Jean-Marc Bonnisseau
email: jean-marc.bonnisseau@univ-paris1.fr

Achis Chery
email: achis.chery@malix.univ-paris1.fr

On the equivalence of financial structures with long-term assets

Keywords: Equivalent financial structures, financial equilibrium, multi-period model, long-term assets, financial sub-structure, reduced forms

 to prove the equivalence between a financial structure and its reduced forms and between to complete financial structures.

Introduction

We consider stochastic financial exchange economies defined on a given finite date-event tree representing time and uncertainty. The financial structures may include long-term assets. We study the equivalence relation on financial structures introduced in [START_REF] Aouani | Equivalent reduced form of a financial structure[END_REF][START_REF] Cornet | A remark on arbitrage free prices in multi-period economy[END_REF], when the portfolios of agents are unconstrained. Two financial structures are equivalent if, for each state price, the marketable payoffs are the same for the arbitrage free asset prices associated to the given state price. In other words, this means that the ranges of the full payoff matrices are identical.

A financial structure allows economic agents to transfer wealth across nodes of the date-event tree through the marketable payoff set. Thereby given a spot price p, the budget set of an agent is fully determined by the marketable payoff set. So the budget sets are the same for two equivalent financial structures. The main consequence is that, regardless of the standard exchange economy Σ, the existence of a financial equilibrium in Σ associated with a financial structure F is equivalent to the existence of equilibrium in Σ associated with any other financial structure F belonging to the equivalence class of F. Furthermore, the equilibrium consumption and the equilibrium spot price are the same, only the asset portfolios have to be suitably modified. Hence the importance of studying the notion of equivalence between the financial structures since the existence of a financial equilibrium for a given financial structure is extended to the equivalence class of this financial structure.

INTRODUCTION

3 Equivalent financial structures have been studied, among others, by Aouani-Cornet and Cornet-Ranjan [START_REF] Aouani | Existence of financial equilibria with restricted participations[END_REF][START_REF] Cornet | A remark on arbitrage free prices in multi-period economy[END_REF] in the two-period case. In [START_REF] Cornet | A remark on arbitrage free prices in multi-period economy[END_REF], it is proved that two financial structures are equivalent when the ranges of their payoff matrices are equal. We have generalized this result to the multi-period case if all assets are short-term in [START_REF] Bonnisseau | On the equivalence of financial structures with short-term assets[END_REF]. By means of examples in Section 4, we show that, with long-term assets, equality between the images of payoff matrices of two financial structures is neither necessary nor sufficient to get the equivalence of these structures.

So, to get the equivalence with long-term assets, we introduce an additional assumptions, Assumptions R1, on the payoff matrices of two financial structures. Precisely, Assumption R1 means that, at each emission node ξ, the assets issued at node ξ for the two structures offer the same possibilities of transfer for the successors of ξ. In other words, the marketable payoffs generated by the assets issued at the same node are identical for the two structures. For a two-period economy, this just means that the ranges of the payoff matrices are the same since there is a unique emission node. The main result of the chapter is that Assumption R1 is sufficient to get the equivalence of the financial structures. Nevertheless, note that Assumption R1 is not necessary to get the equivalence as illustrated in an example in Section 4.

We apply this result to the special case of a financial structures with retrading where assets are issued at date 0 and then re-traded at every node like in Magill-Quinzii [START_REF] Magill | Theory of Incomplete Markets[END_REF]. In this case, note that a financial structure is fully described by the primitive payoff matrix of assets issued at date 0. To do the link with the previous model, following [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF], we introduce the re-trading extension of the financial structure by considering that the re-trade of an asset is equivalent to issuing a new asset. We prove that the equality between the ranges of the two primitive payoff matrices of two financial structures with re-trading is equivalent with the fact that they satisfy Assumption R1, so implies the equivalence of their re-trading extensions.

Then, we study the equivalence of a financial structure with their reduced forms. A reduced form is obtained by removing the redundant assets. This concept is extensively studied in the two period case in [START_REF] Aouani | Characterizing reduced financial structures[END_REF][START_REF] Aouani | Equivalent reduced form of a financial structure[END_REF][START_REF] Aouani | Financial Economies with Restricted Participation[END_REF]. The interest for studying this question comes from the methodology to prove the existence of a financial equilibrium. Indeed, we need a fixed point argument, so we need the compactness of attainable portfolios. With a financial structure, we may not have bounded attainable portfolios due to the presence of redundant assets. So, a way to get an equilibrium is: considering a reduced form by removing redundant assets; obtaining bounded attainable portfolios for the reduced form; proving the existence of an equilibrium for the reduced form; getting an equilibrium for the original economy by equivalence.

We provide an example showing that a structure may not be equivalent with their reduced form. To get the equivalence, we borrow Assumption R from [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]. A financial structure satisfies Asssumption R if the returns of the assets issued at a node ξ are not redundant with the returns of the assets issued previously. We show that a financial structure satisfying this assumption is equivalent to their reduced forms.

Finally, using a result of [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], we prove that two complete financial structures Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 4 are equivalent if both satisfy Assumption R.

In Section 2, we describe the general framework of a financial exchange economy and we define a financial equilibrium. In Section 3, we state the definition of equivalence between two financial structures and we prove the key result on the link between financial equilibrium for two equivalent financial structures. In Section 4, we present and comment our key assumption R1, we prove the equivalence under R1 and then we develop the applications for the re-trading case, the reduced forms and finally the complete financial structures.

Financial exchange economy and equilibrium

In this section, we present the model and the notations, which are borrowed from Angeloni-Cornet [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF] and are essentially the same as those of Magill-Quinzii [START_REF] Magill | Theory of Incomplete Markets[END_REF].

Time and uncertainty

We 1 consider a multi-period exchange economy with (T + 1) dates, t ∈ T := {0, ..., T }, and a finite set of agents I. The uncertainty is described by a dateevent tree D of length T + 1. The set D t is the set of nodes (also called dateevents) that could occur at date t and the family (D t ) t∈T defines a partition of the set D; for each ξ ∈ D, we denote by t(ξ) the unique date t ∈ T such that ξ ∈ D t .

At date t = 0, there is a unique node ξ 0 , that is D 0 = {ξ 0 }. As D is a tree, each node ξ in D\{ξ 0 } has a unique immediate predecessor denoted pr(ξ) or ξ -. The mapping pr maps D t to D t-1 . Each node ξ ∈ D \ D T has a set of immediate successors defined by ξ + = ξ ∈ D : ξ = ξ-.

For τ ∈ T \ {0} and ξ ∈ D \ ∪ τ -1 t=0 D t , we define pr τ (ξ) by the recursive formula: pr τ (ξ) = pr pr τ -1 (ξ) . We then define the set of successors and the set of predecessors of ξ as follows:

D + (ξ) = {ξ ∈ D : ∃τ ∈ T \ {0} | ξ = pr τ (ξ )} D -(ξ) = {ξ ∈ D : ∃τ ∈ T \ {0} | ξ = pr τ (ξ)} For each ξ ∈ D, we note by D (ξ) the union of ξ with D + (ξ). If ξ ∈ D + (ξ) [resp. ξ ∈ D (ξ)], we use the notation ξ > ξ [resp. ξ ≥ ξ]. Note that ξ ∈ D + (ξ) if and only if ξ ∈ D -(ξ ) and similarly ξ ∈ ξ + if and only if ξ = (ξ )
-.

1 We use the following notations. A (D × J )-matrix A is an element of R D×J , with entries (a j ξ ) (ξ∈D,j∈J ) ; we denote by A ξ ∈ R J the ξ-th row of A and by A j ∈ R D the j-th column of A. We recall that the transpose of A is the unique (J × D)-matrix t A satisfying (Ax)

• D y = x • J t Ay for every x ∈ R J , y ∈ R D , where • D [resp. • J ] denotes the usual inner product in R D [resp. R J ].
We denote by rankA the rank of the matrix A and by Vect (A) the range of A, that is the linear sub-space spanned by the column vectors of A. For every subset D ⊂ D and J ⊂ J , the matrix A J D is the ( D× J )-sub-matrix of A with entries a j ξ for every (ξ, j) ∈ ( D× J ). Let x, y be in R n ; x ≥ y (resp. x y ) means x h ≥ y h (resp. x h > y h ) for every h = 1, . . . , n and we let

R n + = {x ∈ R n : x ≥ 0}, R n ++ = {x ∈ R n : x 0}.
We also use the notation x > y if x ≥ y and x = y. The Euclidean norm in the different Euclidean spaces is denoted . and the closed ball centered at x and of radius r > 0 is denoted B(x, r)

:= {y ∈ R n | y -x ≤ r}.
Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

A simple example of D • ξ 1 • ξ 11 • ξ 2 • ξ 12 • ξ 13 • ξ 22 • ξ 21 • ξ 0 t = 0 t = 1 t = 2
: D = {ξ 0 , ξ 1 , ξ 2 , ξ 11 , ξ 12 , ξ 13 , ξ 21 , ξ 22 }, T = 2, the length of D is 3, D 2 = {ξ 11 , ξ 12 , ξ 13 , ξ 21 , ξ 22 }, ξ + 1 = {ξ 11 , ξ 12 , ξ 13 }, D + (ξ 2 ) = {ξ 21 , ξ 22 }, t(ξ 11 ) = t(ξ 12 ) = t(ξ 13 ) = t(ξ 21 ) = t(ξ 22 ) = 2, D -(ξ 11 ) = {ξ 0 , ξ 1 }.

The financial structure

At each node ξ ∈ D, there is a spot market on which a finite set H = {1, . . . , H} of divisible and physical goods are exchanged. We assume that each good is perishable, that is, its life does not last more than one date. In this model, a commodity is a pair (h, ξ) of a physical good h ∈ H and the node ξ ∈ D at which the good is available. Then the commodity space is R L , where

L = H × D. An element x ∈ R L is called a consumption, that is to say x = (x (ξ)) ξ∈D ∈ R L , where x (ξ) = (x (h, ξ)) h∈H ∈ R H for each ξ ∈ D.
We denote by p = (p(ξ)) ξ∈D ∈ R L the vector of spot prices and p (ξ) = (p (h, ξ)) h∈H ∈ R H is called the spot price at node ξ. The spot price p (h, ξ) is the price at the node ξ for immediate delivery of one unit of the physical good h. Thus the value of a consumption x (ξ) at node ξ ∈ D (measured in unit account of the node ξ) is

p (ξ) • H x (ξ) = h∈H p (h, ξ) x (h, ξ) .
The financial structure is constituted by a finite set of assets denoted J = {1, . . . , J}. An asset j ∈ J is a contract issued at a given and unique node in D denoted ξ(j) and called issuance node of j. Each asset is bought or sold only at its issuance node ξ(j) and yields payoffs only at the successor nodes ξ of D + (ξ(j)). To simplify the notation, we consider the payoff of asset j at every node ξ ∈ D and we assume that it is zero if ξ is not a successor of the issuance node ξ(j). The payoff may depend upon the spot price vector p ∈ R L and is Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 6 denoted by V j ξ (p). Formally, we assume that V j ξ (p) = 0 if ξ / ∈ D + (ξ (j)). An asset is a short term asset if it has a non-zero payoff only at the immediate successors of the issuance node, that is, V j ξ (p) = 0 if ξ / ∈ ξ + . In the following, we consider only non trivial assets, that is each asset has a non zero return in at least one node. z = (z j ) j∈J ∈ R J is called the portfolio of agent i. If z j > 0 [resp. z j < 0], then |z j | is the quantity of asset j bought [resp. sold] by agent i at the issuance node ξ (j).

To summarize a financial structure F = J , (ξ (j)) j∈J , V consists of a set of non trivial assets J , a node of issuance ξ(j) for each asset j ∈ J , a payoff mapping V : R L → R D×J which associates to every spot price

p ∈ R L the (D × J )-payoff matrix V (p) = V j ξ (p) ξ∈D,j∈J
and satisfies the condition

V j ξ (p) = 0 if ξ / ∈ D + (ξ (j)).
The price of asset j is denoted by q j ; it is paid at its issuance node ξ(j). We let q = (q j ) j∈J ∈ R J be the asset price vector.

The full payoff matrix W (p, q) is the (D × J )-matrix with the following entries:

W j ξ (p, q) := V j ξ (p) -δ ξ,ξ(j) q j , where δ ξ,ξ = 1 if ξ = ξ and δ ξ,ξ = 0 otherwise. So, given the prices (p, q), the full flow of returns for a given portfolio z ∈ R J is W (p, q)z and the full return at node ξ is

[W (p, q)z] (ξ) := W ξ (p, q) • J z = j∈J V j ξ (p) z j - j∈J δ ξ,ξ(j) q j z j = {j∈J | ξ(j)<ξ} V j ξ (p) z j - {j∈J | ξ(j)=ξ} q j z j ,
We now recall that for a given spot price p, the asset price q is an arbitrage free price if it does not exist a portfolio z ∈ R J such that W (p, q)z > 0. q is an arbitrage free price if and only if it exists a so-called state price vector λ ∈ R D ++ such that t W (p, q)λ = 0 (see, e.g. Magill-Quinzii [START_REF] Magill | Theory of Incomplete Markets[END_REF]). Taken into account the particular structure of the matrix W (p, q), this is equivalent to ∀j ∈ J , λ ξ(j) q j = ξ∈D + (ξ(j))

λ ξ V j ξ (p).
Conversely, for a given state price vector λ ∈ R D ++ , there exists a unique associated arbitrage free price denoted q(λ) satisfying t W (p, q)λ = 0, which is defined by the above formula.

Some additional notations

Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

FINANCIAL EXCHANGE ECONOMY AND EQUILIBRIUM

7

We now introduce some additional notations. For all ξ ∈ D \ D T , J (ξ) is the set of assets issued at the node ξ, that is J (ξ) = {j ∈ J | ξ (j) = ξ} and J (D -(ξ)) is the set of assets issued at a predecessor of ξ, that is J (D -(ξ)) = {j ∈ J | ξ (j) < ξ}. D e is the set of nodes at which there is the issuance of at least one asset, that is, ξ ∈ D e if J (ξ) = ∅. If ξ / ∈ D e , J (ξ) = ∅ and, by convention, we let ImV J (ξ) (p) = {0}.

In all our numerical examples, we assume that there is a unique good at each node of the tree and the price of one unit of the good is equal to 1. Consequently, we will denote the payoff matrix (resp. the full payoff matrix) by V (resp. W (q)).

The stochastic exchange economy

We consider a finite set of consumers I = {1, . . . , I}. Each agent i ∈ I has a consumption set X i ⊂ R L , which consists of all possible consumptions. An allocation is an element x ∈ i∈I X i and we denote by x i the consumption of agent i, which is the projection of x on X i . The tastes of each consumer i ∈ I are represented by a strict preference correspondence P i : j∈I X j -→ X i , where P i (x) defines the set of consumptions that are strictly preferred to x i for agent i, given the consumption x j for the other consumers j = i. P i represents the consumer tastes, but also his behavior with respect to time and uncertainty, especially his impatience and attitude toward risk. If consumer preferences are represented by utility functions u i : X i -→ R for each i ∈ I, the strict preference correspondence is defined by

P i (x) = {x i ∈ X i |u i (x i ) > u i (x i )}.
Finally, for each node ξ ∈ D, every consumer i ∈ I has a node endowment e i (ξ) ∈ R H (contingent on the fact that ξ prevails) and we denote by e i = (e i (ξ)) ξ∈D ∈ R L the endowments for the whole set of nodes. The exchange economy Σ can be summarized by Σ = D, H, I, (X i , P i , e i ) i∈I .

Financial equilibrium

We now consider a financial exchange economy, which is defined as the couple of an exchange economy Σ and a financial structure F. It can thus be summarized by

(Σ, F) := D, H, I, (X i , P i , e i ) i∈I , J , (ξ (j)) j∈J , V . Given the price (p, q) ∈ R L × R J , the budget set of consumer i ∈ I is B i F (p, q) defined by 2 : (x i , z i ) ∈ X i × R J : ∀ξ ∈ D, p (ξ) • H [x i (ξ) -e i (ξ)] ≤ W ξ (p, q) • J z i or (x i , z i ) ∈ X i × R J : p2 (x i -e i ) ≤ W (p, q) z i .
We now introduce the definition of a financial equilibrium:

8 Definition 2.
1. An equilibrium of the financial exchange economy (Σ, F) is a list of strategies and prices (x, z, p, q

) ∈ R L I × R J I × R L \ {0} × R J such that (a)
for every i ∈ I, (x i , zi ) maximizes the preferences P i in the budget set B i F (p, q), in the sense that

(x i , zi ) ∈ B i F (p, q) and P i (x) × R J B i F (p, q) = ∅;
(b) i∈I xi = i∈I e i and i∈I zi = 0.

We recall that the equilibrium asset price is arbitrage free under the following Non-Satiation Assumption:

Assumption NS (i) (Non-Saturation at Every Node) For all x ∈ i∈I X i such that i∈I xi = i∈I e i , for every i ∈ I, for every ξ ∈ D, there exists

x i ∈ X i such that, for each ξ = ξ, x i (ξ ) = xi (ξ ) and x i ∈ P i (x). (ii) if x i ∈ P i (x), then [x i , xi [∈ P i (x). Proposition 2.1. (Magill-Quinzii [9], Angeloni-Cornet [1]) Under Assumption (NS), if (x, z, p, q
) is an equilibrium of the economy (Σ, F) then the asset price q is arbitrage free i.e., there exists a state price λ ∈ R D ++ such that t W (p, q)λ = 0.

Equivalent financial structures

In this section we will define an equivalence relation on financial structures. We will show that the existence of an equilibrium in an exchange economy associated with a given financial structure is equivalent to the existence of equilibrium in exchange economy associated with any other financial structure equivalent to the first one. So equivalence allows to extend the existence results for financial equilibrium to a whole class of financial structures. Hence the importance of studying the notion of equivalence between the financial structures.

Definition 3.1. Let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 )
be two financial structures. We say that F 1 is equivalent to F 2 with respect to a given spot price p (denoted by

F 1 p F 2 ) if for all state price λ = (λ ξ ) ξ∈D ∈ R D ++ , ImW 1 (p, q 1 (λ)) = ImW 2 (p, q 2 (λ))
where q 1 (λ) and q 2 (λ) are the unique arbitrage free prices associated with λ.

We say that F 1 is equivalent to the F 2 if for all spot price vector p ∈ R L ,

F 1 p F 2 .
The intuition behind this definition is that the financial structures allow agents to transfer wealth across nodes of the date-event tree. Thereby given a spot price p, their budget set is determined by the set of marketable payoffs that is the range of the full payoff matrix. To be equivalent, two financial structures must provide the same set of marketable payoffs whatever is the state price and the associated arbitrage free asset prices. The proof is left to the reader.

The main consequence of this definition is given below and states that, regardless of the standard exchange economy Σ, consumption equilibria are the same when agents carry out their financial activities through two different equivalent structures F 1 and F 2 . Proposition 3.2. Let Σ be an exchange economy satisfying Assumption NS.

Let F 1 = J 1 , (ξ(j)) j∈J1 , V 1 and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two equivalent financial structures.
Let (x, z, p, q1 ) be an equilibrium of (Σ, F 1 ). Then there exists ẑ and q2 such that (x, ẑ, p, q2 ) is an equilibrium of (Σ, F 2 ).

Proof of Proposition 3.2. Since (x, z, p, q1 ) is an equilibrium, Proposition 2.1 implies that q1 is an arbitrage free price. So, there exists a state price λ = (λ ξ ) ξ∈D ∈ R D ++ such that t W 1 (p, q1 )λ = 0. Let q2 be the unique arbitrage free price for the financial structure F 2 associed with λ. Since F 1 F 2 , we have

ImW 1 (p, q1 ) = ImW 2 (p, q2 ) (See Definition 3.1). For all i = 1, let ẑ ∈ R J2I be such that W 1 (p, q1 )z i = W 2 (p, q2 )ẑ i . Such ẑi exists because ImW 1 (p, q1 ) = ImW 2 (p, q2 ). Let ẑ1 = - i∈I;i =1 ẑi .
We now show that (x, ẑ, p, q2 ) is an equilibrium of (Σ, F 2 ). Indeed, for all i ∈ I,

W 2 (p, q2 )ẑ i = W 1 (p, q1 )z i This is obvious for i = 1 and if i = 1, as i∈I zi = 0, W 2 (p, q2 )ẑ 1 = W 2 (p, q2 )(- i∈I;i =1 ẑi ) = - i∈I;i =1 [W 2 (p, q2 )ẑ i ] = - i∈I;i =1 [W 1 (p, q1 )z i ] = W 1 (p, q1 )(- i∈I;i =1 zi ) = W 1 (p, q1 )z 1
With this remark, we easily prove that B i F1 (p, q1 ) = B i F2 (p, q2 ) and (x i , ẑi ) ∈ B i F2 (p, q2 ) all i, which is enough to conclude since the feasibility conditions are satisfied.

Examples of equivalent financial stuctures

We will give some examples of equivalent financial structures. The proofs are given in Appendix.

Example 3.1. ( Scalar multiplicator) Let F = (J , (ξ(j)) j∈J , V ) be a finan- cial structure. For each α ∈ R \ {0}, the α-product of F, F α = (J , (ξ(j)) j∈J , V α = αV )
) Let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two financial structures. The financial struc- ture 3 F := F 1 ∪ F 2 := (J := J 1 J 2 , (ξ(j)) j∈J , V = [V 1 , V 2 ]) is called the Union of F 1 and F 2 , . F 1 ∪ F 2 F 2 ∪ F 1 and if F 1 p F 2 , then F 1 ∪ F 2 p F 1 .
Combining the two previous examples we obtain :

Example 3.3. Let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 )
be two financial structures such that F 1 p F 2 with respect to the spot price vector p. For each pair (α,

β) ∈ R * × R * , the structure F α,β := αF 1 ∪ βF 2 = (J = J 1 J 2 , (ξ(j)) j∈J , V α,β = [αV 1 , βV 2 ]
) is equivalent to F 1 and to F 2 with respect to p.

) Let F 1 , F 2 , F 3 , F 4 be four financial structures. Let a spot price p ∈ R L , such that F 1 p F 2 and F 3 p F 4 then F 1 ∪ F 3 p F 2 ∪ F 4 .
Definition 3.2. Let F = (J , (ξ(j)) j∈J , V ) be a financial structure. We call sub-structure of F any financial structure F = (J , (ξ(j)) j∈J , V ) such that J ⊂ J and V = V J .

The following proposition is a consequence of Example 3.4.

Proposition 3.3. Given a spot price p ∈ R L , let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 )
and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two financial structures such that there is a sub-structure F 3 of F 2 which is equivalent to F 1 with respect to a spot price p. Then we can complete the structure F 1 to get a new financial structure F such that F p F 2 .

Sufficient conditions for equivalence

In this section we provide sufficient conditions on the payoff matrices for the equivalence of financial structure with long-term assets. In the two-period case, two financial structures are equivalent if the images of their payoff matrices are equal,(see [START_REF] Cornet | A remark on arbitrage free prices in multi-period economy[END_REF]). In the multi-period case, if all assets are short-term, we have generalized this result in [START_REF] Bonnisseau | On the equivalence of financial structures with short-term assets[END_REF].

In the multi-period case, if there are long-term assets, the equivalence between two financial structure does not imply that the images of payoff matrices of two financial structures are equal (see below Remark 4.1) and equality between the images of payoff matrices of two financial structures does not imply that these two financial structures are equivalent (see below Remark 4.2).

Remark 4.1. Let D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 }, J 1 = j 1 1 , j 2 1 , j 3 1 , J 2 = j 1 2 , j 2 2 , j 3 2 . ξ(j 1 1 ) = ξ(j 1 2 ) = ξ 0 , ξ(j 2 1 ) = ξ(j 2 2 ) = ξ 1 and ξ(j 3 1 ) = ξ(j 3 2 ) = ξ 2 . Let λ = (λ 0 , λ 1 , λ 2 , λ 3 ) ∈ R 4
++ be a state price and (q 1 , q 2 ) ∈ R J1 × R J2 be the couple of arbitrage free prices for the two financial structures associated to λ. The payoff matrices and the full payoff matrices are:

• ξ 0 • ξ 1 • ξ 2 • ξ 3 t = 0 t = 1 t = 2 t = 3
V 1 =     0 0 0 1 0 0 1 1 0 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3 and W 1 (q 1 ) =     -λ1+λ2+λ3 λ0 0 0 1 -λ2+λ3 λ1 0 1 1 -λ3 λ2 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3 V 2 =     0 0 0 1 0 0 1 0 0 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3 and W 2 (q 2 ) =     -λ1+λ2+λ3 λ0 0 0 1 -λ3 λ1 0 1 0 -λ3 λ2 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3
It is clear that rankW 1 (q 1 ) = rankW 2 (q 2 ) = 3, so ImW 2 (q 2 ) = ImW 1 (q 1 ) = λ ⊥ . So the two structures are equivalent although

ImV 1 = ImV 2 because rankV 1 = 3 = rankV 2 = 2.
Remark 4.2. Consider two financial structures such that each contains three assets and D = {ξ ) , ξ 1 , ξ 2 , ξ 11 , ξ 12 , ξ 13 , ξ 21 }. ξ(j 1 1 ) = ξ(j 1 2 ) = ξ(j 2 2 ) = ξ 0 and ξ(j 2 1 ) = ξ(j 3 1 ) = ξ(j 3 2 ) = ξ 1 . 12

• ξ 1 • ξ 0 • ξ 11 • ξ 12 • ξ 13 • ξ 2 • ξ 21 t = 0 t = 1 t = 2
V 1 = V 2 =           0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0           ξ 0 ξ 1 ξ 2 ξ 11 ξ 12 ξ 13 ξ 21
With λ = (1, 1, 1, 1, 1, 1, 1) and q 1 and q 2 the two associated arbitrage free prices, we have:

W 1 (q 1 ) =           -4 0 0 1 -1 -1 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0           ξ 0 ξ 1 ξ 2 ξ 11 ξ 12 ξ 13 ξ 21 and W 2 (q 2 ) =           -4 -1 0 1 0 -1 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0           ξ 0 ξ 1 ξ 2 ξ 11 ξ 12 ξ 13 ξ 21
The two structures are not equivalent since ImW 1 (q 1 ) = ImW 2 (q 2 ) even if the two payoff matrices have the same image. Indeed, we can check that the second column vector of the matrix W 2 (q 2 ) does not belong to ImW 1 (q 1 ).

Equivalence under Assumption R1

To get the equivalence, we need an additional assumption on the payoff matrices that we now introduce:

Let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two financial structures defined on the same date-event tree D and p ∈ R L be a spot price vector.

Assumption R1. ∀ξ ∈ D e 1 ∪ D e 2 , ImV 1 J1(ξ) (p) = ImV 2 J2(ξ) (p).
Assumption R1 means that at each emission node ξ, both structures offer the same possibilities of transfer between successor nodes to ξ. In the two-period case, Assumption R1 at p simply means ImV 1 (p) = ImV 2 (p) since there is only one emission node, ξ 0 , and ImV 1 (p) = ImV 1 J1(ξ0) (p). So Assumption R1 can be seen as the natural extension of the standard assumption on the equality of the range of the payoff matrices when there are more than one issuance node. We provide additional comments on Assumption R1 in Sub-section 4.2.

Note that since trivial assets are excluded, if Assumption R1 is satisfied, then the issuance nodes are the same for both financial structures.

We now state the main result of this paper on equivalence with long-term assets when markets are incomplete or not. Conversely, let λ ∈ R D ++ be a state price and let q 1 and q 2 be the two associated arbitrage free prices. Let y ∈ ImW 1 (p, q 1 ). There exists

z 1 ∈ R J1 such that y = W 1 (p, q 1 )z 1 = ξ∈D e   j∈J1(ξ) W 1,j (p, q 1 )z 1,j   .
Let ξ ∈ D be given. We have:

j∈J1 W 1,j ξ (p, q 1 )z 1,j = j∈J1 V 1,j ξ (p)z 1,j if ξ / ∈ D e j∈J1\J1(ξ) V 1,j ξ (p)z 1,j -j∈J1(ξ) q 1 j z 1,j if ξ ∈ D e
Since, by Assumption, for all η ∈ D e , ImV 1 J1(η

) (p) = ImV 2 J2(η) (p), there exists z 2 ∈ R J2 such that, for all η ∈ D e , j∈J1(η) V 1,j (p)z 1,j = j∈J2(η) V 2,j (p)z 2,j . This implies that j∈J1 W 1,j ξ (p, q 1 )z 1,j = j∈J2 V 2,j ξ (p)z 2,j if ξ / ∈ D e j∈J2\J2(ξ) V 2,j ξ (p)z 2,j -j∈J1(ξ) q 1 j z 1,j if ξ ∈ D e
But, since t W 1 (p, q 1 )λ = 0, with ξ ∈ D e , j∈J1(ξ)

q 1 j z 1,j = j∈J1(ξ)   ξ ∈D + (ξ) λ ξ λ ξ V 1,j ξ (p)   z 1,j = 1 λ ξ ξ ∈D + (ξ)   λ ξ   j∈J1(ξ) V 1,j ξ (p)z 1,j     . Since for all η ∈ D e , j∈J1(η) V 1,j (p)z 1,j = j∈J2(η) V 2,j (p)z 2,j , for each ξ ∈ D + (ξ) j∈J1(ξ) V 1,j ξ (p)z 1,j = j∈J2(ξ) V 2,j ξ (p)z 2,j .
Consequently, since t W 2 (p, q 2 )λ = 0, Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

1 λ ξ ξ ∈D + (ξ)   λ ξ   j∈J1(ξ) V 1,j ξ (p)z 1,j     = 1 λ ξ ξ ∈D + (ξ)   λ ξ   j∈J2(ξ) V 2,j ξ (p)z 2,j     = j∈J2(ξ)   ξ ∈D + (ξ) λ ξ λ ξ V 2,j ξ (p)   z 2,j = j∈J2(ξ) q 2 j z 2,j . Hence j∈J1 W 1,j ξ (p, q 1 )z 1,j = j∈J2 V 2,j ξ (p)z 2,j if ξ / ∈ D e j∈J2\J2(ξ) V 2,j ξ (p)z 2,j -j∈J1(ξ) q 2 j z 2,j if ξ ∈ D e = j∈J2 W 2,j ξ (p, q 2 )z 2,j
So for all ξ ∈ D, y ξ = j∈J1 W 1,j ξ (p, q 1 )z 1,j = j∈J2 W 2,j ξ (p, q 2 )z 2,j . Consequently y ∈ ImW 2 (p, q 2 ) hence ImW 1 (p, q 1 ) ⊂ ImW 2 (p, q 2 ). With a similar reasoning, we can show that ImW 2 (p, q 2 ) ⊂ ImW 1 (p, q 1 ). So ImW 2 (p, q 2 ) = ImW 1 (p, q 1 ), that is

F 1 p F 2 .
Remark 4.3. Assumption R1 is not necessary for the equivalence of financial structures. Indeed, we provide now two equivalent financial structures, which do not satisfy Assumption R1. Let D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 } as in Remark 4.1,

J 1 = j 1 1 , j 2 1 , j 3 1 , J 2 = j 1 2 , j 2 2 , j 3 2 . ξ(j 1 1 ) = ξ(j 2 1 ) = ξ(j 1 2 ) = ξ(j 2
2 ) = ξ 0 , ξ(j 3 1 ) = ξ 2 and ξ(j 3 2 ) = ξ 1 . Let λ = (λ 0 , λ 1 , λ 2 , λ 3 ) be a state price and let (q 1 , q 2 ) ∈ R J1 × R J2 be the couple of associated arbitrage free prices.

The payoff matrices and the full payoff matrices are:

V 1 =     0 0 0 1 0 0 0 1 0 0 1 1     ξ 0 ξ 1 ξ 2 ξ 3 and W 1 (q 1 ) =     -λ1 λ0 -λ2-λ3 λ0 0 1 0 0 0 1 -λ3 λ2 0 1 1     ξ 0 ξ 1 ξ 2 ξ 3 V 2 =     0 0 0 1 0 0 0 1 -1 1 -1 0     ξ 0 ξ 1 ξ 2 ξ 3 and W 2 (q 2 ) =     -λ1-λ3 λ0 -λ2+λ3 λ0 0 1 0 λ2 λ1 0 1 -1 1 -1 0     ξ 0 ξ 1 ξ 2 ξ 3
Assumption R1 is not satisfied since the issuance nodes are not the same for the two financial structures. We have rankW 1 (q 1 ) = rankW 2 (q 2 ) = 3. Indeed, the rank of the square sub-matrix A 1 (resp. A 2 ) composed by the three last Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 rows of W 1 (q 1 ) (resp. W 2 (q 2 )) is equal to 3 because DetA 1 = 1 + λ3 λ2 (resp. DetA 2 = -1 -λ2 λ1 ) which is always different from zero. So, one can conclude that ImW 1 (q 1 ) = ImW 2 (q 2 ) = λ ⊥ , hence the financial structures are equivalent.

Comments on Assumption R1

We first remark that Assumption R1 is stronger than assuming the equality of the images of the two payoff matrices. Indeed, in Example 4.3, ImV 1 = ImV 2 but Assumption R1 is not satisfied. The converse is true as already noticed at the beginning of the proof of Proposition 4.1.

With short term assets, Assumption R1 is equivalent to the equality of the images of the payoff matrices. Lemma 4.1. Let p ∈ R L be a spot price vector. Let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two financial structures with only short-term assets. Then ImV 1 (p) = ImV 2 (p) if and only if Assumption R1 holds true at the spot price p.

The proof of Lemma 4.1 is given in Appendix.

The next proposition provides sufficient conditions under which Assumption R1 holds true. We first recall the assumption introduced by Bonnisseau and Chéry in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]. Given a spot price vector p ∈ R L , and a payoff matrix V (p).

Assumption R. ∀ξ ∈ D e , Vect V J (D -(ξ)) D + (ξ) (p) Vect V J (ξ) D + (ξ) (p) = {0} .
Assumption R means that the returns of the assets issued at a node ξ are not redundant with the returns of the assets issued at a predecessor node of ξ. So, the issuance of additional assets at ξ are a true financial innovation since the payoffs in the successors of ξ cannot be replicated by the payoffs of a portfolio built with the assets issued before ξ. Proposition 4.2. Let p ∈ R L be a spot price vector. Let F 1 and F 2 be two financial structures. They satisfy Assumption R1 at the spot price p if one of the following conditions is satisfied: (i) All assets are issued at the same date τ 1 and ImV 1 (p) = ImV 2 (p).

(ii) For all ξ ∈ D e 1 ∪ D e 2 , ImV 1 J1(ξ) (p) ⊂ ImV 2 J2(ξ) (p)
, Assumption R is satisfied at the spot price p by both financial structures and rankV 1 (p) = rankV 2 (p).

The proof is given in Appendix.

We show in the next sub-section that Assumption R1 is satisfied when we consider re-trading extensions of financial structures with emission of primitive assets only at node ξ 0 and equality of the ranges of the payoff matrices associated to the primitive assets.

Documents de

Equivalence with re-trading

With more than two periods, the financial structure is often built upon primitive assets issued at the initial nodes providing payoffs for the future periods and then re-traded at the successive periods. See Magill-Quinzii [START_REF] Magill | Theory of Incomplete Markets[END_REF] for a complete description. In Angeloni-Cornet [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF], it is shown that the re-trading of an asset can be interpreted as the issuance of a new asset with the payoff being the truncation of the payoffs of the initial asset for the future nodes. They have also proved that we get equivalent financial structures by considering a payoff matrix à la Magill-Quinzii where the asset prices appears two times since the asset can be sold and bought at each non-terminal node or the payoff matrix à la Angeloni-Cornet, where each asset is exchanged only once at the issuance node.

For the sake of simplicity and consistency with the previous sections, we consider the Angeloni-Cornet approach. We show that the equality of the range of the payoff matrices for the primitive assets is sufficient for the equivalence of the re-trading extension.

Let F = J , (ξ(j)) j∈J , V be a financial structure such that all assets are issued at node ξ 0 . Suppose that each asset j, once issued, is re-traded at all succeeding nodes except terminal nodes. Each re-traded asset at a node ξ ∈ D - is considered as a new asset j ξ issued at node ξ. The new financial structure thus constituted is called the re-trading extension of the primitive financial structure. Definition 4.1. Let F = J , (ξ(j)) j∈J , V be a financial structure where ξ(j) = ξ 0 for all j. The re-trading of asset j ∈ J at node ξ , denoted j ξ , is the asset issued at ξ , that is, ξ(j ξ ) = ξ , and whose flow of payoffs is given by

Ṽ j ξ ξ (p) = V j ξ (p), if ξ ∈ D + (ξ ); Ṽ j ξ ξ (p) = 0 otherwise.
The re-trading extension of F is the the new financial structure

J = J × D -, (ξ(j )) j ∈ J , Ṽ ,
which consists of all primitive assets j ∈ J and of all re-trading assets (j ξ ) to all nodes ξ ∈ D + (ξ 0 )\{D T }.

A simple example

Let F be a financial structure constituted of two financial assets {j 1 , j 2 } issued at the first date such that

V j 1 = t (1, 3, -2, 1, 1, 4) and V j 2 = t (2, -2, 3, -1, 1, 1)
and D = {ξ 0 , ξ 1 , ξ 2 , ξ 11 , ξ 12 , ξ 21 , ξ 22 }.

The payoff matrix of the re-trading extension F of F is: Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

• ξ 1 • ξ 11 • ξ 2 • ξ 21 • ξ 22 • ξ 12 • ξ 0 t = 0 t = 1 t = 2 Figure 4: the tree D Ṽ =           0 0 0 0 0 0 1 2 0 0 0 0 3 -2 0 0 0 0 -2 3 -2 3 0 0 1 -1 1 -1 0 0 1 1 0 0 1 1 4 1 0 0 4 1           ξ 0 ξ 1 ξ 2 ξ 11 ξ 12 ξ 21 ξ 22
Notations:

For each ξ ∈ D -, we denote by qj ξ the price of asset j ξ (i.e., the re-trading of asset j at node ξ ), which is sometimes also called the re-trading price of asset j at node ξ . So, for the financial structure F, both the asset price vector q = (q j ξ ) j ξ ∈ J =J ×D -and the portfolio z = (z j ξ ) (j ξ )∈ J =J ×D -now belong to

R J ×D -.
Given a spot price p ∈ R L an asset price vector q ∈ R J ×Dand a portfolio z ∈ R J ×D -, the full financial return of z for the financial structure F at node ξ ∈ D is given by: So, the full payoff matrix of the previous example is:

if ξ = ξ 0 , W F ξ (p, q) • J z = -j ξ 0 ∈ J (ξ0)=J ×{ξ0} qj ξ 0 z j ξ 0 if ξ ∈ D -∩ D + , W F ξ (p, q) • J z is equal to j ξ ∈ J =J ×D - Ṽ j ξ ξ (p)z j ξ -j ξ ∈ J =J ×D -δ ξ,ξ(j ξ ) qj ξ z j ξ = {j ξ ∈ J | ξ(j ξ )<ξ} Ṽ j ξ ξ (p)z j ξ -j ξ ∈ J (ξ)=J ×{ξ} qj ξ z j ξ = ξ ∈D -(ξ) j ξ ∈ J Ṽ j ξ ξ (p)z j ξ -j ξ ∈J ×{ξ} qj ξ z j ξ = ξ ∈D -(ξ) j∈J V j ξ (p)z j ξ -j ξ ∈J ×{ξ} qj ξ z j ξ finally if ξ ∈ D T , W F ξ (p, q) • J z is equal to: ξ ∈D -(ξ)   j ξ ∈ J Ṽ j ξ ξ (p)z j ξ   = ξ ∈D -(ξ)   j∈J V j ξ (p)z j ξ
W F (q) =            -q j 1 ξ 0 -q j 2 ξ 0 0 0 0 0 1 2 -q j 1 ξ 1 -q j 2 ξ 1 0 0 3 -2 0 0 -q j 1 ξ 2 -q j 2 ξ 2 -2 3 -2 3 0 0 1 -1 1 -1 0 0 1 1 0 0 1 1 4 1 0 0 4 1            ξ 0 ξ 1 ξ 2 ξ 11 ξ 12 ξ 21 ξ 22
We now state a simple condition under which the re-trading extensions of two primitive financial structures satisfy Assumption R1. Actually, the necessary and sufficient condition is that the ranges of the payoff matrices of the primitive financial structures be equal. Proposition 4.3. Let p ∈ R L be a spot price. Consider two financial structures

F 1 = J 1 , (ξ(j)) j∈J1 , V 1 and F 2 = J 2 , (ξ(j)) j∈J2 , V 2
where all assets are is-

sued at ξ 0 . Let F1 = J1 , ξ( j) j∈ J1 , Ṽ 1 [resp. F2 = J2 , ξ( j) j∈ J2 , Ṽ 2 ] be the re-trading extension of F 1 [resp. F 2 ]
. The following assertions are equivalent

1) F 1 p F 2 ; 2) ImV 1 (p) = ImV 2 (p);
3) Assumption R1 is satisfied with respect to p by the financial structures F1 and F2 ;

Proof of Proposition 4.3. 1) implies 2). Let λ ∈ R D ++ and (q 1 , q 2 ) the couple of associated arbitrage free prices. Then, by definition of equivalence between financial structure, ImW 1 (p, q 1 ) = ImW 2 (p, q 2 ). Let y ∈ ImV 1 (p). There exists z ∈ R J1 such that y = V 1 (p)z. Since all assets are issued at ξ 0 , (-q 1 • z, y) = W 1 (p, q 1 )z ∈ ImW 1 (p, q 1 ). So, there exists ζ ∈ R J2 so that

(-q 1 •z, y) = W 2 (p, q 2 )ζ = (-q 2 •ζ, V 2 (p)ζ). Hence, y = V 2 (p)ζ, so y ∈ ImV 2 (p).
Consequently ImV 1 (p) ⊂ ImV 2 (p) and one proves the converse inclusion in the same way.

2) implies 3). The matrix Ṽ 1, J1(ξ) (p) (resp. Ṽ 2, J2(ξ) (p)) is obtained from the matrix V 1 (p) (resp. V 2 (p)) by replacing the rows for the nodes η / ∈ D + (ξ) by rows with zero entries. So, the range of Ṽ 1, J1(ξ) (p) (resp. Ṽ 2, J2(ξ) (p)) is just the projection of the range of V 1 (p) (resp. V 2 (p)) on the space of vectors in R D with zero entries for η / ∈ D + (ξ). Hence, since the ranges of V 1 (p) and V 2 (p) are equal, the ranges of Ṽ 1, J1(ξ) (p) and Ṽ 2, J2(ξ) (p) are also equal, so Assumption R1 is satisfied with respect to p by the financial structures F1 and F2 .

3) implies 1). Indeed, if Assumption R1 is satisfied with respect to p by the financial structures F1 and F2 , for the node ξ 0 , Im Ṽ 1, J1(ξ0) (p) = Im Ṽ 2, ˜J2(ξ0) (p). Thanks to the definition of a re-trading extension and the fact that all assets are issued at ξ 0 for the financial structures

F 1 and F 2 , Ṽ 1, J1(ξ0) (p) = V 1 (p) and Ṽ 2, J2(ξ0) (p) = V 2 (p). So ImV 1 (p) = ImV 2 (p).
A direct consequence of the above proposition and Proposition 4.1 is the following sufficient condition for the equivalence of the re-trading extensions of two primitive financial structures. Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 Corollary 4.1. Let p ∈ R L be a spot price. Consider two financial structures F 1 = J 1 , (ξ(j)) j∈J1 , V 1 and F 2 = J 2 , (ξ(j)) j∈J2 , V 2 where all assets are issued at ξ 0 . Let F1 [resp. F2 ] be the re-trading extension of F 1 [resp.

F 2 ]. Then ImV 1 (p) = ImV 2 (p) implies F1 p F2 .

Equivalence with reduced forms

In this subsection, we will define the concept of a reduced form of a financial structure. Then we will give a result on the equivalence of a financial structure with its reduced forms. An important motivation for studying the reduced forms of a financial structure is the following.

To get existence of a financial equilibrium, we use a fixed point argument, so we need the compactness of attainable portfolios. With a financial structure, we may have unbounded attainable portfolios due to redundant assets 4 . So, a way to get an equilibrium is:

-considering a reduced form by removing redundant assets; -then obtaining bounded attainable portfolios for the reduced form; -then proving the existence for the reduced form; -by equivalence, getting an equilibrium for the original economy. Definition 4.2. Let p ∈ R L be a spot price vector. Let F = (J , (ξ(j)) j∈J , V ) be a financial structure. We call a reduced form of F with respect to p, any financial sub-structure 5

F = (J , (ξ(j)) j∈J , V ) of F such that rankV (p) = rankV (p) = #J .
If for all p ∈ R L we have rankV (p) = rankV (p) = #J , we say then that F is a reduced form of F.

We note that according to Definition 4.2, we obtain a reduced form of a financial structure with respect to a spot price p by simply eliminating the maximum number of redundant assets for p. Remark 4.4. This example provides a financial structure which is not equivalent to a reduced form. Indeed, let F be a financial structures with D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 } as in Remark 4.1, J = j 1 , j 2 , j 3 , ξ(j 1 ) = ξ 0 , ξ(j 2 ) = ξ 1 and ξ(j 3 ) = ξ 2 . The payoff matrix is

V =     0 0 0 1 0 0 1 0 0 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3
The matrix of a reduced form

F of F is V =     0 0 1 0 1 0 1 1     . 20 Let λ = (λ 0 , λ 1 , λ 2 , λ 3 ) ∈ R 4
++ and (q, q ) ∈ R 3 × R 2 the associated arbitrage free prices. We have

W (q) =     -λ1+λ2+λ3 λ0 0 0 1 -λ3 λ1 0 1 0 -λ3 λ2 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3
and rankW (q) = 3

while the rank of the full payoff matrix W (q ) is at most equal to two, so ImW (q) = ImW (q ) hence F is not equivalent to F .

The next proposition shows that Assumption R borrowed from [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF] and stated in Subsection 4.2 is a sufficient condition to get the equivalence between a financial structure and its financial sub-structures when the ranges of the payoff matrices are equal. The proof is given later. Proposition 4.4. Let p ∈ R L be a spot price vector. Let F = (J , (ξ(j)) j∈J , V ) be a financial structure satisfying Assumption R at the spot price p and let F = (J , (ξ(j)) j∈J , V ) be a financial sub-structure of F. The following assertions are equivalent:

1) F p F ;
2) For every state price λ ∈ R D ++ , ImW (p, q) = ImW (p, q ) where q and q are the associated arbitrage free prices;

3) For every state price λ ∈ R D ++ , rankW (p, q) = rankW (p, q ) where q and q are the associated arbitrage free prices; Note that [START_REF] Bonnisseau | On the equivalence of financial structures with short-term assets[END_REF] means that F and F satisfy Assumption R1 at p. So, under Assumption R, a sub-structure is equivalent if we remove only redundant assets.

The following corollary is a direct consequence of the previous proposition because we consider a reduced form which is a particular case of a sub-structure and the rank of the payoff matrices are the same by definition. Corollary 4.2. Given a spot price vector p ∈ R L , let F = (J , (ξ(j)) j∈J , V ) be a financial structure satisfying Assumption R at the spot price p. Then, F is equivalent to each of its reduced forms.

Since Assumption R is always true in the case of a financial structure consisting only of short-term assets, we deduce from Corollary 4.2 the following corollary: Corollary 4.4. Given a spot price vector p ∈ R L , let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two financial structure satisfying Assumption R at the spot price p. Then for all reduced forms F 1 of F 1 with respect to p and for all reduced forms F 2 of F 2 with respect to p we have

F 1 ∪ F 2 p F 1 ∪ F 2 .
Remark 4.5. The following example shows that Assumption R is not necessary to get the equivalence between a financial structure and its reduced forms. Indeed, let F = (J , (ξ(j)) j∈J , V ) be a financial structures such that D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 } as in Remark 4.1, J = j 1 , j 2 , j 3 , j 4 and ξ(j 1 ) = ξ(j 2 ) = ξ(j 3 ) = ξ 0 , ξ(j 4 ) = ξ 2 . The payoff matrix is

V =     0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1     ξ 0 ξ 1 ξ 2 ξ 3 Assumption R is not satisfied since Vect V J (D -(ξ2)) D + (ξ2) ∩ Vect V J (ξ2) D + (ξ2) = R = {0}.
The financial structure F has exactly three reduced forms denoted by F 1 , F 2 and F 3 :

V 1 =     0 0 0 1 0 0 0 1 0 0 1 1     , V 2 =     0 0 0 1 0 0 0 1 1 0 0 1     and V 3 =     0 0 0 1 0 0 0 1 0 0 0 1    
The full payoff matrices are as follows:

W (.) =     . . . 0 1 0 0 0 0 1 1 . 0 1 0 1     , W 1 (.) =     . . 0 1 0 0 0 1 . 0 1 1     , W 2 (.) =     . . . 1 0 0 0 1 1 0 0 1     and W 3 (.) =     . . 0 1 0 0 0 1 . 0 0 1     Let λ = (λ 0 , λ 1 , λ 2 , λ 3 ) ∈ R 4
++ and (q, q 1 ) ∈ R 4 × R 3 be the associated arbitrage free prices. Then, ImW (q) = ImW 1 (q 1 ). Indeed, the column vectors of the matrix W 1 (q 1 ) are also column vectors of the matrix W (q) so, Vect W 1 (q 1 ) ⊂ Vect W (q). Moreover, rankW 1 (q 1 ) = rankW (q) = 3. Hence F 1 is equivalent to F. Similarly, we can show that the structures F 2 and F 3 are equivalent to F.

Proof of Proposition 4.4. 1) implies 2). By definition of the equivalence of two financial structures. 2) implies 3). Because rankW (p, q) = dim ImW (p, q) and rankW (p, q ) = dim ImW (p, q ). 3) implies 4). Thanks to Assumption R ( See Proposition 3.5 and Lemma 3.1 in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]). 4) implies 5). Because by definition Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 22 of a sub-structure of a financial structure, we have always ImV (p) ⊂ ImV (p). 5) implies 6). By definition of a sub-structure of a financial structure, we have always for all ξ ∈ D e , ImV J (ξ) (p) ⊂ ImV J (ξ) (p), hence rankV J (ξ) (p) ≤ rankV J (ξ) (p). Furthermore, in the proof of Proposition 3.5 in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], it is shown that Assumption R implies rankV (p) = ξ∈D e rankV J (ξ) (p). F satisfies Assumption R as sub-structure of a financial structure satisfying assumption R, so, we also have rankV (p) = ξ∈D e rankV J (ξ) (p). Hence ξ∈D e rankV J (ξ) (p) = ξ∈D e rankV J (ξ) (p). Combining with the above inequalities, we conclude that rankV J (ξ) (p) = rankV J (ξ) (p) for all ξ ∈ D e , and finally ImV J (ξ) (p) = ImV J (ξ) (p); 6) implies 1): thanks to Proposition 4.1.

Equivalence for complete financial structures

In Remark 4.2, the example shows that even if Assumption R is satisfied by both financial structures, ImV 1 = ImV 2 and the sets of issuance nodes are equal i.e., D e 1 = D e 2 , the financial structures may not be equivalent. Note that the two financial structures satisfy also the stronger condition of Cornet and Ranjan [START_REF] Cornet | A remark on arbitrage free prices in multi-period economy[END_REF], namely

∀ξ ∈ D and ∀j, ∈ J , V j ξ V ξ = 0 if ξ(j) = ξ( )
Nevertheless, we can obtain a more accurate result for complete financial structures. Indeed, Assumption R is then sufficient to get the equivalence. Completeness means that all financial transfers are possible through the financial structure.

Definition 4.3. Let F = (J , (ξ(j)) j∈J , V ) be a financial structure. Given a spot price p ∈ R L and an arbitrage free price q, the financial structure F is complete at (p, q) if dim ImW (p, q) = #D -1. If dim ImW (p, q) < #D -1 the financial structure F is incomplete at (p, q). The following proposition, which is a direct consequence of Proposition 3.5 in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], gives a characterization of market completeness under Assumption R. Proposition 4.5. (Complete Markets) Given a spot price p ∈ R L , let F = (J , (ξ(j)) j∈J , V ) be a financial structure satisfying Assumption R at the spot price p. If rankV (p) = #D -1 then, for all state price λ ∈ R D ++ , the financial structure F is complete at (p, q), where q is the arbitrage free asset price associated to λ.

The following proposition gives a result of equivalence among complete financial structures under Assumption R. Proposition 4.6. Given a spot price p ∈ R L , let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two financial structures satisfying Assumption R at the spot price p. Let us assume that rankV 1 (p) = #D -1. The following assertions are equivalent:

(a) F 1 p F 2 ; Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 (b) For all state price λ ∈ R D ++ , the financial structure F 1 [resp. F 2 ] is complete at (p, q 1 ) [resp. (p, q 2 )] where (q 1 , q 2 ) is the couple of arbitrage free prices associated to λ;

(c) rankV 1 (p) = rankV 2 (p); (d) ImV 1 (p) = ImV 2 (p).
Proof of Proposition 4.6. (a) implies (b). F 1 p F 2 implies for all state price λ ∈ R D ++ , ImW 1 (p, q 1 ) = ImW 2 (p, q 2 ). Since Assumption R holds true for both financial structure, thanks to Proposition 3.5 in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]

, #D -1 = rankV 1 (p) = rankW 1 (p, q 1 ) = dim ImW 1 (p, q 1 ) = dim ImW 2 (p, q 2 ) = rankV 2 (p). So, the financial structure F 1 [resp. F 2 ] is complete at (p, q 1 ) [resp. (p, q 2 )].
(b) implies (c) thanks to Proposition 3.5 in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF].

Let us now show that (c) implies (d). By assumption, rankV 1 (p) = #D -1 and, since the first row of V 1 (p) has only zero entries, ImV 1 (p) = {y ∈ R D | y ξ0 = 0}. The same reasoning applies to V 2 (p) so ImV 1 (p) = ImV 2 (p).

Let us show now that (d) implies (a). Let λ ∈ R D ++ be a state price. Let q1 ∈ R J1 and q2 ∈ R J2 the two arbitrage free prices associated with λ. Since Assumption R holds true for both financial structure, thanks to Proposition 3.5 in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]

, (d) implies that #D -1 = rankV 1 (p) = rankW 1 (p, q1 ) = rankV 2 (p) = rankW 2 (p, q2 ). So since ImW 1 (p, q1 ) and ImW 2 (p, q2 ) are included in λ ⊥ , one concludes that ImW 1 (p, q1 ) = ImW 2 (p, q2 ) = λ ⊥ . Consequently, F 1 p F 2 .
Remark 4.6. In the two-period case, if there exists an arbitrage free price for which the financial market is complete, the financial structure is complete for any other arbitrage free price. With several periods, if the assets are all shortterm, this result is deduced of Proposition 5.2 of Angeloni and Cornet in [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF]. But if there are long-term assets in the financial structure, Proposition 4.6 is no more true without Assumption R. Indeed, consider a stochastic economy with T = 2 and tree nodes, namely D = {ξ 0 , ξ 1 , ξ 2 } and let F 1 = (J 1 , (ξ(j)) j∈J1 , V 1 ) and F 2 = (J 2 , (ξ(j)) j∈J2 , V 2 ) be two financial structures. In the first structure there are two assets j 1 1 and j 2 1 with ξ(j 1 1 ) = ξ 0 and ξ(j 2 1 ) = ξ 1 . In the second structure there are two assets j 1 2 and j 2 2 with ξ(j Then the payoff matrices are:

2 ) = ξ(j 2 2 ) = ξ 0 . • ξ 0 • ξ 1 • ξ 2 t = 0 t = 1 t = 2 1
V 1 =   0 0 -1 0 1 1   ξ 0 ξ 1 ξ 2 and V 2 =   0 0 1 0 0 1   ξ 0 ξ 1 ξ 2
rankV 1 (p) = rankV 2 (p) = #D -1 = 2.
Let λ = (1, 1, 1), q1 = (0, 1) [resp. q2 = (1, 1)] is the arbitrage free price of F 1 [resp. F 2 ] associated with λ. The full payoff matrices associated with q1 and q2 are:

W 1 (q 1 ) =   0 0 -1 -1 1 1   ξ 0 ξ 1 ξ 2 and W 2 (q 2 ) =   -1 -1 1 0 0 1   ξ 0 ξ 1 ξ 2
We have rankW 1 (q 1 ) = 1 and rankW 2 (q 2 ) = 2 this implies that ImW 1 (q 1 ) = ImW 2 (q 2 ). So the structures F 1 and F 2 are not equivalent and the financial structure F 1 is not complete at q1 . Remark 4.7. Without the completeness, even if Assumption R is satisfied in the two financial structures and there is a state price λ ∈ R D ++ such that ImW 1 (p, q 1 ) = ImW 2 (p, q 2 ) for the associated arbitrage free asset prices, we have no guarantee that the two structures are equivalent at p. Indeed, let D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 , ξ 4 }, J 1 = j 1 1 , j 2 1 , j 3 1 , J 2 = j 1 2 , j 2 2 , j 3 2 . ξ(j 1 1 ) = ξ(j 2 1 ) = ξ(j 1 2 ) = ξ 0 and ξ(j 3 1 ) = ξ(j 2 2 ) = ξ(j 3 2 ) = ξ 1 . The payoff matrices and the full payoff matrices for the state price λ = (1, 1, 1, 1, 1) are:

• ξ 0 • ξ 1 • ξ 2 • ξ 3 • ξ 4 t = 0 t = 1 t = 2 t = 3 t = 4
V 1 =       0 0 0 1 -1 0 -1 1 1 0 -1 1 0 1 -2       ξ 0 ξ 1 ξ 2 ξ 3 ξ 4 and W 1 (q 1 ) =       0 0 0 1 -1 0 -1 1 1 0 -1 1 0 1 -2       ξ 0 ξ 1 ξ 2 ξ 3 ξ 4 V 2 =       0 0 0 1 0 0 -1 1 1 0 -1 1 0 1 -2       ξ 0 ξ 1 ξ 2 ξ 3 ξ 4 and W 2 (q 2 ) =       0 0 0 1 -1 0 -1 1 1 0 -1 1 0 1 -2       ξ 0 ξ 1 ξ 2 ξ 3 ξ 4
Assumption R is satisfied by the two financial structures and it is clear that ImW 1 (q 1 ) = ImW 2 (q 2 ) since W 1 (q 1 ) = W 2 (q 2 ).

W 1 (q 1 ) =       0 -1 0 1 -1 2 -1 1 1 0 -1 1 0 1 -2       ξ 0 ξ 1 ξ 2 ξ 3 ξ 4 and W 1 (q 2 ) =       0 0 0 1 -2 2 -1 1 1 0 -1 1 0 1 -2       ξ 0 ξ 1 ξ 2 ξ 3 ξ 4
It is clear that ImW 1 (q 1 ) = ImW 2 (q 2 ) since W 1 ξ0 (q 1 ) = 0 and W 2 ξ0 (q 2 ) = 0. Consequently the financial structures F 1 and F 2 are not equivalent at p. Proof of Lemma 4.1. As already noticed, Assumption R1 implies the equality of the ranges of the payoff matrices. We now prove the converse.

First, we prove that the equality of the ranges of the matrices implies that the node of issuance are the same for both structures. Let us assume that there exists ξ ∈ D e 1 and ξ / ∈ D e 2 . Let j 1 ∈ J 1 such that ξ(j 1 ) = ξ. Since there is no trivial asset, there exists η ∈ ξ + such that V 1j1 η (p) = 0. But, since ξ / ∈ D e 2 and all assets are short term, for all j ∈ J 2 , V 2j η (p) = 0. Consequently, V 1j1 (p) is not a linear combination of V 2j (p) j∈J2 , which contradicts the fact that ImV 1 (p) = ImV 2 (p). In the same way, one shows that there is no node in D e 2 which does not belong to D e 1 . So, the issuance nodes are the same for both structures. Now we show that the condition ImV 1 (p) = ImV 2 (p) implies that Assumption R1 holds true at the spot price p. From [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], since there is only short-term assets, the two structures satisfies Assumption R, so ImV 1 (p) = ⊕ ξ∈D e ImV 1,J1(ξ) (p) = ⊕ ξ∈D e ImV 2,J2(ξ) (p) = ImV 2 (p) Let ξ ∈ D e and y ∈ ImV 1,J1(ξ) (p). Since we have only short-term assets, y η = 0 for η / ∈ ξ + . From the previous equality, there exists a unique couple (z ξ , z) ∈ ImV 2,J2(ξ) (p) × ⊕ ξ ∈D e ,ξ =ξ ImV 2,J2(ξ ) (p) such that y = z ξ + z. Once again, since we have only short-term assets, for all η ∈ ξ + , z η = 0 since ξ = ξ. For all η / ∈ ξ + , y η = 0 = z ξ η hence, z η = 0. So, z = 0 and y = z ξ ∈ ImV 2,J2(ξ) (p), which proves that ImV 1,J1(ξ) (p) ⊂ ImV 2,J1(ξ) (p). The converse inclusion is proved in the same way, so Assumption R1 is satisfied.

Proof of Proposition 4.2 (i) This is obvious since all assets are issued at the same node.

(ii) From Proposition 3.5 in [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], Assumption R implies

Figure 1 :

 1 Figure 1: the tree D Here: D = {ξ 0 , ξ 1 , ξ 2 , ξ 11 , ξ 12 , ξ 13 , ξ 21 , ξ 22 }, T = 2, the length of D is 3, D 2 = {ξ 11 , ξ 12 , ξ 13 , ξ 21 , ξ 22 }, ξ + 1 = {ξ 11 , ξ 12 , ξ 13 }, D + (ξ 2 ) = {ξ 21 , ξ 22 }, t(ξ 11 ) = t(ξ 12 ) = t(ξ 13 ) = t(ξ 21 ) = t(ξ 22 ) = 2, D -(ξ 11 ) = {ξ 0 , ξ 1 }.

Proposition 3 . 1 .

 31 For each spot price p ∈ R L , the relation p defined above is an equivalence relation. Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

  is equivalent to F. Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 10 Example 3.2. (Union of financial structures

Example 3 . 4 .

 34 (Stability of the equivalence by reunion.

Figure 2 :

 2 Figure 2: the tree D

Figure 3 :

 3 Figure 3: the tree D

Proposition 4 . 1 . 1 ImV 1 2 ImV 2

 411122 Given a spot price vector p ∈ R L . Let F 1 and F 2 be two financial structures satisfying Assumption R1 at the spot price p ∈ R L . ThenF 1 p F 2 .Documents deTravail du Centre d'Economie de la Sorbonne -2014.81 13 Proof of Proposition 4.1. Since ImV 1 (p) = ξ∈D e J1(ξ) (p) and ImV 2 (p) = ξ∈D e J2(ξ) (p) and since Assumption R1 implies that D e 1 = D e 2 = D e , one concludes that ImV 1 (p) = ImV 2 (p) under Assumption R1.

  Travail du Centre d'Economie de la Sorbonne -2014.81

  16



  Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 4 SUFFICIENT CONDITIONS FOR EQUIVALENCE 18

4 ) 5 )

 45 rankV (p) = rankV (p); ImV (p) = ImV (p); 6) ∀ξ ∈ D e , ImV J (ξ) (p) = ImV J (ξ) (p).

Corollary 4 . 3 .

 43 A financial structure consisting only of short-term assets is equivalent to each of its reduced forms. Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 21 Corollary 4.3 generalizes Proposition 1.5 of the thesis Zaier Aouani [2], which deals with the two-period case. The following corollary is deduced from Corollary 4.2 and Example 3.4.

Figure 5 :

 5 Figure 5: the tree D

Figure 6 :

 6 Figure 6: the tree D

  Now let λ = (1, 1, 1, 1, 2) be another state price. Then, Documents de Travail du Centre d'Economie de la Sorbonne -2014.81 4 SUFFICIENT CONDITIONS FOR EQUIVALENCE 25

rankV 1 (p) = ξ∈D e 1 rankV 1 2 rankV 2 1 ∪D e 2 dim 1 ∪D e 2 rankV 1 1 ∪D e 2 rankV 2 1 ∪D e 2 dimD e 2 ,

 1112212121122122 J1(ξ) (p) and rankV 2 (p) = ξ∈D e J2(ξ) (p)Since by assumption rankV 1 (p) = rankV 2 (p), we have:ξ∈D e ImV 1 J1(ξ) (p) = ξ∈D e J1(ξ) (p) = ξ∈D e J2(ξ) (p) = ξ∈D e ImV 2 J2(ξ) (p)and by assumption we have for allξ ∈ D e 1 ∪ ImV 1 J1(ξ) (p) ⊂ ImV 2 J2(ξ) (p) thus this implies that ξ ∈ D e 1 ∪ D e 2 , dim ImV 1 J1(ξ) (p) ≤ dim ImV 2 J2(ξ) (p).Combined with the previous equality, this implies thatdim ImV 1 J1(ξ) (p) = dim ImV 2 J2(ξ) (p), which together with ImV 1 J1(ξ) (p) ⊂ ImV 2 J2(ξ) (p) leads to ImV 1 J1(ξ) (p) = ImV 2 J2(ξ) (p).So Assumption R1 is true at the spot price p. Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

For x = (x (ξ)) ξ∈D , p = (p (ξ)) ξ∈D in R L = R H×D (with x(ξ), p(ξ) in R H ) we let p2x = (p(ξ) • H x(ξ)) ξ∈D ∈ R D .Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

J 1 J 2 is the union of assets of F 1 and of F 2 where, the common assets in J 1 ∩ J 2 are counted twice in the new structure, ifJ 1 ∩ J 2 = ∅. The matrix [V 1 , V 2 ] is the (D × (J 1 J 2 ))matrix whose first J 1 columns are those of V 1 and the last J 2 columns are those of V 2 . Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

Recall that an asset j 0 of a financial structure F = (J , (ξ(j)) j∈J , V ) is redundant at the spot price p if the column vector V j 0 (p) representing its payoffs on D is a linear combination of the other column vectors of the matrix V .

See Definition 3.2. Documents de Travail du Centre d'Economie de la Sorbonne -2014.81

Appendix

Proof of Examples of Subsection 3.1 Example 3.1. Let p be a spot price, λ ∈ R D ++ be a state price, α be one non-zero real number and (q, q α )R J ×R J be the couple of arbitrage free prices associated to λ. From the arbitrage free condition, q α = αq, so W α (p, q α ) = αW (p, q), which implies that ImW α (p, q α ) = ImW (p, q) since α = 0. So, F F α .

Example 3.2. Let p be a spot price, λ ∈ R D ++ be a state price and (q 1 , q 2 ) ∈ R J1 × R J2 be the couple of arbitrage free prices associated to λ. From the structure of V , the arbitrage free price q associated to λ for the payoff matrix

then ImW 1 (p, q 1 ) = ImW 2 (p, q 2 ) and ImW 1 (p, q 1 )+ImW 2 (p, q 2 ) = ImW 1 (p, q 1 ) = ImW 2 (p, q 2 ). So,

++ be a state price and (q 1 , q 2 , q 3 , q 4 )R J1 × R J2 × R J3 × R J4 be the arbitrage free prices associated to λ. Like in the previous example, we remark that the full payoff matrix associated to

and ImW 3 (p, q 3 ) = ImW 4 (p, q 4 ).

Hence Im[W 1 (p, q 1 ), W 3 (p, q 3 )] = Im[W 2 (p, q 2 ), W 4 (p, q 4 )], which shows that

Proof of Proposition 3.3 Let F 4 be the financial structure with the assets of F 2 , which are not in F 3 , that is,


Michel Agoyan
email: michel.agoyan@cea.fr

Jean-Max Dutertre
email: dutertre@emse.fr

Amir-Pasha Mirbaha
email: mirbaha@emse.fr

David Naccache
email: david.naccache@ens.fr

Anne-Lise Ribotta
email: ribotta@emse.fr

Assia Tria
email: assia.tria@cea.fr

How to Flip a Bit?

This note describes laser fault experiments on an 8bit 0.35µm microcontroller with no countermeasures. We show that reproducible single-bit faults, often considered unfeasible, can be obtained by careful beam-size and shot-instant tuning.

I. INTRODUCTION

Fault attacks consist in using hardware malfunction to infer secrets from the target's faulty outputs. Within fault attacks, Differential Fault Analysis [START_REF] Biham | Differential fault analysis of secret key cryptosystems[END_REF] (DFA) is a particular analysis technique exploiting differences between correct and faulty outputs. We refer the reader to [START_REF] Tria | Fault attacks from theory to practise: what is possible to do?, 2-nd Canada-France Workshop on Foundations & Practice of Security[END_REF] for more information on fault injection techniques.

This note describes laser faults experiments on an 8-bit 0.35µm RISC microcontroller with no countermeasures. We show that reproducible single-bit faults, often considered unfeasible, can be obtained by careful beam-size and shot-instant tuning. Moreover, we obtain such faults even when the beam's impact area exceeds a single SRAM cell. This underlines the need to protect small data objects, such as pointers, counters or flags, against "surgical" faults targeting a single-bit on a specific byte in memory and nothing else.

II. THE ADVANCED ENCRYPTION STANDARD

We assume that the reader is familiar with the AES [START_REF]Announcing the advanced encryption standard (AES)[END_REF] that we recall here for the ease of reference.

The AES-128 (hereafter AES) encrypts 128-bit plaintexts into 128-bit ciphertexts using a 128-bit key K. The algorithm performs 10 rounds (after a short initial round) and consists of two separated processes: a key schedule that derives round keys and the encryption routine itself. During decryption key schedule is reversed and encryption is replaced by a very similar decryption process.

The initial round uses K 0 = K as a round key; for all subsequent rounds, new round keys K i are calculated from their predecessors K i-1 . Figure 1 illustrates the AES' structure.

In most implementations the K i s are computed and stored in memory before encryption starts. Encryption treats the 16byte plaintext M as a 4×4 byte matrix. Each round, except the initial and the final, includes four steps: A substitution of the matrix's contents using a lookup table (SubBytes), a rotation

III. LASER FAULT INJECTION

Laser (Light Amplification by Stimulated Emission of Radiation) is a stimulated-emission electromagnetic radiation in the visible or the invisible domain. Laser light is monochromatic, unidirectional, coherent and artificial (i.e. laser does not spontaneously exist in nature). Laser light can be generated as a beam of very small diameter (a few µm). The beam can pass through various material obstacles before impacting a target during a very short duration.

Laser impacts on electronic circuits are known to alter functioning. Current chip manufacturing technologies are in Fig. 2. Architecture of a typical SRAM cell. the nanometers range. This, and the laser's brief and precise reaction time, makes laser a particularly suitable fault injection means.

A. Photoelectrical Effects of Laser on Silicon

SRAM (Static Random Access Memory) laser exposure is known to cause bit-flips [START_REF] Skorobogatov | Optical fault induction attacks, Cryptographic Hardware and Embedded Systems[END_REF], [START_REF] Darracq | Single-event sensitivity of a single SRAM cell[END_REF], [START_REF] Bar-El | The sorcerer's apprentice guide to fault attacks[END_REF], [START_REF] Canivet | Analyse des effets d'attaques par fautes et conception sécurisée sur plate-forme reconfigurable[END_REF], a phenomenon called Single Event Upset (SEU). By tuning the beam's energy level below a destructive threshold, the target will not suffer any permanent damage.

A conventional one-bit SRAM cell (Figure 2) is made of two cross-coupled inverters. Every cell has two additional transistors controlling the cell's content access during write and read. As every inverter is made of two transistors, an SRAM cell contains six MOS.

In each cell, the states of four transistors encode the stored value. By design, the cell admits only two stable states: a "0" or a "1". In each stable state, two transistors are at an ON state and two others are OFF.

If a laser beam hits the drain/bulk reversed-biased PN junctions of a blocked transistor, the beam's energy may create pairs of electrons as the beam passes through the silicon. The charge carriers induced in the collection volume of the drainsubstrate junction of the blocked transistor are collected and create a transient current that inverts logically the inverter's output voltage. This voltage inversion is in turn applied to the second inverter that switches to its opposite state: all in all, a bit flip happens [START_REF] Darracq | Single-event sensitivity of a single SRAM cell[END_REF], [START_REF] Bar-El | The sorcerer's apprentice guide to fault attacks[END_REF].

From the opponent's perspective, an additional advantage of laser fault injection is reproducibility. Identical faults can be repeated by carefully tuning the laser's parameters and the target's operating conditions.

B. Different Parameters in a Fault Attack by Laser

In a laser attack, the opponent usually controls the beam's diameter, wavelength, amount of emitted energy, impact coordinates (attacked circuit part) and the exposure's duration. Sometimes, the opponent may also control the impact's moment1 , the target's clock frequency, V cc and temperature. Finally, laser attacks may indifferently target the chip's front side or back side.

However, the chip's front and back sides have different characteristics when exposed to a laser beam:

1) Front side attacks: are particularly suited to green wavelength (∼ 532nm). The visibility of chips components makes positioning very easy in comparison to backside attacks. But because of the metallic interconnects' reflective effect, it is difficult to target a component with enough accuracy. In addition, progress in manufacturing technologies results in both a proliferation of metal interconnects and much smaller chips. All in all, it becomes increasingly difficult to hit a target area.

2) Backside attacks: are more successful at the infrared wavelength (∼ 1064nm) as the laser needs to deeply enter the silicon. Positioning is more difficult for lack of visibility. Nonetheless, backside attacks allow to circumvent the reflective problem of metallic surfaces.

IV. GIRAUD'S BIT DFA Differential Fault Analysis [START_REF] Biham | Differential fault analysis of secret key cryptosystems[END_REF] (DFA) is an analysis technique exploiting differences between correct and faulty outputs. Several byte-level and bit-level AES DFA variants exist (e.g. [START_REF] Piret | A differential fault attack technique against SPN structure with application to the AES and KHAZAD[END_REF], [START_REF] Ch | DFA on AES[END_REF], [START_REF] Moradi | A generalized method of differential fault attack against AES cryptosystem[END_REF], [START_REF] Dusart | Differential fault analysis on A[END_REF], [START_REF] Blömer | Fault based cryptanalysis of the Advanced Encryption Standard (AES)[END_REF]). Given the dependence of these attacks on precise "surgical" fault injection, the feasibility of bit/byte-level DFA remained somewhat unclear.

[8] describes a bit-level and a byte-level DFA on AES. The bit-level attack requires the injection of a single-bit fault into a specific byte of the temporary ciphertext result of the penultimate round (M 9 ) (e.g. to inject such a fault into the 9-th round AddRoundKey or into the temporary ciphertext result just before the SubBytes input to the 10-th round).

To discover one byte of K 10 , the attack requires to repeat a single-bit fault for at least three different plaintexts. The three faulty results are then compared to their corresponding correct ciphertexts to infer key information. We show that this attack can be implemented, even when the laser spot is wider than the SRAM's cell.

During normal processing, the value of each ciphertext (C) cell is calculated by xoring a corresponding K 10 value with a temporary value resulting from the application of SubBytes (SB) and ShiftRows (SR) to (M 9 ):

C = SR[SB(M 9 )] ⊕ K 10 (1)
As shown in Figure 3, upon single-byte fault injection in K 9 (regardless the number of faulty bits) the faulty message will feature only one faulty byte that will leak information on one byte of K 10 . Figure 4 shows the consequences of an injected fault in K 9 throughout the 9-th and 10-th rounds.

For the sake of clarity, we consider all subsequent equations bytewise thereby abstracting away ShiftRowss that do not affect individual byte values. Thus, (1) becomes (2):

C = SB(M 9 ) ⊕ K 10 (2)
By considering the injection of the single-bit fault e on the 10-th round SubBytes input, the faulty ciphertext (D) can be expressed as (3):

(4) will yield a set of hypotheses on possible M 9 and e value-pairs. Using (5), a corresponding K 10 value can be replaced for each pair of (M 9 , e) values.

K 10 = SB(M 9 ⊕ e) ⊕ D (5)
By repeating the fault injection for at least three different plaintexts, the opponent creates three sets of hypotheses on the corresponding K 10 byte value. Then, sets are intersected to spot the single hypothesis that reveals one K 10 byte. With a probability of about 97%, three plaintexts suffice to discover a byte of K 10 [START_REF] Ch | DFA on AES[END_REF]. Otherwise, the opponent iterates the process for more plaintexts to until the sets' intersection reaches a singleton. After finishing this operation for one byte of K 10 , the procedure is repeated to discover K 10 's remaining bytes. Finally, K = K 0 is inferred by reversing the key schedule.

V. PRACTICAL SINGLE-BIT FAULT INJECTION

Outline: After chip decapsulation and a mapping of the chip's components, we selected a large target area, given our knowledge of the implementation. Using automated search on the chip's front-side, we modified the impact's coordinates, the beam parameters and timing until a single bit fault was obtained. Finally, Giraud's bit DFA was performed.

The target is an 8-bit 0.35 µm 16 MHz RISC microcontroller with an integrated 4KB SRAM and no countermeasures. The device runs SOSSE (Simple Operating System for Smartcard Education [START_REF] Brstle | SOSSE -Simple Operating System for Smartcard Education[END_REF]) to which we added some commands, most notably for feeding-in cleartexts and retrieving ciphertexts. K was embedded in the code. As encryption starts, the K i s are derived and stored in SRAM. The laser, shown in Figures 10 and11, is equipped with a YAG laser emitter in three different wavelengths: green, infrared and ultraviolet.

The spot's diameter can be set between 0 and 2500µm. As the beam passes through a lens, it gets reduced by the lens' zoom factor and loses a big part of its energy. Our experiments were conducted with a 20× Mitutoyo lens, a green2 beam of ∅4µm and ≃ 15pJ per shot 3 . The circuit is installed on a programmable Prior Scientific X-Y positioning table 4 . The X-Y table, card reader, laser and an FPGA trigger board, were connected via RS-232 to a control PC. The FPGA trigger board receives an activation signal from the reader and sends a trigger signal to the laser after a delay defined by the control PC.

Experiments were conducted in ambient temperature and at V cc = 5V . These parameters are within the device's normal operating conditions 2.7V ≤ V cc ≤ 5.5V .

The chip was decapsulated by chemical etching using a Nisene JetEtch automated acid decapsulator. The decapsulator can be programmed for the chemical opening of different chip types using different ratios of nitric acid (HNO 3 ) and sulfuric acid (H 2 SO 4 ), at a desired temperature and during a specified time. For opening our chip, we used only nitric acid at 80 • C for 40 seconds. The etched chip (Figure 5) successfully passed functional tests before and during fault injection.

As it is very difficult to target the chip's (ALU) (Arithmetic Logic Unit) during a very specific instant between the end of 9-th round and the beginning of the 10-th round, we decided to target K 9 .

Finding the SRAM area containing K 9 and properly tuning the laser's parameters is very time consuming. The number of faults in C, their position and their contents indicate which round key has been hit. MixColumns will amplify any singlebit/byte fault occurring in any K i preceding K 9 and result in Fig. 6. Decapsulated chip (closeup on SRAM). a multi-byte or a fully faulty ciphertext (we call such a bad event an "early fault"). As shown in Figure 3, a single-bit/byte fault on K 8 changes 4 bytes and on any previous K results in a completely faulty ciphertext while a fault in K 9 or K 10 changes only one byte. However, injected faults are not always limited to a single byte and/or to a single K i . When more than 3 ciphertext bytes are faulty, it is difficult to determine if the observed result is due to an early fault or to several faults in K 9 and K 10 (Table I). Despite fine-grained energy and spatial control we detected faults in keys neighboring K 9 . To overcome this problem, we isolated K 10 from faults and restricted the shot to a 100µs interval between the use of K 8 and K 9 (Figure 9).

< < < < < < < < < < < < < < < < < < < < < < < < < < < < < <
Figure 7 shows how we could confine faults to a singlebit of K 9 . When physically more than one single-bit faulty byte existed, we could logically obtain a single-bit fault by controlling the laser's shooting time. Figure 7 is just a model of the real SRAM (Figure 6) to describe our technique and does not correspond to real address allocation. We could successfully inject a single-bit fault into each of the 16 bytes of K 9 and iterate the process for 4 different texts. This sufficed to succeed Giraud's bit DFA.

As shown in the topmost part of Figure 7, we searched K 9 's precise storage area by monitoring the number and the type of faults in the ciphertext. Then (middle part of Figure 7), by a precise beam localization, we managed to inject faults only into K 9 . This, however, did not turn out to be fully deterministic as sometimes we would also inflict faults to previous round keys. At that point (lowermost part of Figure 7), by fine-tuning spatial and temporal beam localization (just after the use of K 8 ), we managed to restrict the injected faults only to K 9 . This is the exact assumption of Giraud's scenario for single-bit fault injection.

VI. CONCLUSION

We implemented single-bit Giraud's attack [START_REF] Ch | DFA on AES[END_REF] using laser fault injection. Whilst this is not the most effective fault attack on AES, this scenario is usually regarded as the most difficult as it requires to limit the attack to one single bit. This is much more stringent that most other AES fault attacks (e.g. [START_REF] Piret | A differential fault attack technique against SPN structure with application to the AES and KHAZAD[END_REF], [START_REF] Moradi | A generalized method of differential fault attack against AES cryptosystem[END_REF], [START_REF] Dusart | Differential fault analysis on A[END_REF], [START_REF] Blömer | Fault based cryptanalysis of the Advanced Encryption Standard (AES)[END_REF]) that target an entire byte, regardless the number of faulty bits. The experiments reported in this paper also apply to other attacks (e.g. [START_REF] Piret | A differential fault attack technique against SPN structure with application to the AES and KHAZAD[END_REF], [START_REF] Moradi | A generalized method of differential fault attack against AES cryptosystem[END_REF]) and underline the possibility to precisely modify even flags, counters, pointers and other single memory cells that control program flow, in the absence of countermeasures, even at a single-bit level.

In summary, this note's main conclusions are:

• It is possible to implement a single-bit laser fault attack on an AES round key.

• Even when it is physically impossible to target a singlebit on one byte because the beam hits a few other bytes, careful spatial and temporal coordination may allow to deceive the encryption process to consider logically only a single-bit fault that corresponds to Giraud's bit scheme.

• It is possible to reproduce the same faults on different plaintexts. This assesses the reality of bit-level Giraud's scenario on unprotected chips.

Fig. 1 .

 1 Fig. 1. The AES -General Outline.

Fig. 3 .Fig. 4 . 3 )[ 8 ]

 3438 Fig. 3. Effects of one faulty round key byte at different rounds.

Fig. 5 .

 5 Fig. 5. Decapsulated chip (SRAM is on the left middle and bottom side).

Fig. 7 .

 7 Fig. 7. Exploration process.

Figure 8

 8 Figure 8 compares a 1µm laser spot and SRAM cells in

Fig. 8 .Fig. 9 .

 89 Fig. 8. 1µm laser spot (dotted circle) vs. technology sizes [12].

Fig. 10 .

 10 Fig. 10. Laser and target (general overview).

Fig. 11 .

 11 Fig. 11. Laser and target (closeup).

TABLE I POTENTIAL

 I FAULTY K i S AS FUNCTION OF OBSERVED FAULTY As technology advances, transistor density per µm grows. With several transistors are packed into 1µm areas, single-bit fault injection will require much more precise equipment and are likely to become unfeasible using cheap lasers.

	CIPHERTEXT BYTES.
	number of faulty C bytes	potential faulty round keys
	K 10	K 9	K 8	previous round keys
	1, 2, 3
	4, . . . , 15
	16
	different technology sizes.

i.e. the impact's synchrony with a given clock cycle of the target.

532nm wavelength.

At the laser source emitter, before passing through the lens.

Motorized stepper stage for upright microscopes with 0,1 µm steps.