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

Achis Chéry
email: cheryachis07@yahoo.fr

Stability of marketable payoffs with re-trading

Keywords: Incomplete markets, financial equilibrium, multi-period model, long-term assets, re-traded assets, financial structures with re-trading. JEL codes: D5, D4, G1

We consider a stochastic financial exchange economy with a finite dateevent tree representing time and uncertainty and a financial structure with possibly long-term assets. We have addressed in Bonnisseau-Chéry [3] the question of the stability of financial structures thanks to a suitable Assumption R which is based only on the return of the assets independently of the price of the assets. However, Assumption R is never satisfied in the structures with re-trading (See [3]), which is a very common feature in many papers (See [5]). The main purpose of this new paper is to deepen the study of the stability of financial structures with long-term assets and especially to address the stability in cases where there is a re-trading of assets after their issuance at different nodes and different dates. We exhibit a newsufficient condition, which allows to have the equality between the kernel of payoffs matrices, and, thus the stability of the marketable payoffs. This sufficient condition is then used to get the stability result for the re-trading extension of a financial structure. We also show that the situation is more complex when all initial assets are not issued at the initial date and we then provide a stronger condition to cover this case.

INTRODUCTION

1 Introduction

In Bonnisseau-Chéry [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], we have studied the question of stability of the set of marketable payoffs with respect to the arbitrage free asset prices in presence of long-term asset. The stability means that the set-valued mapping of marketable payoffs from the set of non-arbitrage asset prices to the set of payoffs is lower semi-continuous and has a closed graph. This is a key property for the functioning of the financial markets since it implies that the demand for regular consumers is continuous. Then, we can get the existence of equilibrium prices under reasonable standard assumptions. Otherwise, we can observe a divergence of the trades under very small variations of the asset prices to get the same payoffs. Actually, the stability is a consequence of the equality of the kernel of the payoff matrix and the kernel of the full payoff matrix of the financial structure for all non-arbitrage asset prices. But this assumption cannot be checked on the fundamentals of the economy.

In [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], we provide a sufficient condition (called R) on the payoff matrix describing the returns of the assets. Under this condition we get the stability of marketable payoffs. Assumption R is stated in a framework without re-trading whereas in many contributions, (see, in particular, the seminal book of Magill and Quinzii [START_REF] Magill | Theory of Incomplete Markets[END_REF]), an initial set of assets are issued at the initial node and are then re-traded at all successive date. Angeloni-Cornet ( [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF]) have shown that the retrading can be equivalently described as the emission of a new asset with payoffs being the truncated payoffs of the initial asset. So, using the Angeloni-Cornet ( [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF]) transformation, we can check Assumption R in presence of re-trading. Unfortunately, it appears that is not satisfied, except on the very particular case where there is only short-term assets. Hence the need to introduce new Assumptions on the payoff matrix to address this issue of stability of the set of marketable payoffs of a financial structure with re-trading, which is the main purpose of this paper.

We start by working on a given financial structure and then we consider the case of the re-trading extension of a financial structure. Our objective is to provide sufficient conditions on the initial payoff matrix to get the stability for the re-trading extension.

We introduce a first assumption, Assumption H which implies that the kernel of the full payoff matrix is included in the kernel of the payoff matrix. However this inclusion can be strict as illustrated by a simple example. Note that we nevertheless get the equality in absence of redundant assets. Thanks to another Assumption K, we get the other inclusion. It should be noted that H and K are not incompatible and one does not implies the other. As a by-product, we prove that the set of arbitrage free prices is a cone under Assumption H. This result is always true in the two-period case. Cornet-Ranjan (see [START_REF] Cornet | A remark on the set of arbitrage-free prices in a multi-period model[END_REF]) have obtained this result in a multi-period framework with another assumption, which also implies the convexity of the set of arbitrage free prices is a convex cone. But our Assumption H is not sufficient to have the convexity.

We pursue the study by providing a third assumption K', which is stronger than K but is sufficient for the stability without H. In words, Assumption K' means that the distribution of returns of the assets issued at a given node is constant over time but the magnitude may vary. Documents de travail du Centre d'Economie de la Sorbonne -2018.24

THE T -PERIOD FINANCIAL EXCHANGE ECONOMY

The purpose of the last section is to apply these preliminary results to the case of an economy with re-trading. First we consider that all assets are issued at the same node, the initial one of the date-event tree representing the uncertainty. This is the usual case in the literature. we prove that Assumption K' on the initial payoff matrix is inherited by the re-trading extension when the assets have non-zero returns between their emission node and their terminal nodes.

We then remark that these conditions are not enough when assets are not issued at the same node. So we state a stronger condition, Assumption K", which is then sufficient to get the stability of the re-trading extension. Assumption K" supposes that the magnitude leading the evolution of the payoffs of the assets over time is independent of the issuance node of the asset, whereas Assumption K' leaves the possibility to have different magnitudes for different issuance nodes.

After introducing notations and the model of a financial structure borrowed from [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF][START_REF] Magill | Theory of Incomplete Markets[END_REF] in Section 2 , we formulate our Assumptions on the payoff matrix V (p) in Section 3 and we state the results on stability without re-trading. In Section 4, we introduce the re-trading extension of a financial structure and we prove the two results on stability with a unique node of issuance and with several nodes. Some proofs have been postponed in Appendix.

The T -period financial exchange economy

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 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 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:

R n + = {x ∈ R n : x ≥ 0}, R n ++ = {x ∈ R n : x 0}.
D + (ξ) = {ξ ∈ D : ∃τ ∈ T \ {0} | ξ = pr τ (ξ )} D -(ξ) = {ξ ∈ D : ∃τ ∈ T \ {0} | ξ = pr τ (ξ)} If ξ ∈ D + (ξ) [resp. ξ ∈ D(ξ) := D + (ξ) ∪ {ξ}], we shall 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 ) -.
In the example of the event tree of Figure 1,

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 }, D(ξ 2 ) = {ξ 2 , ξ 21 , ξ 22 }. • ξ 1 • ξ 11 • ξ 2 • ξ 12 • ξ 13 • ξ 22 • ξ 21 • ξ 0 t = 0 t = 1 t = 2

The financial structure without re-trading

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), 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). Documents de travail du Centre d'Economie de la Sorbonne -2018.24

THE T -PERIOD FINANCIAL EXCHANGE ECONOMY

6

The payoff may depend upon the spot price vector2 p ∈ R L and is denoted by V j ξ (p). Formally, we assume that V j ξ (p) = 0 if ξ / ∈ D + (ξ (j)).

For each consumer, z i = (z j i ) j∈J ∈ R J is called the portfolio of agent i. If z j i > 0 [resp. z j i < 0], then |z j i | is the quantity of asset j bought [resp. sold] by agent i at the issuance node ξ (j). We assume that each consumer i is endowed with a portfolio set Z i ⊂ R J , which represents the set of admissible portfolios for agent i. For a discussion on this concept we refer to Angeloni-Cornet [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF], Aouani-Cornet [START_REF] Aouani | Existence of financial equilibria with restricted participations[END_REF] and the references therein.

To summarize a financial structure F = J , (Z i ) i∈I , (ξ (j)) j∈J , V consists of a set of assets J , a collection of portfolio sets (Z i ⊂ R J ) i∈I , 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 payoffs for a given portfolio z ∈ R J is W (p, q) z and the full payoff 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 assume that they are no participation constraints, i.e., for all i ∈ I,

Z i = R J .
We are now able to define the set of marketable payoffs for (p, q) as:

H(p, q) = {w ∈ R D | ∃z ∈ R J , w = W (p, q)z}
which is the range of the matrix W (p, q). 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).
In our financial structures there are no trivial assets i.e., there is no asset that gives zero payoffs to all nodes of the event tree D.

In all our numerical examples, we assume that there is only one good at each node of the tree and the spot price of the unique good is equal to 1. Consequently, for the sake of simpler notations, we omit the price p and note the payoff matrix (resp. full payoff matrix) by V (resp. W (q)).

Stability of the set of marketable payoffs

In this section we will give conditions on the payoff matrices of our financial structures to get the stability of the marketable payoffs with respect to the arbitrage free asset prices . In [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], we have studied this question. The general abstract sufficient condition is the equality of kernels of payoff matrices. We show that this general condition is satisfied under Assumption R (See [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]). Assumption R means that the returns of the assets issued at a node ξ are not redundant 3 with the returns of the assets issued at a predecessor node of ξ. So, the issuance of additional assets issued at ξ leads to 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.

We recall that the continuity of the set of marketable payoffs, in mathematical terms, means that the correspondence q → H(p, q) has a closed graph and is lower semi-continuous for a given commodity price p. We show easily that in a multi-period economy, for each spot price p, the correspondence H(p, .) is lower semi-continuous (See [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]). This is a direct consequence of the fact that the matrix W (p, q) depends continuously on q. We illustrate (See [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]) by means of a numerical example that the closedness of the graph of the set of marketable payoffs is not granted in a multi-period economy. Equality between the kernels of the payoffs matrices regardless of the non-arbitrage price is sufficient to get the closure of H(p, .) (See, Proposition 3.2 of Bonnisseau-Chéry [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]).

Assumption R is not suitable to study financial structures with re-trading. Indeed, in [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF], it is shown that a financial structure with re-trading is equivalent to a financial structure without re-trading by considering that a re-trade is equivalent to the issuance of a new asset. So we remark that if the financial structure has long-term assets with re-trading, then Assumption R may hold true without re-trading but not with re-trading. We give an example in the Section 3 of Bonnisseau-Chéry [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF]). Hence the need to develop new Assumptions on the payoff matrix which will allow us to address the issue of stability of the set of marketable payoffs of a financial structure with re-trading. This will allow us to extend the domain of validity of the payoff matrices with continuous marketable payoffs.

Some additional notations

We first 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}.
For all t ∈ {0, . . . , T -1}, we denote by J t the set of assets issued at date t, that is,

J t = {j ∈ J | ξ (j) ∈ D t }. Let (τ 1 , . . . , τ k ) such that 0 ≤ τ 1 < τ 2 < . . . < τ k ≤ T -1 be
the dates at which there is at least the issuance of one asset, that is J τκ = ∅. For κ = 1, . . . , k, let D e τκ be the set of nodes at date τ κ at which there is the issuance of at least one asset. D e = ∪ k κ=1 D e τκ is the set of nodes at which there is the issuance of at least one asset. We remark that τ ∈{0,...,T -1}

J τ = κ∈{1,...,k} J τκ = J
and for all τ ∈ {τ 1 , . . . , τ k } , ξ∈Dτ J (ξ) = J τ .

Assumptions on V F (p)

We posit some assumptions on the payoff matrices.

Assumptions

Let F = {J , R J , (ξ(j)) j∈J , V } be a financial structure given.

Let p ∈ R L be a spot price.

Assumption H:

for all j ∈ J , V j ξ (p) = 0 if ξ ∈ D e ∩ D(ξ(j)).
Assumption K:

for each η ∈ D e , KerV J (η) η + (p) = KerV J (η) D + (η) (p)
Assumptions K': for each η ∈ D e and for each ξ

∈ η + , ∃α ξ = (α ξ ξ ) ξ ∈D + (ξ) ∈ (R + ) D + (ξ) such that ∀ξ ∈ D + (ξ), V J (η) ξ (p) = α ξ ξ V J (η) ξ (p)
Assumption H means that all assets give zero payoffs at the nodes at which there is the issuance of at least one asset i.e., at the nodes ξ ∈ D e . Assumption K means that if the portfolio of assets purchased today at node ξ gives zero payoff at the next date (i.e., at the nodes ξ ∈ ξ + ) then this portfolio gives zero payoffs at all nodes of the event-tree D. Assumption K' means that the array of returns of assets issued at the same node η at a node ξ ∈ D + (η) is proportional to the array of returns of these assets at the immediate successor node of η which is a predecessor of ξ . In other words, the distribution of returns of assets issued at the same node is constant over time but the magnitude may vary. Documents de travail du Centre d'Economie de la Sorbonne -2018.24 Remark 3.1. Assumptions K and H are not incompatible and they do not imply Assumption K'. Indeed, consider the financial structure

F = J , R J , (ξ(j)) j∈J , V
with D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ 6 , ξ 7 , ξ 8 , ξ 9 , ξ 10 }, J = j 1 , j 2 , j 3 , j 4 and that the first two assets are issued at node ξ 0 , The third asset is issued at node ξ 4 and the fourth is issued at node ξ 5 . The payoff matrix is:

• ξ 0 • ξ 2 • ξ 1 • ξ 3 • ξ 5 • ξ 4 • ξ 6 • ξ 9 • ξ 7 • ξ 8 • ξ 10 t = 0 t = 1 t = 2 t = 3
V =                   0 0 0 0 1 2 0 0 -1 3 0 0 4 -3 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 2 3 0 3 6 1 0 -3 9 0 2 12 -9 0 0                   ξ 0 ξ 1 ξ 2 ξ 3 ξ 4 ξ 5 ξ 6 ξ 7 ξ 8 ξ 9 ξ 10 D e = {ξ 0 , ξ 4 , ξ 5 }, D + (ξ 4 ) = ξ 4 + , D + (ξ 5 ) = ξ 5 + and D + (ξ 1 ) = {ξ 4 , ξ 7 , ξ 8 }. We have V ξ0 = V ξ4 = V ξ5 = 0 so Assumption H is satisfied in F. We also remark that KerV J (ξ0) ξ0 + = KerV J (ξ0) D + (ξ0) = {0}, KerV J (ξ4) ξ4 + = KerV J (ξ4) D + (ξ4) = {0}, KerV J (ξ5) ξ5 + = KerV J (ξ5) D + (ξ5) = {0} so Assumption K is satisfied in F. Since regardless α ∈ R + , we have V J (ξ0) ξ4 = 1 2 = αV J (ξ0) ξ1 = α 1 1 so Assumption K' is not satisfied in F.
Remark 3.2. Assumption K' is not stronger than Assumption H. Indeed, let F = (J , R J , (ξ(j)) j∈J , V ) be financial structure such that J = {j 1 , j 2 }, Documents de travail du Centre d'Economie de la Sorbonne -2018.24 D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 } and that the first asset is issued at node ξ 0 and the second asset is issued at node ξ 1 .

• ξ 0 • ξ 1 • ξ 2 • ξ 3 t = 0 t = 1 t = 2 t = 3
Figure 3: the tree D

The payoff matrix is:

V =     0 0 1 0 1 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3
Here we have:

D e = {ξ 0 , ξ 1 }, ξ + 0 = {ξ 1 }, ξ + 1 = {ξ 2 }, ξ + 2 = {ξ 3 }, α ξ1 = (1, 1), α ξ2 = (1), so Assumption K' is satisfied in F. Since V ξ1 = 1 0 = 0 0 so Assumption H is not satisfied in F. Remark 3.3. Assumption K' is equivalent to: for each η ∈ D e , for each ξ ∈ η + , ∃α ξ = (α ξ ξ ) ξ ∈D + (ξ) ∈ (R + ) D + (ξ) such that ∀ξ ∈ D + (ξ), V j ξ (p) = α ξ ξ V j ξ (p)
, ∀j ∈ J (η). Remark 3.4. Assumption K' is satisfied by all financial structures consisting only of short-term assets. Indeed, it suffices to take α ξ = 0, for all ξ ∈ D e . Lemma 3.1. All financial sub-structure4 of a financial structure which satisfies Assumption K' satisfies Assumption K'.

The proof of Lemma 3.1 is obvious. Lemma 3.2. Assumption K' is stronger than Assumption K.

Proof of Lemma 3.2 is given in Appendix.

Remark 3.5. If there is only two dates i.e., if T = 1, Assumption H is satisfied.

If financial assets are issued at the same date, Assumption H is satisfied.

Assumption K is satisfied by the financial structures consisting only of shortterm assets.

Assumption K is equivalent to each of the two following assumptions:

• for each η ∈ D e , KerV J (η) η + (p) = KerV J (η) (p). • for each η ∈ D e , Im t V J (η) η + (p) = Im t V J (η) D + (η) (p).
Lemma 3.3. Let p ∈ R L be a spot price vector. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption K at the spot price p. Then F has no trivial assets5 if and only if ∀η ∈ D e and ∀j ∈ J (η) there exists ξ ∈ η + such that V j ξ (p) = 0.

Proof of Lemma 3.3 is given in Appendix.

Equality between the kernel of payoff matrices

The main purpose of this section is to provide a new sufficient conditions compatible with long term assets under which the kernels of the payoff and full payoff matrices are equal for every arbitrage free price. Hence the stability of the financial structure holds true because in Bonnisseau-Chéry [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], we have shown that the equality between kernels of payoffs matrices whatever the price of non arbitrage implies the stability in this financial structure.

Let F = {J , R J , (ξ(j)) j∈J , V } be a financial structure. For each spot price p ∈ R L , we note by Q p the set of arbitrage free prices with respect to p for F where Q p = {q ∈ R J ; ∃λ ∈ R D ++ satisfying t W (p, q)λ = 0}. Under Assumption H, the following Proposition shows that the kernel the payoff matrix is always contains in the kernels of the full payoff matrices whatever the arbitrage free price. We get the other inclusion thanks to Assumption K (See Proposition 3.3). In addition, if there are no redundant assets, we have equality without Assumption K. Proposition 3.1. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure and p ∈ R L be a spot price vector. If F satisfies Assumption H at the spot price p, then KerW (p, q) ⊂ KerV (p), ∀q ∈ Q p and rankV (p) ≤ rankW (p, q), ∀q ∈ Q p .

The proof of Proposition 3.1 is given after the following Corollary.

Corollary 3.1. Let p ∈ R L be a spot price. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption H at the spot price p. If there is no redundant assets or, equivalently, rankV (p) = J then

KerW (p, q) = KerV (p) and rankV (p) = rankW (p, q).
Proof of Proposition 3.1. Let λ ∈ R D ++ be a state price and let q be the arbitrage free price associated with λ. For all z ∈ KerW (p, q) we have:

0 = W (p, q)z = j∈J W j (p, q)z j = η∈D e [ j∈J (η) W j (p, q)z j ]. This im- plies that ∀ξ ∈ D, 0 = j∈J W j ξ (p, q)z j = η∈D e   j∈J (η) W j ξ (p, q)z j   ( * )
Thanks to the structure of the matrix W (p, q) and Assumption H, for each ξ ∈ D we have j∈J

W j ξ (p, q)z j =    j∈J V j ξ (p)z j if ξ / ∈ D e -j∈J (ξ) q j z j if ξ ∈ D e
and we also have

j∈J V j ξ (p)z j =    j∈J V j ξ (p)z j if ξ / ∈ D e 0 if ξ ∈ D e
Since for all ξ ∈ D, j∈J W j ξ (p, q λ )z j = 0 (See ( * )) we have for all ξ ∈ D, j∈J V j ξ (p)z j = 0 i.e., z ∈ KerV (p). Hence KerW (p, q) ⊂ KerV (p). If moreover, rankV (p) = J , we have

{0} ⊂ KerW (p, q) ⊂ KerV (p) = {0}.
and this implies that KerW (p, q) = KerV (p). From the rank Theorem in linear algebra we have rankV (p) ≤ rankW (p, q). Remark 3.6. The inclusion KerW (p, q) ⊂ KerV (p) of Proposition 3.1 may be strict. Indeed, consider the financial structure F = (J , R J , (ξ(j)) j∈J , V ) such that D = ξ ) , ξ 1 , ξ 2 , J = j 1 , j 2 and that the first asset is issued at node ξ 0 and the second asset is issued at node ξ 1 . The payoff matrix and the full payoff matrix are:

• ξ 0 • ξ 1 • ξ 2 t = 0 t = 1 t = 2
V =   0 0 0 0 1 1   ξ 0 ξ 1 ξ 2 and W(q) =   -1 0 0 -1 1 1   ξ 0 ξ 1 ξ 2
It is clear that Assumption H is satisfied and we have:

{0} = KerW (q) = KerV = {α(1, -1); α ∈ R}.
The following proposition states that, under Assumption H, the set of arbitrage free prices is a cone. This result is always true in two-period case. Cornet and Ranjan (See [START_REF] Cornet | A remark on the set of arbitrage-free prices in a multi-period model[END_REF]) had proved a similar result in multi-period economy under another Assumption. They have showed with their Assumption that the set of arbitrage free prices is a convex cone. But Assumption H is not sufficient to have the convexity. Documents de travail du Centre d'Economie de la Sorbonne -2018.24 Proposition 3.2. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure and p ∈ R L be a spot price vector. If F satisfies Assumption H at the spot price p, then, the set Q p which is the set arbitrage free prices associated with spot price p is a cone.

Proof of proposition 3.2. Q p is a cone if for all q ∈ Q p and t ∈]0, +∞[, tq ∈ Q p . For each q ∈ Q p , from the characterization of arbitrage free price, we have λ ξ(j) q j = ξ∈D + (ξ(j))

λ ξ V j ξ (p), ∀j ∈ J .
Let t ∈]0, +∞[. So, to show that tq ∈ Q p it suffices to show that there exists λ = (λ ξ ) ξ∈D ∈ R D ++ such that:

λ ξ(j) (tq j ) = ξ∈D + (ξ(j)) λ ξ V j ξ (p), ∀j ∈ J .
Choose λ according to the following manner: for each ξ ∈ D we have

λ ξ =    tλ ξ if ξ / ∈ D e λ ξ if ξ ∈ D e
We easily verify that, for each j ∈ J ,

λ ξ(j) (tq j ) = t(λ ξ(j) q j ) = t   ξ∈D + (ξ(j)) λ ξ V j ξ (p)   = ξ∈(D + (ξ(j))∩D e ) (tλ ξ )V j ξ (p) + ξ∈(D + (ξ(j))\D e ) (tλ ξ )V j ξ (p)
Assumption H implies that for all ξ ∈ (D + (ξ(j)) ∩ D e ), V j ξ (p) = 0 hence we have ξ∈(D + (ξ(j))∩D e ) λ ξ V j ξ (p) = 0 and ξ∈(D

+ (ξ(j))∩D e ) λ ξ V j ξ (p) = 0. So λ ξ(j) (tq j ) = ξ∈(D + (ξ(j))\D e ) (tλ ξ )V j ξ (p) = ξ∈(D + (ξ(j))\D e ) λ ξ V j ξ (p) = ξ∈D + (ξ(j)) λ ξ V j ξ (p). Remark 3.7.
If Assumption H is not satisfied, Q p may not be a cone even if there is equality between the kernels of the payoff matrices. Indeed, consider the financial structure F = J , R J , (ξ(j)) j∈J , V such that: D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 }, J = j 1 , j 2 and that the first asset is issued at node ξ 0 and the second asset is issued at node ξ 1 .

The payoff matrix is:

V =     0 0 -2 0 1 -1 1 0     ξ 0 ξ 1 ξ 2 ξ 3
Documents de travail du Centre d'Economie de la Sorbonne -2018.24 Here Assumption H is not satisfied in F and we have KerV = KerW (q) = {0} for any asset price q. With λ = (1, 1, 1, 1) we have q λ = (0, -1) while 2q λ = (0, -2) / ∈ Q p .

• ξ 0 • ξ 1 • ξ 3 • ξ 2 t = 0 t = 1 t = 2
In the following Proposition we show that, under Assumption K, the kernel of the payoff matrix is always included in the kernels of the full payoff matrices whatever the arbitrage free price. Proposition 3.3. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure and p ∈ R L be a spot price vector. If F satisfies Assumption K at the spot price p.

Then

KerV (p) ⊂ KerW (p, q), ∀q ∈ Q p and rankW (p, q) ≤ rankV (p), ∀q ∈ Q p .

The proof of Proposition 3.3 is given below.

If the kernel of the payoff matrix of financial structure is included in the kernel of the full payoff matrix any arbitrage free price given, Proposition 3.4 states that, for each spot price p ∈ R L given, the set of arbitrage free prices associated with p is orthogonal at the kernel of the payoff matrix. Proposition 3.4. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure and p ∈ R L be a spot price vector. Suppose that for each q ∈ Q p , KerV (p) ⊂ KerW (p, q) then

Q p ⊂ (KerV (p)) ⊥ Proof of Proposition 3.4 is given in Appendix.
The following Corollary is deduced from Proposition 3.3 and Proposition 3.4. Corollary 3.2. Let p ∈ R L be a spot price. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption K at the spot price p. We have

Q p ⊂ (KerV (p)) ⊥ .
z ∈ KerV (p) ⇔   0 = j∈J V j (p)z j = η∈D e j∈J (η) V j (p)z j   ∀ξ ∈ D, 0 = j∈J V j ξ (p)z j = η∈D e   j∈J (η) V j ξ (p)z j  
Let us show that: for all η ∈ D e τ1 , z J (η) = (z j ) j∈J (η) ∈ KerV J (η) (p). Let η ∈ D e τ1 . Given the structure of the matrix V (p), noticing that τ 1 < τ κ for all κ = 1, we have for all j / ∈ J (η):

V j ξ (p) = 0, ∀ξ ∈ η +
So, for each η ∈ D e τ1 , ∀ξ ∈ η + we have:

0 = j∈J V j ξ (p)z j = j∈J (η) V j ξ (p)z j
and this means that z J (η) ∈ KerV (p) = 0. This implies that z J (η) ∈ KerV J (η) (p). Now, since for every η ∈ D e τ1 , z J (η) ∈ KerV J (η) (p), taking z ∈ KerV (p), we get

0 = η∈D e   j∈J (η) V j (p)z j   = η∈(D e \D e τ 1 )   j∈J (η) V j (p)z j   ∀ξ ∈ D, 0 = j∈J V j ξ (p)z j = η∈(D e \D e τ 1 )   j∈J (η) V j ξ (p)z j  
We do the same reasoning to get z J (η) ∈ KerV J (η) (p) for all η ∈ D e τ2 . Then for each (in increasing order) κ ∈ {2, . . . , k} and for each node η ∈ D e τκ and finally, we get that, for all κ ∈ {1, . . . , k} and for all η ∈ D e τκ , z J (η) = (z j ) j∈J (η) ∈ KerV J (η) (p).

• Let λ ∈ R D ++ be a given state price and let q the arbitrage free price associated with λ.

Let κ ∈ {1, . . . , k} and η ∈ D e τκ . Let us show that z J (η) ∈ KerV J (η) (p) ⇒ z J (η) ∈ KerW J (η) (p, q). Indeed, if z J (η) ∈ KerV J (η) (p) then, for each ξ ∈ D, we have:

W J (η) ξ (p, q) • z J (η) = j∈J (η) W j ξ (p, q)z j =    j∈J (η) V j ξ (p)z j if ξ = η -j∈J (η) q j z j if ξ = η
Since q is the arbitrage free price associated with λ, we have j∈J (η)

q j z j = j∈J (η)   1 λ η ξ ∈D + (η) λ ξ V j ξ (p)   z j = 1 λ η ξ ∈D + (η) λ ξ   j∈J (η) V j ξ (p)z j   But z J (η) ∈ KerV J (η) (p) implies j∈J (η) V j ξ (p)z j = 0, ∀ξ ∈ D so j∈J (η) q j z j = 1 λ ξ ∈D + (η) λ ξ [0] = 0.
Consequently, W J (η) ξ (p, q) • z J (η) = 0 for all ξ or, in other words, z J (η) ∈ KerW J (η) (p, q). Since for all κ ∈ {1, . . . , k} and for all η ∈ D e τκ (see Claim 3.1) we have z J (η) ∈ KerV J (η) (p), this implies that for all κ ∈ {1, . . . , k} and for all η ∈ D e τκ we have z J (η) ∈ KerW J (η) (p, q) and this implies that z ∈ KerW (p, q). Hence KerV (p) ⊂ KerW (p, q). Corollary 3.3. Let p ∈ R L be a spot price. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption K at the spot price p. Then the linear subspaces of the family {ImV J (ξ) (p)} ξ∈D e are linearly independent 7 and 8

ImV (p) = ξ∈D e

ImV J (ξ) (p).

7 Linear subspaces U 1 , U 2 , . . . , Um (with U = {0} for all ) are linearly independent if from u 1 + u 2 + . . . + um = 0, where u ∈ U it follows that u = 0 for all . Sum of the linearly independent linear subspaces is called a direct sum of these linear subspaces and is denoted by Corollary 3.4. Let p ∈ R L be a spot price. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption K at the spot price p. Then the two following conditions are equivalent:

(i) for every ξ ∈ D e , the family {V j (p)} j∈J (ξ) is linearly independent;

(ii) the family {V j (p)} j∈J is linearly independent.

Remark 3.8. The inclusion KerV (p) ⊂ KerW (p, q) of Proposition 3.3 may be strict. Indeed, let a financial structure with three dates such that D = {ξ 0 , ξ 1 , ξ 2 , ξ 11 , ξ 12 }, J = {j 1 , j 2 , j 3 }, ξ(j 1 ) = ξ 0 , ξ(j 2 ) = ξ 1 and ξ(j 3 ) = ξ 2 .

ξ 0 • • ξ 1 • ξ 2 • ξ 11 ξ 21 • t = 0 t = 1 t = 2
Figure 6: the tree D

The payoff matrices are:

V =       0 0 0 1 0 0 -1 0 0 -1 -1 0 1 0 1       ξ 0 ξ 1 ξ 2 ξ 11 ξ 21
Assumption K is satisfied in our financial structure. Taking λ = (1, 1, 1, 1, 1), the full payoff matrix is:

W(q) =       0 0 0 1 1 0 -1 0 -1 -1 -1 0 1 0 1       ξ 0 ξ 1 ξ 2 ξ 11 ξ 21
We have: KerV = {0} because rankV = 3 and z = t (1, -1, -1) ∈ KerW (q) = {0}. Documents de travail du Centre d'Economie de la Sorbonne -2018.24

The following Corollary which is deduced of Proposition 3.3 and Proposition 3.1, gives a result of equality between the kernel the payoff matrix and the kernel full payoff matrice at any arbitrage free prices even with redundant assets. Corollary 3.5. Let p ∈ R L be a spot price. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption K and H at the spot price p. Then

KerV (p) = KerW (p, q), ∀q ∈ Q p and rankV (p) = rankW (p, q), ∀q ∈ Q p .
The following Proposition shows that under Assumption K' the kernel of the payoff matrix of the financial structure is equal to the kernel of the full payoff matrix regardless the arbitrage free price. It is a complementary result since there are no implications between Assumptions H and K' (See Remarks 3.1 and 3.2). Proposition 3.5. Given a spot price vector p ∈ R L . If the payoff matrix V satisfies Assumption K' at the spot price p, then for all arbitrage free price q ∈ R J , KerV (p) = KerW (p, q). The main difficulty of the proof of Proposition 3.5 relies on the fact that there is not necessarily emission of assets at each date t ∈ {0, . . . , T -1} of the tree D i.e., we have not necessarily: for all t ∈ {0, . . . , T -1}, D t ∩ D e = ∅. And, even if t is a date where assets are issued, all nodes at date t are not issuers of assets i.e., if D t ∩ D e = ∅ this does not imply that D t = D e t .

Proof of Proposition 3.5. Let q ∈ R J be an arbitrage free price and let λ ∈ R D ++ be a state price satisfying t W (p, q)λ = 0. Thanks to Lemma 3.2 Assumption K' is stronger than Assumption K. From Proposition 3.3 we have the inclusion KerV (p) ⊂ KerW (p, q). It remains to show that KerW (p, q) ⊂ KerV (p). Let z = (z j ) j∈J ∈ R J such that z ∈ KerW (p, q). This is equivalent to

j∈J W j (p, q)z j = 0 ⇔ [ j∈J W j ξ (p, q)z j = 0 for all ξ ∈ D]
Let us show that: for each ξ ∈ D e , j∈J (ξ) V j ξ (p)z j = 0 for all ξ ∈ D, which implies z ∈ KerV (p).

Note that, thanks to the structure matrix V (p), we always have

for each ξ ∈ D e , j∈J (ξ) V j ξ (p)z j = 0 for all ξ / ∈ D + (ξ).
So it remains to show that

for each ξ ∈ D e , j∈J (ξ)
V j ξ (p)z j = 0 for all ξ ∈ D + (ξ).

Documents de travail du Centre d'Economie de la Sorbonne -2018.24

The following Lemma will facilitate the understanding of the proof of Proposition 3.5. Lemma 3.4. Let p ∈ R L be a given spot price vector. Let F = {J , R J , (ξ(j)) j∈J , V } be a financial structure. Let λ = (λ ξ ) ξ∈D ∈ R D ++ be a given state price. Then for all arbitrage free price q associated with λ we have:

∀η ∈ D, ∀z ∈ R J , ξ∈D(η)   j∈J λ ξ W j ξ (p, q)z j   = ξ∈D(η)   β∈(D e ∩D -(η)) j∈J (β) λ ξ W j ξ (p, q)z j  
The proof of Lemma 3.4 is given in Appendix.

Claim 3.2. For all ξ ∈ D e τ1 we have j∈J (ξ) V j ξ (p)z j = 0, ∀ξ ∈ D + (ξ). Proof of Claim 3.2. let ξ ∈ D e τ1 . Let η ∈ ξ + be given. Thanks to Lemma 3.4, we have

ξ∈D(η)   j∈J λ ξ W j ξ (p, q)z j   = ξ∈D(η)   β∈(D e ∩D -(η)) j∈J (β) λ ξ W j ξ (p, q)z j   Since η -= ξ ∈ D e τ1 , D e ∩ D -(η) = {ξ}, the sum above is equal to ξ∈D(η)   j∈J (ξ) λ ξ W j ξ (p, q)z j  
thanks to the structure of the full payoff matrix W (p, q) we have

ξ∈D(η)   j∈J (ξ) λ ξ W j ξ (p, q)z j   = ξ∈D(η)   j∈J (ξ) λ ξ V j ξ (p)z j   But, since z ∈ KerW (p, q) we have j∈J W j ξ (p, q)z j = 0 for all ξ ∈ D So 0 = ξ∈D(η)   j∈J λ ξ W j ξ (p, q)z j   = ξ∈D(η)   j∈J (ξ) λ ξ V j ξ (p)z j   9 ( * )
Thanks to Assumption K' there exists

α η = (α η ξ ) ξ∈D(η) ∈ R D + (η) + ( with α η η = 1) such that ξ∈D(η)   j∈J (ξ) λ ξ V j ξ (p)z j   = ξ∈D(η)   j∈J (ξ) λ ξ (α η ξ V j η (p))z j   =   ξ∈D(η) λ ξ α η ξ     j∈J (ξ) V j η (p)z j   According to ( * ) we have 0 =   ξ∈D(η) λ ξ α η ξ     j∈J (ξ) V j η (p)z j   But ξ∈D(η) λ ξ α η ξ > 0 because α η ξ ≥ 0, ∀ξ ∈ D + (η), α η η = 1 and λ ξ > 0, ∀ξ ∈ D(η), so j∈J (ξ) V j η (p)z j = 0
and this implies that

j∈J (ξ) α η ξ V j η (p)z j = 0, ∀ξ ∈ D(η) ⇓ j∈J (ξ) V j ξ (p)z j = j∈J (ξ) α η ξ V j η (p)z j = 0, ∀ξ ∈ D(η)
Since η is arbitrary in ξ + , we have so

for each η ∈ ξ + , j∈J (ξ)
V j ξ (p)z j = 0, ∀ξ ∈ D(η)

Consequently:

j∈J (ξ) V j ξ (p)z j = 0, ∀ξ ∈ D + (ξ).
Since ξ is arbitrary in D e τ1 , we have: for each ξ ∈ D e τ1 , j∈J (ξ) V j ξ (p)z j = 0, ∀ξ ∈ D + (ξ) If all assets are issued on the same date i.e., τ k = τ 1 , the proof is complete.

Otherwise, we iterate the process using the following claim:

Claim 3.3. For all ξ ∈ D e τ2 we have j∈J ( ξ) V j ξ (p)z j = 0, ∀ξ ∈ D + ( ξ).
Documents de travail du Centre d'Economie de la Sorbonne -2018.24

Proof of Claim 3.3. let ξ ∈ D e τ2 . Let η ∈ ξ + be given. Thanks to Lemma 3.4, we have

ξ∈D(η)   j∈J λ ξ W j ξ (p, q)z j   = ξ∈D(η)   β∈(D e ∩D -(η)) j∈J (β) λ ξ W j ξ (p, q)z j   Two possible cases: either D e ∩ D -(η) = { ξ} or D e ∩ D -(η) = {ξ, ξ} (with ξ ∈ D e τ1 ).
First case: If D e ∩D -(η) = { ξ}, thanks to the structure of the full matrix W (p, q), we have

ξ∈D(η)   β∈(D e ∩D -(η)) j∈J (β) λ ξ W j ξ (p, q)z j   = ξ∈D(η)   j∈J ( ξ) λ ξ W j ξ (p, q)z j   = ξ∈D(η)   j∈J ( ξ) λ ξ V j ξ (p)z j  
With the same reasoning as in the previous claim, we show that

j∈J ( ξ) V j ξ (p)z j = 0 for all ξ ∈ D(η) Second case If D e ∩ D -(η) = {ξ, ξ} (with ξ ∈ D e τ1
), thanks to the structure of the full matrix W (p, q), we have

ξ∈D(η)   β∈(D e ∩D -(η)) j∈J (β) λ ξ W j ξ (p, q)z j   = ξ∈D(η)   j∈J (ξ) λ ξ W j ξ (p, q)z j + j∈J ( ξ) λ ξ W j ξ (p, q)z j   = ξ∈D(η)   j∈J (ξ) λ ξ V j ξ (p)z j + j∈J ( ξ) λ ξ V j ξ (p)z j   According to Claim 3.2, j∈J (ξ) V j ξ (p)z j = 0 for all ξ ∈ D(η). Con- sequently ξ∈D(η)   j∈J λ ξ W j ξ (p, q)z j   = ξ∈D(η)   j∈J ( ξ) λ ξ V j ξ (p)z j  
With the same reasoning as for Claim 3.2 we show that j∈J ( ξ)

V j ξ (p)z j = 0 for all ξ ∈ D(η) Documents de travail du Centre d'Economie de la Sorbonne -2018.24

Since this is true for all η ∈ ξ + , we have

for each η ∈ ξ, j∈J ( ξ)
V j ξ (p)z j = j∈J ( ξ) α η ξ V j η (p)z j = 0, ∀ξ ∈ D(η)
Consequently:

j∈J ( ξ) V j ξ (p)z j = 0, ∀ξ ∈ D + ( ξ).
Since this is true for all ξ ∈ D e τ2 , we have, for each ξ ∈ D e τ2 , j∈J ( ξ) V j ξ (p)z j = 0, ∀ξ ∈ D + ( ξ). If there were only two dates of issuance of asset i.e., τ k = τ 2 , the proof is complete thanks to Claim 3.3. Otherwise, we repeat the same reasoning for the next dates of emission to get at the end, for each

ξ ∈ D e , j∈J (ξ)
V j ξ (p)z j = 0, for all ξ ∈ D.

So we deduce that: for all z ∈ KerW (p, q), z ∈ KerV (p). Hence KerW (p, q) ⊂ KerV (p).

Corollary 3.6. Given a spot price p ∈ R L . Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption K' at p, then for all arbitrage free price q, rankV (p) = rankW (p, q). Remark 3.9. Corollary 3.6 is a generalization of Proposition 5.2. b) and c) in Angeloni-Cornet [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF] and of Magill-Quinzii [START_REF] Magill | Theory of Incomplete Markets[END_REF] where only short-term assets are considered.

Corollary 3.7 is a generalization of Corollary 3.6 when we restrict the portfolios to belong to a subspace G of R J . The proof of Corollary 3.7 is given in Bonnisseau-Chéry [START_REF] Bonnisseau | Stability of marketable payoffs with longterm assets[END_REF], it suffices to replace Assumption R in the proof of Corollary 3.4 by each of the conditions of Corollary 3.7. Corollary 3.7. Given a spot price p ∈ R L . Let F = J , R J , (ξ(j)) j∈J , V be a financial structure given and let G be a linear subspace of R J . Suppose that one of the following three conditions is satisfied: In this section, we introduce the notion of the re-trading extension of the financial structure as in Angeloni and Cornet [START_REF] Angeloni | Existence of financial equilibria in a multi-period stochastic economy[END_REF]. Actually, it consists in considering that re-trade an asset after its date of issuance is as issuing a new asset with the payoffs being the truncated payoffs of the initial asset. We do not only consider the standard case where all assets are issued at the initial node, but we also present the extension of a financial structure when the assets of the initial structure are issued at different nodes and dates. The new financial structure thus constituted is called the re-trading extention of the initial financial structure. When all assets are issued at the same node for a financial structure, we show that Assumption K' is inherited by the re-trading extension. We show that this is no more the case when the initial assets are issued at different periods. So we introduce a stronger assumption, Assumption K". If the initial financial structure satisfies this condition, then the re-trading extension satisfies Assumption K'. So we obtain stability result or continuity of the marketable payoffs for re-trading extension of financial structure. Definition 4.1. Let F = J , R J , (ξ(j)) j∈J , V be a financial structure. The re-trading of asset j ∈ J at node ξ ∈ D(ξ(j) \ D T , 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.
Note that the asset j ξ(j) is just the asset j. We associate to F its re-trading extension defined by: F = J = ξ∈D e J (ξ) × (D(ξ)\D T ) , R J , ξ( j) j∈ J , Ṽ which consists of all assets j ∈ J and of all re-trading assets (j ξ ) to all nodes ξ ∈ D(ξ) \ D T . Hence J = ξ∈D e [J (ξ) × (D(ξ)\D T )] and the D × J -matrix Ṽ (p) has for coefficients Ṽ j ξ ξ (p), as defined above. We denote by q j ξ the price of asset j ξ (i.e., the retrading of asset j at node ξ ), which is sometimes also called the price of asset j at node ξ . So, for the financial structure F, both the asset price vector q = q j ξ j∈J ,ξ ∈ ξ∈D e (D(ξ)\D T ) and the portfolio z = z j ξ j∈J ,ξ ∈ ξ∈D e (D(ξ)\D T ) now belong to R J .

A simple example

Let F = J , R J , (ξ(j)) j∈J , V be a financial structure such that D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 }, J = {j 1 , j 2 }, ξ(j 1 ) = ξ 0 , ξ(j 2 ) = ξ 1 , and the payoff matrix is

V =     0 0 2 0 1 3 2 1     ξ 0 ξ 1 ξ 2 ξ 3
Documents de travail du Centre d'Economie de la Sorbonne -2018.24

4 STABILITY WITH RE-TRADING 24 • ξ 0 • ξ 1 • ξ 2 • ξ 3 t = 0 t = 1 t = 2 t = 3
Figure 7: the tree D

We have De = {ξ 0 , ξ 1 , ξ 2 }, J = j 1 ξ0 , j 1 ξ1 , j 2 ξ1 , j 1 ξ2 , j 2 ξ2 (because the asset j 1 is retraded at nodes ξ 1 and ξ 2 and j 2 is retraded at node ξ 2 ). The payoff matrix of the re-trading extension financial structures of F is:

Ṽ =     0 0 0 0 0 2 0 0 0 0 1 1 3 0 0 2 2 1 2 1     ξ 0 ξ 1 ξ 2 ξ 3
The full payoff matrix of the re-trading extension financial structures of F at the asset price q = q 1 ξ0 , q 1 ξ1 , q 2 ξ1 , q 1 ξ2 , q 2 ξ2 is:

W (q) =     -q 1 ξ0 0 0 0 0 2 -q 1 ξ1 -q 2 ξ1 0 0 1 1 3 -q 1 ξ2 -q 2 ξ2 2 2 1 2 1     ξ 0 ξ 1 ξ 2 ξ 3
Remark 4.1. Even if Assumption K' is satisfied in the financial structures F, it may not be satisfied in F. Indeed, in the previous example, it is clear that F satisfies Assumption K' and in the re-trading extension F of F Assumption K' is not satisfied because regardless α ∈ R, Ṽ J (ξ1)

ξ3

= α Ṽ J (ξ1) ξ2 .

About Asssumption K' for re-trading

In this sub-section, we will make some remarks on Assumption K' with respect to the stability with of the re-trading extension. The previous example shows that the re-trading extension may not satisfy Assumption K' even if the initial financial structure satisfies it. We will show that Assumption K' is satisfied by the re-trading extension of a financial structure when there is a unique node of issuance of all initial asset, Assumption K' is satisfied by the initial financial structure and the payoffs of an asset do not vanish between the date of issuance and the expiration date of the asset. So, we get the stability of the re-trading extension under the same hypothesis.

Remark 4.2. Assumption K' may not be inherited by the re-trading extension even if there is a signe asset issued at the initial node. Indeed, let F = J , R J , (ξ(j)) j∈J , V be a financial structure such that D = {ξ 0 , ξ 1 , ξ 2 , ξ 3 }, J = {j} and ξ(j) = ξ 0 , Documents de travail du Centre d'Economie de la Sorbonne -2018.24

4 STABILITY WITH RE-TRADING 25 • ξ 0 • ξ 1 • ξ 2 • ξ 3 t = 0 t = 1 t = 2 t = 3
Figure 8: the tree D and the payoff matrix is

V =     0 1 0 1     ξ 0 ξ 1 ξ 2 ξ 3
It is clear that F satisfies Assumption K'. In the re-trading extension F of F we have De = {ξ 0 , ξ 1 , ξ 2 } and the asset j is retraded at nodes ξ 1 and ξ 2 . The payoff matrix of the re-trading extension financial structures of

F Ṽ =     0 0 0 1 0 0 0 0 0 1 1 1     ξ 0 ξ 1 ξ 2 ξ 3
We find that Assumption K' is not satisfied by F because Ṽ j ξ 1 ξ2 = 0 and that regardless α ∈ R,

Ṽ j ξ 1 ξ3 = α Ṽ j ξ 1 ξ2 .

Re-trading with a single issuance node

Suppose that T ≥ 2 because if T = 1 there is no re-trading. Consider the financial structure F = (J , R J , (ξ(j)) j∈J , V ) such that all the assets are issued at node ξ 0 . Suppose that each asset j, once issued, is re-traded at all succeeding nodes of the node ξ 0 except terminal nodes. In this case De = D -= D\D T and J = J × D -. Proposition 4.1 shows that Assumption K' is satisfied by the re-trading extensions if D e = {ξ 0 } and that the asset returns to non-terminal nodes are non-zero. Proposition 4.1. Given a spot price vector p ∈ R L . Consider the financial structure F = J , R J , (ξ(j) = ξ 0 ) j∈J , V such that V (p) satisfies Assumption K' and for all j ∈ J , if V j ξ (p) = 0, then V j ξ (p) = 0 for all ξ > ξ. Then the re-trading extension F = J , R J , ξ( j) j∈ J , Ṽ of F satisfies Assumption K' at price p.

ζ ξ ∈ R + such that V J ξ (p) = α ζ ξ V J ζ (p) and V J ξ (p) = α ζ ξ V J ζ (p). If α ζ ξ > 0, we let θ ξ ξ = α ζ ξ α ζ ξ
, and we get Ṽ J (η)

ξ (p) = θ ξ ξ Ṽ J (η) ξ (p).
If α ζ ξ = 0, then V J ξ (p) = 0 and from our assumption, V J ξ (p) also vanish since ξ ∈ D + (ξ). Then, let θ ξ ξ = 0, and we obviously get 0 = Ṽ J (η)

ξ (p) = θ ξ ξ Ṽ J (η) ξ (p).
Consequently the financial structure F satisfies Assumption K'.

From Proposition 4.1 and the results of the previous section, we deduce the following Corollary.

Corollary 4.1. Let F = J , R J , (ξ(j) = ξ 0 ) j∈J , V be a financial structure and p ∈ R L a spot price vector. such that V (p) satisfies Assumption K' and for all j ∈ J , if V j ξ (p) = 0, then V j ξ (p) = 0 for all ξ > ξ. Let F = J , R J , ξ( j) j∈ J , Ṽ be the re-trading extension of F. Then

• Ker Ṽ (p) = KerW F (p, q) for all arbitrage free asset price q.

• the marketable payoffs correspondence defined by q → H(p, q) = {ω ∈ R D |∃z ∈ R J , ω = W F (p, q)z} has a closed graph.

Re-trading with different dates of issuance

Consider now a financial structure F = J , R J , (ξ(j)) j∈J , V where assets may be issued at any non terminal node. Given that in this case Assumption K' does not allow to have stability, we exhibit the following Assumption K"( which is a slightly stronger than K') to deal with the question of stability.

Assumption K":

There

is α = (α ) ∈D t≥2 ∈ R D t≥2
++ such that for each η ∈ D e , for each ξ ∈ η + , for each ξ , ξ ∈ D(ξ) such that ξ ∈ ξ + , V

J (η) ξ (p) = α ξ V J (η) ξ (p).
Assumption K" means that the ratio of the returns at two successive nodes after the issuance node dos not depend neither on the asset nor on the issuance node. It could be interpreted as if the returns of the assets are driven overt time by a common financial index represented by the parameter α. The difference with Assumption K is that in this later case, the ratio could vary with respect to the issuance node. Proof of Lemma 3.3. [⇐]. It is clear that if for all j ∈ J , there exists ξ ∈ ξ + (j) such that V j ξ (p) = 0 then F contains no trivial assets. [⇒]. By contradiction. Suppose that there exists an asset j 0 ∈ J such that: ∀ξ ∈ ξ + (j 0 ), V j0 ξ (p) = 0. We assume without any loss of generality that the column V j0 (p) of V (p) is ranked in such a way that it is the first column of the submatrix V J (ξ(j0)) (p). Let z = (z j ) j∈J (ξ(j0)) ∈ KerV • First given the structure of the matrix W (p, q), for all β / ∈ D ∪ (η), and for all ξ ∈ D(η) we have W j ξ (p, q) = 0 for all j ∈ J (β) So λ ξ W j ξ (p, q)z j   .

Thanks to the structure of the matrix W (p, q), for all ξ ∈ (D(η)\D(β)) we have W j ξ (p, q) = 0 for all j ∈ J (β) so λ ξ W j ξ (p, q)]z j

For each j ∈ J (β), W j ξ (p, q) = -q j if ξ = β and W j ξ (p, q) = V j ξ (p) if ξ = β. Since q is the arbitrage free price associated with λ, for each j ∈ J (β), λ β q j = ξ∈D + (β) λ ξ V j ξ (p). So, for each j ∈ J (β) we have ξ∈D(β) λ ξ W j ξ (p, q) = 0 hence ξ∈D(β) j∈J (β) λ ξ W j ξ (p, q)z j = 0. Consequently we have V j ξ (p)z j and

W ξ (p, q) • z =      j∈J (D -(ξ)) V j ξ (p)z j if ξ / ∈ D e j∈J (D -(ξ)) V j ξ (p)z j -j∈J (ξ) q j z j if ξ ∈ D e
Since W (p, q)z = 0, one has j∈J (ξ) q j z j = 0, ∀ξ ∈ D e . So q • z = j∈J q j z j =

  We also use the notation x > y if x ≥ y and x = y. The Euclidean norm in the Euclidean different 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 -2018.24 2 THE T -PERIOD FINANCIAL EXCHANGE ECONOMY 5 successors defined by ξ + = ξ ∈ D : ξ = ξ-.

Figure 1 :

 1 Figure 1: the tree D

Figure 2 :

 2 Figure 2: the tree D

Figure 4 :

 4 Figure 4: the tree D

Figure 5 :

 5 Figure 5: the tree D

•

  Now we give the proof of Proposition 3.3. Proof of Proposition 3.3. Let z ∈ R J such that z ∈ KerV (p). Documents de travail du Centre d'Economie de la Sorbonne -2018.24 Let us show, first, the following claim : Claim 3.1. z ∈ KerV (p) if and only if [∀κ ∈ {1, . . . , k}, ∀η ∈ D e τκ , z J (η) = (z j ) j∈J (η) ∈ KerV J (η) (p)]. 6 Proof of Claim 3.1 The implication [⇐] is obvious. Let us show the other implication.

J

  (η) η + (p) and, Assumption K implies that z J (η) ∈ KerV J (η) D + (η) (p). So, thanks to the structure of V (p), we have: for all ξ / ∈ D + (η), V J (η) ξ

  Linear subspace U is said to be equal to m =1 U = U 1 . . . Um if any u from U can be represented as u = u 1 + . . . + um, u ∈ U uniquely. Documents de travail du Centre d'Economie de la Sorbonne -2018.24 The proof of Corollary 3.3 is deduced from Claim 3.1. The following Corollary are deduced from Corollary 3.3.

i)4

  Assumption K' at p ii) Assumption K and Assumption H at p iii) Assumption H at p and rankV (p) = J .Then for all arbitrage free price q, dim [W (p, q)G] = dim(V (p)G).Documents de travail duCentre d'Economie de la Sorbonne -2018.24 4 STABILITY WITH RE-TRADING 23 Stability with re-trading

Proof of Proposition 4 . 1 .

 41 Let η ∈ D -and ξ ∈ η + . If η ∈ D T -1 there is nothing to demonstrate because D + (ξ) = ∅. This implies that, if T = 2, Proposition 4.1 is obvious. Now, if T > 2, let us take η ∈ D -\D T -1 , ξ ∈ η + and ξ ∈ D + (ξ). Then Ṽ J (η) ξ (p) = V J ξ (p) and Ṽ J (η) ξ (p) = V J ξ (p)because ξ and ξ belong to D + (η) Documents de travail du Centre d'Economie de la Sorbonne -2018.24 4 STABILITY WITH RE-TRADING 26 (See Definition 4.1). Since ξ > ξ > η > ξ 0 , there exists a unique ζ ∈ ξ + 0 such that η ≥ ζ. Thanks to Assumption K' on F there exist α ζ ξ ∈ R + and α

Remark 4 . 3 .

 43 It is clear that Assumption K" implies K'. It suffices to define the parameter α ξ ξ of AssumptionK as α ξ1 α ξ2 . . . α ξ where (ξ 1 , ξ 2 , . . . , ξ ) is the unique path connecting ξ to ξ . Documents de travail du Centre d'Economie de la Sorbonne -2018.24that: ∀j ∈ J (η), V j ξ (p) = α η ξ V j η (p); so j∈J (η) V j ξ (p)z j = j∈J (η) α η ξ V j η (p)z j = α η ξ ( j∈J (η) V j η (p)z j )Since η ∈ η + and z ∈ KerVJ (η) η + (p), we have so ∀j ∈ J (η), V j ξ (p) = 0. Consequently, j∈J (η) V j ζ (p)z j = 0, ∀ζ ∈ D + (η) i.e., z ∈ KerV J (η) D + (η) (p). Hence KerV J (η) η + (p) ⊂ KerV J (η) D + (η) (p).It is easy to see that for each η ∈ D e , KerV J (η) D + (η) (p) ⊂ KerV J (η) η + (p) because ∀η ∈ D e , η + ⊂ D + (η).

J

  (ξ(j0)) ξ(j0) + (p). For all ζ ∈ R, z = (ζ, (z j ) j∈J (ξ(j0))\{j0} ) ∈ KerV J (ξ(j0)) ξ(j0) + (p). Assumption K implies that z ∈ KerV J (ξ(j0)) D + (ξ(j0)) (p). This implies that ∀ξ ∈ D(ξ(j 0 )), V j0 ξ (p) = 0, thus V j0 (p) = 0, i.e.,j 0 is a trivial asset. It is a contradiction. Proof of Lemma 3.4. Let η ∈ D and z ∈ R J be given. We have ξ∈D(η) Note by D ∪ (η) The union of η with all its predecessors and his successors i.e., D ∪ (η) = D -(η) ∪ D(η).

•

  e ∩D ∪ (η)) j∈J (β) λ ξ W j ξ (p, q)z j   Second, for all β ∈ D e ∩ D(η) we have ξ∈D(η) λ ξ W j ξ (p, q)z j  = Documents de travail du Centre d'Economie de la Sorbonne -2018.24



  e ∩D -(η)) j∈J (β) λ ξ W j ξ (p, q)z jProof of Proposition 3.4. Let z ∈ KerV (p), since by Assumption KerV (p) ⊂ KerW (p, q), we have W (p, q)z = 0. Thanks to the structures of the matrices V (p) and W (p, q), ∀ξ ∈ D,0 = V ξ (p) • z = j∈J V j ξ (p)z j = j∈J (D -(ξ))

  Documents de travail du Centre d'Economie de la Sorbonne -2018.24

L = H × D where H is a finite set of divisible and physical goods exchanged at each node ξ ∈ D. Documents de travail du Centre d'Economie de la Sorbonne -2018.24

Recall that an asset j 0 of a financial structure F = (J , R J , (ξ(j)) j∈J , V ) is said to be redundant at spot price p if the column vector V j 0 (p) representing its payoffs on D is a linear combination column vectors representing the payoffs of other assets p i.e., if there exists α = (α j ) j∈J \{j 0 } ∈ R J \{j 0 } such that V j 0 (p) = j∈J \{j 0 } α j V j (p). Documents de travail du Centre d'Economie de la Sorbonne -2018.24

Let F = (J , R J , (ξ(j)) j∈J , V ) be a given financial structure. We call sub-structure of F any financial structure F = (J , R J , (ξ(j)) j∈J , V ) such that J ⊂ J . Documents de travail du Centre d'Economie de la Sorbonne -2018.24

A trivial asset is an asset that gives zero returns at each node. Documents de travail du Centre d'Economie de la Sorbonne -2018.24

τκ is a date in which there is issuance of at least one asset and τ 1 << . . . < τ k . D e τκ is set of nodes where there is the isssuance of at least one asset at date τκ. Documents de travail du Centre d'Economie de la Sorbonne -2018.24

We remind that for each j ∈ J , -λ ξ(j) q j + ξ∈D + (ξ(j)) λ ξ V j ξ (p) = 0 because q j is an arbitrage free price associated to the state price λ. Documents de travail du Centre d'Economie de la Sorbonne -2018.24

Proposition 4.2 shows that Assumption K" is satisfied by re-trading extensions.

Proposition 4.2. Given a spot price vector p ∈ R L . Consider a financial structure F = J , R J , (ξ(j)) j∈J , V such that V (p) satisfies Assumption K". Then the re-trading extension F = J , R J , (ξ(j)) j∈ J , Ṽ of F satisfies Assumption K" at price p.

We deduce the following results on kernels equality of payoffs matrices in the extension of financial structures where assets are issued at different dates and different nodes. The proofs of the following Corollaries are deduced from Remark 4.3 and Proposition 3.5.

Corollary 4.2. Given a spot price p ∈ R L . Let F = J , R J , (ξ(j)) j∈J , V be a financial structure satisfying Assumption K". Let F = J , R J , ξ( j) j∈ J , Ṽ be the re-trading extension of F. Then Ker Ṽ (p) = KerW F (p, q) for all arbitrage free asset price q.

Appendix

We recall that for each ξ ∈ D, we denote by: D + (ξ) the set of successors of ξ;

Proof of Lemma 3.2. Let F = {J , R J , (ξ(j)) j∈J , V } be a financial structure and p ∈ R L be a spot price given. We assume that F satisfies Assumption K' at the spot price p.

Let us show that for each η ∈ D e , KerV


Shan Yongping
email: shanyp@yeah.net

Ping Zeng
email: bzengping71@hotmail.com

Song Yonghui

Guo Yuting

MANERO Marie -Helene

Amino-modified Hydrogen-Bonding Resin and Its Adsorption on Berberine

Keywords: Berberine, Hydrogen-bond, Adsorption, Resin

A hydrogen-bond donor resin HX was obtained by modifying resin H103 . The adsorption effect of resin HX on berberine was determined. The adsorbing kinetic and dynamic mechanisms of resin HX and resin H103 was compared. The results showed that, after modification, resin HX could form hydrogen-bond with berberine molecules. In the adsorption process, two C-O groups on berberine molecules became the electron acceptor of hydrogen-bond and the N-H group on resin molecules became the electron donor of hydrogen-bond. The existence of hydrogen-bond enhanced the adsorption force, and led to bigger amount of adsorption.

Introduction

Berberine is one of the important isoquinoline alkaloids, which has been widely used as an anti-inflammatory medicine. As the research in recent years, berberine could also be used in anti-hyperlipidemic [1], antidepressive [2], anticancer [3], and so on.

As the increase of demand, more and more berberine should be synthetized, and the amount of wastewater during berberine production has become a serious environmental problem. Berberine wastewater is a challenging work because of its high acidity, high concentration of organics and antibacterial characteristics. Research shows that bioreactors as UASB-MBR [4], SBBR [START_REF] Hu | Technical feature, kinetic mechanism and application of SBBR[END_REF], as well as physico-chemical processes like UV-O 3 [START_REF] Weiwei Qin | [END_REF], could decrease the concentration of berberine under certain conditions. But these processes also manifested shortages: wildly fluctuate of wastewater quality would be fatal to UASB-MBR and SBBR as the antibiotic character of berberine could damage microorganisms in bioreactors, and the high energy-consumption features would limit the application of UV-O 3 process.

In recent years, resin adsorption processes have been applied in many important fields. Macroporous resin of polystyrene skeleton is widely used in wastewater treatment due to its large specific surface area, strong adaptability of poor water quality. H103 is one important type of macroporous resins of polystyrene skeleton, and it has excellent acid-and alkali-resistant and heat-resistant. Considering it's characteristic, it was select to treat berberine wastewater with high concentration and low pH.

In this paper, resin H109 was modified, in order to obtain a hydrogen-bonding adsorbent which could lead to significant improvement of berberine adsorption.The berberine adsorption mechanism on resin H103 was studied as well as resin HX. Resin HX was synthesized by adding ammonia functional group to H109, the skeleton resin of resin H103.

Experimental Set-up

Modification of resin with hydrogen-bonding donor. The modification mechanism was shown in Fig. 1 Fig. 1 Modification of polystyrene resin with protonated amino groups Analyses. The berberine concentration in solution was determined by Agilent 1260 HPLC at a wavelength of 345 nm, employing a 0.05M KH 2 PO 4 / acetonitrile 30/70 (v/v) solution as mobile phase. FTIR spectra were determined using a Thermo Scientific Nicolet 6700 instrument. The mesopore size and the specific surface area of resins were determined using a TRISTAR II3020 apparatus.

Kinetic experiments. In typical adsorption experiments, 1.0 g resin HX or resin H103 was added in 200 mL berberine solution (without pH adjustment, the pH value was 3.92) of different concentrations from 500 to 900 mg/L in a 250 mL flask and kept in an incubator shaker at 150 rpm and 25℃. 1mL of solution were taken and filtered with 0.22 µm filter membrane at the time of 24 h.

Adsorption Mechanism experiments. To evaluate the influence of the temperature on berberine removal from a solution, 1.0 g of resin H103 was added in five conical flasks containing 200 mL of berberine solutions (without pH adjustment, the pH value was 3.92) at concentrations of 500 to 900 mg/L, and kept in an incubator shaker at 150 rpm and temperatures of 10, 30, 50 and 70℃.

Results and Discussion

Properties of new protonated polystyrene resin HX. As can be seen by FTIR in Fig. 2, the presence of a N-H bond is revealed by the band at 1110 cm -1 . The existence of N-H proved that the modification of resin with hydrogen-bonding donor has been successful, and the rest functional groups of resin H109 did not change. Berberine adsorption. The amount of berberine adsorbed on resin HX was significantly higher than that of resin H109 (Fig. 3). After 1 h, the berberine amount adsorbed on resin HX was 21.4 mg/g higher than that on resin H109. This value then decreased with time. The adsorption reached equilibrium after 8 h, and resin HX adsorbed 7.9 mg/g more berberine than resin H109. In the first hour, the adsorption of berberine on resin HX was much faster than on resin H109, mainly due to hydrogen bonds forming between berberine and resin HX, which is an interaction much stronger than Van der Waals force promoting berberine adsorption on the pure polystyrene resin as H109. Hydrogen-bond interactions between N-H and the two -OCH 3 functional groups on benzene could be the main cause of the increase amount of berberine adsorption and the rate of adsorption. For resin HX,the correlation coefficient of pseudo secondary dynamics equation (d) was much higher than for the other equations. This indicates that another effect than just physical adsorption is governing the adsorption and could certainly be H-bonding interactions. Kinetic mechanism of adsorption is therefore more complicated on the new HX resin in comparison with on the commercial polystyrene resin H103. Hydrogen-bond affected the kinetic mechanism of resins.

Adsorption mechanisms. In order to better characterize adsorption mechanism of resin H103, thermodynamic analysis was carried out using the Freundlich equation and the Van't Hoff equation.

Freundlich adsorption isotherm, originally proposed as an empirical equation, is used to describe the data for heterogeneous adsorbents.

Q m = K a C e 1/n
Q m is the equilibrium adsorption capacity (mg/g), K a and n are Freundlich characteristic parameters, C e is the equilibrium concentration (mg/L).

The Freundlich equation is consistent with the thermodynamics of heterogeneous adsorption. According to the fitting results of adsorption process performed at different temperatures (Fig. 3), the adsorption isotherms of berberine on resin H103 could be well described by the Freundlich equation. Freundlich adsorption intensity parameter n increased with the increase of temperature, which means that the adsorbate was more active at higher temperature. The adsorption intensity parameter n > 2 indicated that H103 was a good adsorbent of berberine [7][8][9].

Freundlich adsorption isotherm, originally proposed as an empirical equation, is used to describe the data for heterogeneous adsorbents. The Freundlich equation is consistent with the thermodynamics of heterogeneous adsorption. According to the fitting results of adsorption process performed at different temperatures (Fig. 4), the adsorption isotherms of berberine on resin H103 could be well described by the Freundlich equation. Freundlich adsorption intensity parameter n increased with the increase of temperature, which means that the adsorbate was more active at higher temperature. To characterize the thermodynamic properties of berberine adsorption on H103, isosteric adsorption enthalpy was calculated with the Van't Hoff equation:

[ln (C e )] q = -ln(K 0 )+ ∆H/RT Ce is the equilibrium concentration and K 0 is the constant.

Tab. The calculated adsorption enthalpy for berberine adsorption on polystyrene resin H103 was less than 40 kJ/mol (Tab.2), and this confirms that the adsorption was mainly physical adsorption. It indicated that there was no hydrogen bonding, chelating or other chemical adsorption.

FTIR spectra were taken to elucidate the berberine adsorption mechanisms on resin HX. According to Fig. 2, the FTIR spectrum of berberine molecule showed two infrared absorption peaks at 1290~1240 cm -1 and 1050~1000 cm -1 corresponding to the two C-O bonds and the 1150~1085 cm -1 band corresponding to the C-O-C bond. After the adsorption of berberine on resin HX, the two C-O bands moved to higher wavelength for 16 and 10 cm -1 , respectively, which proved the existence of H-bond between berberine molecule and the protonated nitrogen of the resin HX and evidenced that the H-bond acceptor in berberine molecule was the two CH 3 -O bonds (Fig. 5). Furthermore, as the N-H bond of the resin at 1110 cm -1 disappeared after the berberine adsorption, this indicates that the proton of the resin HX is moving close to the C-O bond of the berberine molecule. The electron cloud of N-H moved much closer to oxygen atom, and led to the disappearance of N-H infrared absorption peak. In the hydrogen-bonding mechanism of berberine adsorption on modified-polystyrene resin, the berberine molecule was the electron acceptor and resin HX was the electron donor. Fig. 5 The adsorption mechanism of berberine molecule on resin HX

Conclusions

The modification of a resin with a hydrogen-bonding donor characteristic was successfully realized with a commercial polystyrene resin. The amount of berberine adsorption on this resin was significantly increased. The kinetic mechanism analysis indicated that hydrogen bonding was the cause of the increase of adsorption. The thermodynamic analysis and FTIR spectrum analysis proved the existence of hydrogen bonding between the berberine molecules and the resin. The H-bonding interaction was identified with a proton transfer between the N-H bond of the resin to the O-CH 3 bonds of the berberine molecule.

Fig. 2 FTIR

 2 Fig.2 FTIR of berberine and resins

Fig. 3

 3 Fig.3 Amount of berberine adsorbed on resin HX and resin H109 at an initial berberine concentration of 1000 mg/L Kinetics of adsorption. In order to characterize the mechanisms of the adsorption, the adsorption processes were fitted with: (a) Intraparticle diffusion model, (b) Film diffusion model, (c) Modified pseudo-first order model and (d) Pseudo-second order model. Tab.1. Correlation coefficient of the different equations on kinetic analysis of resins Correlations Equation (a) Equation (b) Equation (c) Equation (d) H103 0.996 0.994 0.926 0.792 HX 0.788 0.773 0.948 0.997 As shown in Tab.1, for polystyrene resin H103, the two equations (a) and (b) well fitted the adsorption isotherm of berberine as correlation coefficients are over 0.99, and both of fitting curves are not zero crossing. It means that intragranular diffusion and liquid film diffusion are the control steps of adsorption at the same time. This indicated that the main reaction of adsorption with resin H103 was physical adsorption.For resin HX,the correlation coefficient of pseudo secondary dynamics equation (d) was much higher than for the other equations. This indicates that another effect than just physical adsorption is governing the adsorption and could certainly be H-bonding interactions. Kinetic mechanism of adsorption is therefore more complicated on the new HX resin in comparison with on the commercial polystyrene resin H103. Hydrogen-bond affected the kinetic mechanism of resins.Adsorption mechanisms. In order to better characterize adsorption mechanism of resin H103, thermodynamic analysis was carried out using the Freundlich equation and the Van't Hoff equation.Freundlich adsorption isotherm, originally proposed as an empirical equation, is used to describe the data for heterogeneous adsorbents. Q m = K a C e

Fig. 4

 4 Fig.4 Freundlich equation to simulate the berberine adsorption process on H103 resin

  2 Adsorption enthalpy under different amount of adsorption q e (mg/g) slope △H (KJ/mol)

				R 2
	40	6536	20.76	0.995
	50	6983	23.56	0.997
	60	7499	26.80	0.998
	70	8109	30.62	0.999
	80	8855	35.31	0.999

Acknowledgement

This work was financial supported by the Sino-France international Scientific and Technological Cooperation Project for petrochemical & pharmaceutical industrial conservation and reduction technical cooperative research (2010DFB90590). The National Key Scientific and Technological Project for Water Pollution Control and Management (2012ZX07202-002,2012ZX07202-005). The authors also acknowledge the French National Agency for Research (ANR) for supporting this study through the convention ANR ECOTECH 2010 project PETZECO (1081C0230/ ANR-10-ECOT-011-03).