewise linear function.

Proof. By assumption, there exist a subdivision (a

i ) i=0,••• ,n of [0, ∞], with a 0 = 0 < a 1 < ... < a n-1 < a n = ∞, such that g t is an affine function on each interval. Let us define x i (S t-1 ) = (k d t-1 S t-1 ∨ a i ) ∧ k u t-1 S t-1 , i = 0, • • • , n. It is straightforward that V t-1 (θ t-1 ) = sup i=0,••• ,n [g t (x i (S t-1 )) -θ t-1 x i (S t-1 )] + θ t-1 S t-1 .
Note that x 0 (S t-1 ) = k d S t-1 and x n (S t-1 ) = k u S t-1 and some terms of the sequence (x i ) i may coincide. Let us define the functions

h i (θ t-1 , S t-1 ) = g t (x i (S t-1 )) + θ t-1 (S t-1 -x i (S t-1 )), i = 0, • • • , n.
The slopes of the affine functions θ t-1 → h i (θ t-1 , S t-1 ) are given by the non decreasing sequence (S t-1 -x i (S t-1 )) i=n,n-1,••• ,0 such that S t-1 -x n (S t-1 ) < 0 and S t-1 -x 0 (S t-1 ) > 0.

By ordering the indices in the decreasing order, we obtain (n + 1) affine functions θ t-1 → h i (θ t-1 ) for i ∈ {n, n -1, ..., 1, 0} with increasing slopes (S t-1x i ) i∈{n,n-1,...,1,0} , such that: S t-1 -x n < 0 and S t-1 -x 0 > 0. Therefore, the mapping

V t-1 : θ t-1 → sup i=n,...,0 h i (θ t-1 , S t-1
) is a piecewise affine function, i.e. there exists a subdivision:

-∞ = α 0 < α 1 (S t-1 ) ≤ ... ≤ α m-1 (S t-1 ) < α m = ∞ such that V t-1 is an affine function of θ t-1 on each interval [α i (S t-1 ), α i+1 (S t-1 )], i = 0, • • • , m -1. Note that the function V t-1 is convex in θ t-1
and the elements of the partition define the intersection points between two distinct and successive graphs of the affine functions h i+1 (θ t-1 ), h i (θ t-1 ). So, there exists

θ * t-1 ∈ [α 1 (S t-1 ) - 1, ..., α m-1 (S t-1 ) + 1] ∩ Z such that: inf θ t-1 ∈Z V t-1 (θ t-1 ) = V t-1 (θ * t-1 ).
It remains to evaluate α 1 (S t-1 ) and α m-1 (S t-1 ). To do so, let us solve the equations h i (α) = h j (α), i, j = 0, ..., m and x i = x j . Since we suppose that x i -x j = 0, we get that α = g t (x i ) -g t (x j )

x i -x j .

We deduce that |α| ≤ L t where L t > 0 is a Lipschitz constant of the piecewise affine function g t . We deduce that θ

* t-1 ∈ [-L t -1, L t + 1] ∩ Z and g t-1 (S t-1 ) = V t-1 (θ * t-1 (S t-1 )) = min θ t-1 ∈[-Lt-1,Lt+1]∩Z sup i=0,••• ,n h i (θ t-1 , S t-1 ).
We conclude that g t-1 is a continuous piecewise affine function as a finite minimum of continuous piecewise affine functions.

Corollary 4.3.3. (The multi-period super-hedging problem) Suppose that, at time T > 0, the payoff is ξ T = g T (S T ) where g T is a continuous piecewise affine function. Moreover, we assume that there exists deterministic numbers k d t-1 ∈ (0, 1) and

k u t-1 ∈ (1, ∞) for each t = 1, • • • , T such that we have supp F t-1 (S t ) = [k d t-1 S t-1 , k u t-1 S t-1 ].
Then, there exists a minimal super-hedging portfolio process (V * t ) t=0,••• ,T such that V * T ≥ ξ T . We have V * t = g(t, S t ) where g(t, •) is a continuous piecewise affine function given by g(t, s) = min

θ∈[-L t+1 -1,L t+1 +1]∩Z sup i=0,••• ,n (t+1) g(t + 1, x (t+1) i (s)) + θ(s -x (t+1) i (s)) , x (t+1) i (s) = (k d t s ∨ a (t+1) i ) ∧ k u t s, i = 0, • • • , n (t+1) ,
where L t+1 is any Lipschitz constant of g(t + 1, •) and (a

(t+1) i ) i=0,••• ,n (t+1) is any par- tition such that a (t+1) 0 = 0 < a (t+1) 1 < ... < a (t+1) n (t+1) -1 < a (t+1) n (t+1) = ∞ and g(t + 1, •) is an affine function on [a (t+1) i , a (t+1) i+1 ], i ≤ n (t+1) -1.
The associated super-hedging strategy θ * is given by the argmin of the minimisation problem defining g(t, •) in the expression above.

Example in the one step problem: the case of the Call option

At time t = T , suppose that the payoff is ξ n T = ng(S T ) where n ≥ 1 and g(x) = (x -K) + , K = 500. We suppose that supp

F T -1 (S T ) = [k d S T -1 , k u S T -1 ] for some constants k d , k u such that 0 < k d < 1 < k u .
Precisely, we suppose that k d = 0.9 and k u = 1.2. Observe that the super-hedging problem

V T -1 + θ T -1 ∆S T ≥ ng(S T ) is equivalent to V T -1 ≥ V T -1 (θ T -1 ) = max k∈{k d ,k u } [ng(kS T -1 ) -θ T -1 kS T -1 ] + θ T -1 S T -1 .
In the following we give the explicit expression of

V T -1 (θ T -1 ) = V n T -1 (θ T -1 ). If k u ≤ K/S T -1 , i.e. S T -1 ≤ K/k u , then V T -1 (θ T -1 ) = θ T -1 S T -1 (1 -k u ) if θ T -1 ≤ 0, θ T -1 S T -1 (1 -k d ) if θ T -1 ≥ 0. Therefore, θ * ,n T -1 (S T -1 ) = 0, and V * ,n T -1 (S T -1 ) = V T -1 (θ * ,n T -1 ) = 0. If k d ≥ K/S T -1 , i.e. S T -1 ≥ K/k d , V T -1 (θ T -1 ) = θ T -1 S T -1 (1 -k u ) + nk u S T -1 -nK if θ T -1 ≤ n, θ T -1 S T -1 (1 -k d ) + nk d S T -1 -nK if θ T -1 ≥ n.
We conclude that θ * ,n T -1 (S T -1 ) = n, and

V * ,n T -1 (S T -1 ) = n(S T -1 -K). If k d ≤ K/S T -1 ≤ k u , i.e. S T -1 ∈ [K/k u , K/k d ], V T -1 (θ T -1 ) = θ T -1 S T -1 (1 -k u ) + nk u S T -1 -nK if θ T -1 ≤ nk u S T -1 -nK S T -1 (k u -k d ) , θ T -1 S T -1 (1 -k d ) if θ T -1 ≥ nk u S T -1 -nK S T -1 (k u -k d ) .
Let us define

α n T -1 (S T -1 ) := nk u S T -1 -nK S T -1 (k u -k d ) , f n (x, S T -1 ) := xS T -1 (1 -k d )1 {x≥α T -1 (S T -1 )} +(xS T -1 (1 -k u ) + nk u S T -1 -nK)1 {x<α T -1 (S T -1 )} .
We denote by α n T -1 (S T -1 ) the lower integer part of α n T -1 . Then,

θ * ,n T -1 (S T -1 ) = α n T -1 (S T -1 ) if f n ( α T -1 (S T -1 ) ) ≤ f n ( α T -1 (S T -1 ) + 1), α n
T -1 (S T -1 ) + 1 otherwise. So, we have:

V * ,n T -1 (S T -1 ) = ( α n T -1 (S T -1 ) S T -1 (1 -k u ) + nk u S T -1 -nK)1 G n T -1 (S T -1 ) +( α n T -1 + 1)S T -1 (1 -k d )1 (G n T -1 ) c (S T -1 )
, where

G n T -1 := {S : f n ( α n T -1 (S) ) ≤ f n ( α n T -1 (S) + 1)} = { α n T -1 (S) ≤ β n T -1 (S)}, β n T -1 (S) := α n T -1 (S) + 1 -k d k d -k u .
A graphic illustration of V * ,n T -1 /n as a function of S T -1 is given in

Numerical illustration

In this section, we illustrate the method developed above when the underlying asset S is the french CAC 40 index and the European claim is the Call option. The historical data is composed of daily observations of the CAC 40 values between the 6th of June 2019 and the 16th of June 2021 (Appendix C [4.4]). We use the two first years of the data set to calibrate the model while we implement the model on the third year. Here, we suppose that supp Ft S t+1 = [k d t S t , k u t S t ] where k d t and k u t are estimated as follows:

k d t = min i=j,••• ,N S (j) t+1 /S (j) t , k u t = max i=j,••• ,N S (j) t+1 /S (j) t ,
where N is the number of training periods and S (j) t are the observed values at time t during the j-th periods. The algorithms are written in Python,see Appendix C [4.4]. The main difficulty is to write a code whose execution time is reasonable. Indeed, recall that the price function g(t, x) is computed backwardly from g(t + 1, x). If this function g(t, x) is naively coded from g(t + 1, x), then the computation may take more than two weeks ! So it is better to approximate, at each step, the function g(t, x) as a numpy array consisting of discretized values following a grid (x i ) Nt i=0 where x i = step * i. Here, we choose step = 0.1 and N t is chosen so that x N 0 ≤ S max 0 where S max 0 is the maximal value for S 0 that we observe in our data. At last, x Nt ≤ S max 0 * (max r≤t k u r ) t . The relative hedging error is defined as T = 100 * (V T -g(T, S T ))/S T where (V * t ) t=0,••• ,T is the optimal super-hedging portfolio process whose initial value is the minimal super-hedging price, as computed in the last section. We present in Figure 4.2.1 the distribution of T when n = 1. Of course, we expect that T ≥ 0 a.s. and this is confirmed on our test data set. Note that, we could have observed some negative values as the model is calibrated from data values anterior to the test data set.

Let us denote by g(t, x, n) the price function at time t of the optimal portfolio process, i.e. V * ,n t = g(t, S t , n) such that V * ,n T ≥ ξ n T a.s., when the European claim is ξ n T := n * (S T -K) + . The natural question is the following: Do we have g(t, x, n) = ng(t, x, 1) ? The answer is yes when real-valued strategies are allowed since the hedging problem is then linear with respect to the number of claims.

In the case of integer-valued strategies, the answer is not trivial and is actually negative, see the first example above. By definition of the infimum super-hedging price, we have g(t, x, n) ≤ ng(t, x, 1). As a first step, we have computed the relative infimum super-hedging prices per unit of claims, i.e. V * ,n 0 /n at time 0, for different values of n on each period of the test data set. Then, computing the average of the V * ,n 0 /n values over all the periods, we get that the empirical average of V * ,n 0 /n is approximately equal to 49.48% for n = 1, 5, 10, 15, 20. Nevertheless, we observe that the price function per unit of unit of claims, i.e. g(0, S 0 , n)/n is non-increasing when n increases, see Figure 4.3. This implies that the equality g(t, x, n) = ng(t, x, 1) does not hold. We conjecture that g(t, x, n)/n converges to the price function ĝ(t, x) of the model where real-valued strategies are allowed, see [START_REF] Carassus | Pricing without no-arbitrage condition in discrete time[END_REF]. This is an open question we suggest. In Figure 4.3, we clearly observe the convergence of x → g(0, x, n)/n as n → ∞.

The same question arises for the optimal strategy associated to V * ,n , i.e. do we have θ * (t, S t , n) = nθ * (t, S t , 1)? Intuitively, this is a priori not the case as θ * (t, S t , 1) = θ * (t, S t , n)/n could be not integer-valued. This is confirmed at time 0 when we compute the optimal strategy θ * (0, S 0 , n)/n per unit of claims. This is illustrated by Figure 4.2.2 where we compare θ * (0, S 0 , n)/n for n = 10 to θ * (0, S 0 , 1). We may observe that the optimal strategy per unit of claims θ * (0, S 0 , n)/n (blue graph) is smaller that θ * (0, S 0 , 1) for n = 10.

Conclusion and future perspectives

In this thesis, the main goal is to explore new approaches of pricing where executable prices are uncertain. This chapter will discuss some potential ideas to explore in future research.

It was possible to determine and compute the infimum of the super-hedging price for general models by considering the conditional random sets and the essential supremum and the essential infimum operators under the AIP condition without any assumptions of no-arbitrage on the market. In fact the AIP condition was introduced to ensure the finiteness of prices. It is important to note that we only needed to know the conditional support of the returns S t+1

St to compute the infimum of the super-hedging prices. We suggested a straightforward model in the one-dimensional case. It is a generalisation of the binomial model as observed in [START_REF] Baptiste | Pricing without martingale measures[END_REF] that works well with real data. However, it is interesting to consider the multivariate case, i.e the case of multiple assets. In this context, several questions arise :

1. How to define the conditional support for the multivariate case to get an efficient calibration on a historical data base both with fast and simple computation?

2. Moreover the advantage of the approach used in chapter 3 lies in its ability to include a large range of potential models, e.g. with transaction costs or limit order book. Therefore, the numerical aspect can be further explored and compared to other models.

3. Another natural question is: what happens if we increase the number of discrete dates?

4. It is also interesting to study the case where the payoff functions are not necessary convex.

On the other hand, regarding the super-hedging problem with integer valuedstrategies, we can observe that the execution is large as soon as the number of dates increases ( more then 6 dates). Also, we only considered simple payoff functions. Then, the multivariate case can be studied with more complex payoff, but we expect that the computation will be more time consuming. Theorem 4.4.8. Let f : R d → R a function. We denote by conv(f ) the convex hull of function f. 1 We suppose that conv(f ) is proper. Then f * and f * * are proper, lsc and convex and we have:

f * * = cl conv(f ).
Moreover, we have f * * ≤ f, and if f is proper, lsc and convex then we have f * * = f.

1 If f : R d → R is a function, then conv(f ) is the greatest convex function majorized by f.

, 0 t,

 0 be the discounted value of V θ t and St = St S the discounted value of S t then ∆ Ṽ θ t = θ t-1 .∆ St ∀t = 1, ..., T.

Lemma 2 . 2 . 10 .

 2210 Let O be an open set in a separable Banach space. For every x ∈ O, B(x, d(x, ∂O)) ⊆ O. Proof. Let us consider r * = sup R where R is the non empty set of all r > 0 such that B(x, r) ⊆ O. It is trivial that B(x, r * ) ⊆ O. So, for all o ∈ ∂O, d(x, o) ≥ r * hence r * ≤ d(x, ∂O). Moreover, by definition of r * , for all n there exists

Figure 3 . 1 :

 31 Figure 3.1: Vinci order book Figure 3.2: Airbus order book Figure 3.3: Examples of order books

Example 3 . 3 . 11 .

 3311 Suppose that d = 1 and cl

  .

4 . 1 ≤ S ask 1 ≤ S * 1 (S bid 1 ≤

 41111 We define S * 1 such that ∆θ 1 (S * 1 ) = 0. If S bid in the green zone {S 1 : ∆θ 1 (S 1 ) ≤ 0}), then S 1 = S bid 1 since ∆θ 1 ≤ 0. If S * 1 ≤ S ask 1 , (the yellow zone,) then S 1 = S ask 1 as ∆θ 1 > 0. Otherwise, if S bid < S * 1 < S ask 1 , we may arbitrarily choose S 1 = S ask 1 or S 1 = S bid 1 . In our model, we make the (arbitrary) choice that, if |S * 1 -S bid 1 | ≤ |S * 1 -S ask 1 |, then S 1 = S ask 1 and S 1 = S bid 1 otherwise.

Figure 3 . 4 2 = S bid 2 ∈

 3422 Figure 3.4

Figure 3 . 5 Figure 3 . 6

 3536 Figure 3.5 Figure 3.6

Figure 3

 3 Figure 3.8 Figure 3.9

Figure 3

 3 Figure 3.10 Figure 3.11

Figure 3 . 15

 315 Figure 3.15

Figure 3 .

 3 Figure 3.16: K=100.

Figure 3 .

 3 Figure 3.17: K=100.Figure3.18: K=100.

Figure 3 .

 3 Figure 3.17: K=100.Figure3.18: K=100.

Figure 4

 4

Figure 4 . 1 :

 41 Figure 4.1: The function x= S T -1 → g(T -1, x, n)/n = V * ,nT -1 /n,K = 500, for n = 1 (black), n = 5 (blue), n = 100 (red).

Figure 4 . 2 :

 42 Figure 4.2: F1.1: Super-hedging errors with n = 1 and K = 3000. F1.2: Comparison of the optimal strategies per unit of claims for n = 1 and n = 10.

Figure 4 . 3 :

 43 Figure 4.3: Super-hedging price mapping x → g(0, x, n)/n of n units of call option per unit of claims for different values of n = 1 (black), n = 3 (grey), n = 5 (green), n = 7 (blue), n = 10 (orange), n = 100 (red).

  / M2 <= S0 * m1 <= K / m2 ) and ( K / m2 <= S0 * M1 ) : # 5 th case of S0 theta0 = (( S0 * M1 -K ) *( M2 -m2 ) -( S0 * m1 * M2 -K ) *(1 -m2 ) ) / ( S0 * ( M2 -m2 ) *( M1 -m1 ) ) S1star = K /( M2 -theta0 *( M2 -m2 ) ) m2 <= S0 * m1 ) and ( K / m2 <= S0 * M1 ) : ' the error value is : ' , eps ) # return eps , epsrel , theta0

Fichier

  = xlrd . open_workbook ( " Data . xlsx " ) # Import our data set data = Fichier . sheet_by_index (0) S =[] for i in range (778) : s = data . cell_value (778 -i ,1) s = s . replace ( ' ' , ' ') s = float ( s ) S . append ( s ) # S is the historical data set pas =0.1 # pas = step S_max = max ( S ) def mon_min (x , y ) : return x *( x <= y ) + y *( y < x ) def mon_max (x , y ) : return x *( x >= y ) + y *( y > x ) NP_test =52 # number of periods NP_calibration =25 # number of periods for calibration T = int (9) S_test =[ S [ i ] for i in range (10* NP_calibration ,10*( NP_calibration + NP_test ) ) ] S_calibration =[ S [ i ] for i in range (10* NP_calibration ) ] kd = np . array ([100.0 for i in range (9) ]) ku = np . array ([0.0 for i in range (9) ]) for i in range (9) : for j in range ( NP_calibration ) : kd [ i ]= min ( kd [ i ] , S_calibration [10* j + i +1]/ S_calibration [10* j + i ])

  Notice that such a conditional interior is necessarily unique by definition. We first show the existence of the conditional interior, when Γ is open-valued. Theorem 2.2.8. Let us consider an F-graph measurable open random set O. Then, there exits a unique H-graph measurable interior of O. Proof. By Proposition 2.2.3, cl O and ∂O := cl O \O are closed F-graph measurable random sets, see Corollary 2.2.4.

Definition 2.2.7. The H-graph measurable interior o(Γ|H) of a set-valued mapping Γ is the largest H-graph measurable random open set Γ such that Γ (ω) ⊆ Γ(ω) a.s..

  graph measurable random subsets of O and we have O = n F n . Let us define the H-graph measurable open random set (see Corollary 2.2.4 and Proposition 2.2.6): Let us show that this is the largest H-graph measurable open random subset of O. To do so, let O H ⊆ O be a H-graph measurable open subset of O. As for O, we may write O H = n H n where H In the contrary case, on a non-null set, there exists o ∈ ∂O such that o ∈ B(x, d(x, ∂O H )). By Lemma 2.2.10, this implies that o ∈ O H ⊆ O which yields a contradiction.

	o(O|H) = int

n m(F n |H) ⊆ O. n := {x : d(x, ∂O H ) ≥ 1/n} ∩ cl O H are H-graph measurable closed subsets of O H .

We claim that H n ⊆ F n . Indeed, if x ∈ H n , it suffices to show that d(x, ∂O H ) ≤ d(x, ∂O) a.s.. As H n is H-graph measurable and closed, H n ⊆ m(F n |H) for all n. We deduce that O H

  Lemma 2.3.4. Let H ∈ H. Then, cl (Γ1 H |H) = 1 H cl (Γ|H), for every F-graph measurable set Γ. Proof. First, observe that P(Γ1 H ⊆ 1 H cl (Γ|H)) = 1 as 1 H cl (Γ|H) = {0} on Ω\H. We deduce that cl (Γ1 H |H) ⊆ 1 H cl (Γ|H) a.s. and cl (Γ1 H |H) = {0} on Ω\H. Let us define Z = cl (Γ1 H |H)1 H + cl (Γ|H)1 Ω\H . As Γ ⊆ Γ1 H on H, we deduce that Γ ⊆ Z a.s. hence cl (Γ|H) ⊆ Z a.s.. Therefore, we deduce that cl (Γ|H)1 H ⊆ Z1 H and finally cl (Γ|H)1 H ⊆ cl (Γ1 H |H) as the latter set is {0} on

It follows that ξ = ess inf H γ 1 ∈ R. By definition of the essential infemum, there is a non-null set on which ess inf H γ 1 + 1

  Consider a H-measurable closed set Γ H containing Γ and let us show that[ess inf H γ 1 , ess sup H γ 2 ] ∩ R ⊆ Γ H . [ess inf H γ 1 , γ 1 ] ∩ R or ξ ∈ [γ 2 , ess sup H γ 2 ] ∩ R.Let us consider the case where ξ ∈ [ess inf H γ 1 , γ 1 ] ∩ R, the other case being similar. If ess inf H γ 1 = γ 1 , then ξ = γ 1 ∈ Γ H , i.e., a contradiction. Therefore, ess inf H γ 1 < γ 1 and, since ξ is H-measurable and ξ ≤ γ 1 a.s., we deduce that ξ ≤ ess inf H γ 1 . H ) > 0. We first consider the case where ess sup H r < ∞. On the set A H

	such that P(ξ / ∈ Γ
	To do so, consider ξ ∈ L 0 ([ess inf H γ 1 , ess sup H γ 2 ] ∩ R, H) and suppose that P(ξ / ∈
	Γ H ) > 0. We suppose w.l.o.g. that P(ξ / ∈ Γ H ) = 1. As Γ ⊆ Γ H , we necessarily
	∈ Γ H . We may suppose have ξ ∈ 2 d(ξ, Γ H ) > γ 1 where d(ξ, Γ H ) > 0 as ξ / that ξ + 1 2 d(ξ, Γ H ) ∈ [γ 1 , γ 2 ] even if we have to change d(ξ, Γ H ) by a smaller F-measurable term. Therefore, ξ+ 1 2 d(ξ, Γ H ) ∈ Γ H and d(ξ, Γ H ) ≤ d(ξ, ξ+ 1 2 d(ξ, Γ H )) ≤ 1 2 d(ξ, Γ H ). We get a contradiction and, finally, cl (Γ|H) = [ess inf H γ 1 , ess sup H γ 2 ]∩R.
	Example 2.3.6. Let γ ∈ M(X , H) and r ∈ M((0, ∞), F). Consider Γ the random
	closed ball B(γ, r) of center γ and radius r. We shall prove that
	cl (Γ|H)) = B(γ, ess sup H r).
	First observe that Γ ⊆ B(γ, ess sup H r) a.s. Moreover, consider Γ H a H-graph mea-
	surable random closed set containing Γ a.s. Suppose that there is
	ξ ∈ L 0 (B(γ, ess sup H r), H)

  As cl (Γ|H) is H-graph measurable and closed-valued, it admits a Castaing representation cl (Γ|H)(ω) = cl{γ n (ω) : n ∈ N} where, for all n, γ n ∈ L 0 (cl (Γ|H), H), by Proposition 2.2.3, see also the lemma above.

	Proof. Step 1. We show that
	sup	h(x) = ess sup H {h(γ), γ ∈ L 0 (cl (Γ|H), H)}.
	x∈cl (Γ|H)
	To see it, notice that by Lemma 2.4.1,
	sup
	x∈cl (Γ|H)
		h(x), a.s..

  ] that inf z∈ B(γ,εn) h(z) is H-measurable. Therefore,

	E( inf z∈ B(γ,εn)	h(z)1 Λn |H) = inf z∈ B(γ,εn)	h(z)P( Λn |H).
	As P( Λn |H) > 0 a.s., we deduce by (2.4.1) that
	ess sup H {h(γ) : γ ∈ L 0 (Γ, F)} ≥ inf z∈ B(γ,εn)	h(z), a.s.
	for every n ≥ 1. Therefore,
	ess sup H {h(γ) : γ ∈ L 0 (Γ, F)} ≥ lim n	inf z∈ B(γ,εn)	h(z), a.s..
	Since B(γ, ε n ) is a.s. compact and h is a.s. lower semi-continuous, we deduce that
	inf z∈ B(γ,εn)

  t (S t ) is only F t+1 -measurable since S t is F t+1 -measurable. Here the notation xy is used to designate the Euler scalar product between two vectors x, y of R d . = θ t-1 ∆S t , where ∆S t := S t -S t-1 . Indeed, this dynamics holds if and only if we have -(θ 0

	In the following, we only consider self-financing portfolio processes V θ, i.e. they
	satisfy by definition:
	∆V θ t := V θ t -V θ t-1 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,

t-1 0 ) where ∆θ t (x) is F t -measurable and the executed quantity ∆θ

  -1 for all y ∈ cl (I t |F t ). 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 construction, α t-1 x + β t-1 ≤ αt-1 x + βn t-1 for all x ∈ cl (I t |F t ), where α t-1 , β t-1 are given in (3.3.8). It follows that αt-1 x + βn t-1 ≥ g t (S 1 , • • • , S t-1 , x), for every x ∈ cl (I t |F t ). By (3.3.9), we deduce that

  .14, empirical results are presented for different values of the strike K and a sample of 10 6 scenarios.

	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 * 10 -6 1, 9 * 10 -7 5.975 * 10 -8 0
	max( R )	0.18	0.19	0.195	0.187	0.187

A sequence (X t ) 0≤t≤T is said to be a martingale relative to the filtration (F t ) 0≤t≤T if every X t is integrable and E(X t | F s ) = X s , ∀s < t.

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

such that

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

], see Figure

3.7. Therefore, V 1 (S 1 ) = g 1 (S 1 ) := 0 and we deduce that θ 1 (S 1 ) = 0.

Acknowledgements

Bibliography 5th case: S

We obtain that (see Figure 3.12):

x

. V 0 (S 0 ) = ϕ 0 (S 0 ) =: g 0 (S 0 )

and

.

We have V 0 (S 0 ) = (S 0 -K) + =: g 0 (S 0 ) and θ 0 (S 0 ) = 1, see Figure 3.13.

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

Appendices

Appendix A

Conditional support and conditional essential supremum Definition 4.4.1. Let (Ω, F, P) be a probability space, H a sub-σ algebra of F and let X ∈ L 0 (R d , F). Then the conditional support of X with respect to H is the random set:

We remark that if H = {∅, Ω}, then supp H X is the support of X.

Proposition 4.4.2. Let H and F be complete σ-algebra such that H ⊂ F and Λ = (λ i ) i∈I be a family of real valued F-measurable random variables. There exists a unique random variable H-measurable λ H ∈ L 0 (R ∪ {∞}, H), denoted by ess sup H (Λ) that satisfies:

Symmetrically the essential infimum is defined as:

Lower semi-continuity Definition 4.4.3. Let f : R d → R be a function.

• We call the effective domain the set dom(f ) defined by:

• We say that the function f is proper if:

1. There exists x ∈ R d , such that f (x) < ∞.

For all

• The epigraph of f is the set:

The lower semi continuity can be characterised as fellow:

Theorem 4.4.5. The following properties of a function f : R d → R are equivalent:

Super-differential

Definition 4.4.6.

[2]

• Let f : R d → R be a convex function. We say that a vector v ∈ R d is a subgradient of f at x if it satisfies the following:

The set of subgradients at x is the sub-differential of f, denoted ∂f (x).

• If f is concave function satisfies the reverse inequality,i.e:

then we say that v is a supergradient of f at x.The set of supergradients is the super-differential and also denoted ∂f (x).

Fenchel-Lengendre conjugate

ABSTRACT

In this thesis we introduce two new types of conditional random sets taking values in a Banach space: the conditional interior and the conditional closure.

In the second part of the thesis, we apply the theoretical results established in the first part to solve the problem of super-hedging European or Asian options for discrete-time financial market models where executable prices are uncertain. We illustrate our method by a numerical example.

In the last part we show that, in discrete-time, it is possible to evaluate the minimal superhedging price when we restrict ourselves to integer-valued strategies. We formulate a dynamic programming principle that can be directly implemented on an historical data, and which also provides the optimal integer-valued strategy.

MOTS CLÉS

Ensemble aléatoire conditionnel, optimisation conditionnelle, problème de sur-réplication, Option européenne, prix de sur-réplication, information différées, incertitude, condition AIP , stratégie à valeurs entières, principe de programmation dynamique.

RÉSUMÉ

Dans cette thèse nous introduisons deux nouveaux types d'ensembles aléatoires conditionnels à valeurs dans un espace de Banach : l'intérieur conditionnel et la clôture conditionnelle des ensembles aléatoires. Dans la deuxième partie, nous appliquons les résultats théoriques établis dans la première partie pour résoudre le problème de sur-réplication des options européennes ou asiatiques pour les modèles de marchés financiers en temps discret où les prix exécutables sont incertains. Nous illustrons notre méthode par un exemple numérique. Dans la dernière partie nous montrons que, dans un modèle de marché en temps discret, il est possible d'évaluer le prix minimal de sur-réplication lorsqu'on se limite aux stratégies à valeurs entières. Nous formulons un principe de programmation dynamique qui peut être directement implémenté sur une donnée historique et qui fournit également la stratégie optimale à valeurs entières.

KEYWORDS

Conditional random set, conditional optimization, super-hedging problem, European option, super-hedging prices, delayed information, uncertainty, AIP condition, integer valued strategies, dynamic programming principle.