Objective Context Example & Model State of the art Approach The multi-level framework Scotch multi-criteria algorithm Results

criteria partitioning tools.

Context

In the field of numerical simulation, one challenge is to reduce the restitution time.

We consider simulations:

Running on distributed memory architectures Using 2D or 3D meshes

Coupling several physics (multi-physics simulations)

Running these simulations efficiently needs to:

1 Balance the workloads of each processor for each phase of the computation (2 computing phases: f and g, coupling the variables X and Y )

for t ∈ [0, t end -1] do X(t + 1) ← f (t + 1, X(t), Y (t)) # First computing phase Y (t + 1) ← g(t + 1, X(t + 1), Y (t))
# Second computing phase end for

Objectives:

1 Balance the workloads of each processor for each phase 2 Minimize communications between processors Our approach: Multi-level multi-criteria algorithm

2ExampleAlgorithm 1

 1 Overlap or minimize communications 3 Take care of memory accesses 4 Exploit full processor characteristics We focus on the first 2 items. | 28/02/2017 | PAGE 2/14 Example of the temporal loop of a multi-physics simulation

  02/2017 | PAGE 6/14

A 3 -

 3 phases algorithm: 1 Coarsening → (Ongoing) Selection of a matching policy improving the quality of the returned solutions 2 Initial partitioning of the coarsened hypergraph → New algorithm focusing on balance constraints algorithm: local optimization of the edgecut with hill-climbing Moves that unbalance the partition more than the tolerance are forbidden ⇒ Finding a balanced partition of the coarsest graph guarantees that the returned partition will respect the balance constraints. Number of parts: 2 Each algorithm is executed 100 times on each instance with random seeds. | 28/02/2017 | PAGE 8/14 Results 1 -How to compare the results f algo (x) = number of valid solutions of algo of cut ≤ x. We want to minimize the cut of each returned solution: left is better mushroom_1 | 28/02/2017 | PAGE 9/14 Results 2 -Comparison on the quality of the returned solutions different parameters on the multi-level An example: settings of the coarsening parameters Heavy-Edge-Matching Particular order of the vertices: sorted by increasing degrees Do not allow to match 2 vertices if the resulting vertex is overweighted | 28/02/2017 | PAGE 11/14 Comparison of Crack on the previous instances will support multi-criteria graph partitioning This new version implements a new initial partitioning and sets the refinement phase to guaranty that the returned solution will respect the balance constraints Scotch results are promising Perspectives: Enforce the algorithm robustness by the study of variations of the algorithms Validation on more instances Validation on a simulation code Set up of a parallel version of the algorithms | 28/02/2017 | PAGE 13/14 Thank you Commissariat à l'énergie atomique et aux énergies alternatives Centre DAM Île-de-France | F-91297 Arpajon T. +33 (0)1 69 26 40 00 Établissement public à caractère industriel et commercial | RCS Paris B

Model

Mesh

Dual Graph G = (V, E) →

Problem : Graph partitioning

We search for a partition P of V respecting:

1 some constraints: load-balancing (imbalance of P is for all criteria smaller than a given tolerance)

2 an objective: minimize the edgecut (sum of the weights of the edges cut by the partition).

NP-Hard Problem.


Elyès Jouini is also affiliated with CREST, CEPR and Ecole Polytechnique de Tunisie. He wrote this presentation when he was visiting the nance department of the Stern School of Business, New York University.

The theory of asset pricing takes its roots in the Arrow-Debreu model (see, for instance, Debreu 1959, Chap. 7), the Black and Scholes (1973) formula, and the [START_REF] Cox | The valuation of options for alternative stochastic processes[END_REF] linear pricing model. This theory and its link to arbitrage has been formalized in a general framework by [START_REF] Harrison | Martingales and arbitrage in multiperiod securities markets[END_REF], Harrison andPliska (1981, 1983), and [START_REF] Duffie | [END_REF]. In these models, security markets are assumed to be frictionless: securities can be sold short in unlimited amounts, the borrowing and lending rates are equal, and there is no transaction cost. The main result is that the price process of traded securities is arbitrage free if and only if there exists some equivalent probability measure that transforms it into a martingale, when normalized by the numeraire. Contingent claims can then be priced by taking the expected value of their (normalized) payoff with respect to any equivalent martingale measure. If this value is unique, the claim is said to be priced by arbitrage and it can be perfectly hedged (i.e. duplicated) by dynamic trading. When the markets are dynamically complete, there is only one such a and any contingent claim is priced by arbitrage. The of each state of the world for this probability measure can be interpreted as the state price of the economy (the prices of $1 tomorrow in that state of the world) as well as the marginal utilities (for consumption in that state of the world) of rational agents maximizing their expected utility.

When there are frictions, including dynamic market incompleteness, the characterization of the no-arbitrage condition is nomore equivalent to the existence of a unique equivalent martingale measure. More precisely, for each equivalent martingale-measure condition equivalent supermartingale-measure condition, equivalent submartingale-measure condition, absolutely continuous martingale-measure condition kind of imperfection, the is replaced by a weaker one : , etc. Besides, we generally have, more than one measure satisfying these conditions. Furthermore, when there are frictions, even if a contingent claim can be duplicated by dynamic trading, it is not necessarily possible to price it by arbitrage. However arbitrage bounds can be computed, for arbitrary contingent claims, taking the expected value of their (normalized) payoff with respect to all the measures that characterize the absence of arbitrage opportunities. These bounds are the minimum amount it costs to hedge the claim and the maximum amount that can be borrowed against it using dynamic strategies. These are the tightest bounds that can be inferred on the price of a contingent claim without knowing the agent s preferences. The determination of these bounds in a dynamic setting leads to a maximization (and/or minimization) program, and, in a dynamic setting, is often transformed into a stochastic optimal control problem. The main assumption in these models is, in fact, a necessary condition for the existence of an equilibrium: the no-arbitrage condition. These preference-free theories give results of great generality without specifying the equilibrium in its full details.

Another important class of valuation theories makes assumptions on preferences and derives more speci c pricing restrictions than the preference-free theory does, even in the presence of imperfections. The price of a given contingent claim, for these theories, is just the expected value of its (normalized) terminal payoff with respect to a probability measure, whose density is proportional to the marginal utility (for consumption) of the considered agent. From a mathematical point of view, starting with a given utility function, the problem is to write the rst order conditions of the agent s utility maximization program, taking into account the potential imperfections in the description of the budget constraints and/or of the strategies in order to characterize the marginal utility for consumption at the nal date. In a multi-period setting, this maximization problem is a stochastic optimal control problem. The main advantage of this approach is that it leads to a unique price for a given contingent claim. The main drawback is that this price depends on the choice of the utility function and on the agent endowment. If the utility function belongs to a given connected class of utility functions, we will obtain an interval of possible prices for that claim. More speci cally, if the considered class is the set of all von Neumann-Morgenstern (VNM) increasing and concave utility functions, the set of possible prices is exactly the set obtained with the arbitrage approach as shown by [START_REF] Jouini | Viability and equilibrium in securities markets with frictions[END_REF]. The unique way to obtain tighter bounds with the utilitymaximization approach seems then to consider speci c functions, or speci c sets of functions, smaller than the set of all the VNM ones.

In fact, there is an interesting link between the two approaches. Since the arbitrage upper bound for a given contingent claim is equal to the minimum amount it costs to hedge it, taking the market frictions into account, the agent s problem (maximization of the utility provided by the terminal payoff among the strategies satisfying a given dynamic budget constraint) can be transformed

P P R R R R ∞ ∞ →∞
F F F ∈ F ∈ F F F ∈ ∈ ∩ { } ( ) ( ) ( ) = 0 ) = (0) = 0 ( ) ( ) = ( ) ( ) = lim ( ) = 0 ( )

bounded losses

We say that the semi-martingale satis es the condition of noarbitrage, (NA), if into a static problem where we maximize the utility among the set of contingent claims satisfying a budget constraint where the classical price functional is replaced by the functional. If we have, for instance, an explicit formula for the arbitrage upper bound, it suffices then to solve a static maximization problem instead of the initial stochastic dynamic control problem. The characterization of the no-arbitrage assumption is therefore crucial in order to solve the contingent claim pricing problem as well as to solve the individual utility maximization problem of each agent in the economy. The last step, if we want to explore all the implications of the Arrow-Debreu model in this nancial setting is then to write the equilibrium conditions in order to ensure that all the individual solutions are compatible .

Let

be a probability space and be a ltration which models our information structure. This ltration is supposed to satisfy the usual conditions, i.e. the ltration is right continuous and contains all negligible sets (if and then We also suppose that the sigmaalgebra , and we consider a real valued semimartingale which models the price process for the marketed claims. In the next, we will denote by real line and by the set of nonnegative ones. Let us de ne, as in Delbaen and Schachermayer (1994), an admissible strategy as follows:

This admissibility condition can be interpreted as a condition for strategies with a zero initial investment.

We consider, as in [START_REF] Stricker | [END_REF], the convex cone in the space of equivalent classes of measurable functions, de ned up to equality almost everywhere, given by admissible and exists a.s.

The set is then the set of all terminal payoffs obtained through some admissible strategy.

Since

represents the set of all admissible terminal payoffs the no-arbitrage condition amounts to say that it is impossible to obtain a non-negative, non-zero payoffs with a zero initial investment.

Assume that it is possible to separate and in the sense that there exists a non-zero linear functional and a real number such that and

R ∞ ∞ ∞ ∞ ∞ ∞ ∞ Q Q P E 1 E 1 Q Q s t s B Q t s B p + 0 0 0 0 0 + + 0 0 + + 0 0 + 0 + 0 0 + 1 1 ∈ ∈ ∈ ∩ ∩ { } ∩ ∩ { } ∩ { } ∩ { } F ( ) [ ] 0 ( ) [( ) ] = 0 ( ) = 0 ( ) = 0 ( ) = 0 = 0 ( )
Theorem 1 [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF])

f L > c f f f K . s < t, B F , , S S K S S , S. S S Q S K L L L L L . K L L L K L L , K L . L , L L S. L S S S
Let be a locally bounded real valued semi-martingale. There is an equivalent local martingale measure for if and only if (where is the space of bounded measurable functions and where the closure is taken with respect to the norm-topology of )

and assume that the separating functional admits a representation as an expectation operator with respect to a probability measure , then under the (NA) condition, is equivalent to and for each in In particular, for each we have , therefore, and is a martingale measure for Unfortunately, the (NA) condition is seldom sufficient to apply a separation theorem. In the case where is locally bounded we have the following: This last condition is called No Free-Lunch with Vanishing Risk (NFLVR) and deals with sequences of strategies such that the negative parts of their terminal payoff tends to zero uniformly instead of strategies with non-negative terminal payoff as in the (NA) condition. Remark that the main difference between these two conditions lies in the fact that we have to consider a closure. Indeed, the condition is weaker than which is equivalent to the (NA) condition If, in the previous condition, we replace the norm-topology closure by the topology closure (where is the space of all integrable measurable functions) we obtain a version of the No Free-Lunch (NFL) condition introduced by [START_REF] Kreps | Arbitrage and equilibrium in economies with in nitelymany commodities[END_REF], and the existence of an equivalent local martingale measure is obtained for a bounded càdlàg, and adapted process Other intermediary concepts, as the No Free-Lunch with Bounded Risk (NFLBR) condition (where the closure is de ned as the set of weak*-limits and where the negative parts of the terminal payoffs tend to zero in probability and remains uniformly bounded), have been introduced in the literature and permit to obtain results similar to the previous theorem in different contexts: nite time set and (NA) condition with Dalang et al. (1989) (see [START_REF] Schachermayer | A Hilbert-space proof of the fundamental theorem of asset-pricing in nite discrete time[END_REF], Kabanov and Kramkov (1994a) and [START_REF] Rogers | Equivalent martingale measures and no-arbitrage[END_REF] for elementary proofs), innite but discrete time set with [START_REF] Schachermayer | Martingale measures for discrete-time processes with in nite horizon[END_REF], continuous and bounded processes in continuous time with [START_REF] Delbaen | Representing martingale measures when asset prices are continuous and bounded[END_REF]. [START_REF] Harrison | Martingales and arbitrage in multiperiod securities markets[END_REF] and [START_REF] Harrison | Martingales and stochastic integrals in the theory of continuous trading[END_REF] used the concept of simple strategies. [START_REF] Kreps | Arbitrage and equilibrium in economies with in nitelymany commodities[END_REF] used a concept of no free-lunch involving the convergence of nets or generalised sequences. [START_REF] Duffie | [END_REF] and [START_REF] Stricker | [END_REF] used convergence. [START_REF] Lakner | Martingale memasures for a class of right-continuous processes[END_REF] used convergence in Orlicz spaces. Furthermore, it appears that the class of semi-martingales is the most general one compatible with this kind of results. Indeed, from the work of [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF] and [START_REF] Ansel | Unicité et existence de la loi minimale[END_REF] we know that No Free-Lunch conditions imply, in some sense, that is a semi-martingale. Conversely, the existence of an equivalent-martingale measure for implies, by Girsanov s theorem, that is a semi-martingale.

As in [START_REF] Harrison | Martingales and arbitrage in multiperiod securities markets[END_REF] and [START_REF] Kreps | Arbitrage and equilibrium in economies with in nitelymany commodities[END_REF] and for a given contingent claim, we de ne the arbitrage pricing interval as the set of all the prices that are

t t easier 0 0 0 + 0 0 + + ∞ ∞ →∞ ∞ ∞ ∞ ∞ ∞ { | } ( ) 0 ( ) = lim ( ) ( ) ( ) ( ) 0 ( ) ( ) (( )) ( ) ( ) K .

H S H H H S H S H S H S H, H H, H H S H S

S S H S V H S H H S V H S K L K L L
compatible with the No Free-Lunch condition, i.e. introducing this contingent claim at one of these prices does not create free-lunches. In all the mentioned papers, it appears that this set is equal to the set of expected values of the considered claim terminal payoff with respect to all the probability-measures which characterize the absence of free-lunch. When there is no imperfection, the cash-ow space can be identi ed with In case of short-sale constraints, it can be identi ed with admissible, and exists a.s.

and when there are short-selling costs, it can be identi ed with admissible, and exists a.s.

where (resp. ) models the long (resp. short) position returns. When there are transaction costs, can be identi ed with admissible, and exists a.s.

where represents the total variation of and the magnitude of the transaction costs. Jouini and Kallal (1995a and b) characterized rst the absence of arbitrage opportunities in these different situations. Other contributions on related subjects are due to Kabanov and Kramkov (1994b), [START_REF] Shirakawa | Pricing of options under the proportional transaction costs[END_REF], [START_REF] Kusuoka | Limit theorem on option replication with transaction costs[END_REF], [START_REF] Cvitanic | Hedging and portfolio optimization under transaction costs : a martingale approach[END_REF], Cvitanic, Pham and Touzi (1999), [START_REF] Kabanov | Hedging and liquidation under transaction costs in currency markets[END_REF]. The differences between all these references are in the choice of the topology (or no topology) in order to de ne the concept of free-lunch, the choice of a space of admissible strategies (discrete strategies, simple strategies,...) and nally the choice of possible imperfections (or no imperfection). This choice is summarized by the choice of a convex cone contained in instead of itself in order to model the opportunity set. In this context, it is to nd a separating hyperplane between that set (or its closure with respect to some topology) and . Jouini and Kallal (1999) extended all the arbitrage, viability and equilibrium classical results to that setting mainly by assuming that the opportunity set is a convex cone (or even a convex set) and the pricing rule is sublinear. In this issue, [START_REF] Yu | The Harrison-Pliska arbitrage pricing theorem under transaction costs[END_REF] propose a generalization of Jouini and Kallal s (1995a) result to the important case of a multi-asset market model where the transaction costs are de ned for each kind of transaction between any pair of assets. They use the geometric formalism developed previously by [START_REF] Kabanov | Hedging and liquidation under transaction costs in currency markets[END_REF] and they characterize the absence of arbitrage opportunities in terms of martingale-like measures. Their result is established in a discrete time and nite set of states of the world framework and they only deal with arbitrages and not with freelunches.

In order to take a large set of possible frictions into account Carassus and Jouini (1997,1998,2000) in discrete time or in a deterministic setting, Jouini and Napp (2000) and Jouini, [START_REF] Jouini | A remark on Kreps-Yan theorem[END_REF]

) in continuous T T ∞ ∞ ( ) + ( ) ( ) ( ) [0 ] H H S c H , c H, H S , F, P
, T , T 3 The utility maximization problem separating kill the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model where his income is generated by returns on assets and these returns or instantaneous growth rates are stochastic time propose to deal directly with the space of possible cash-ows instead of the space of terminal payoffs and they provide a characterization of the No Free-Lunch assumption in terms of the existence of a functional. Napp (2001) develops an arbitrage pricing theory and a super-replication concept in this cash-ow space.

However, all these results are obtained under a convexity condition on the space of attainable payoffs. This last assumption is not satis ed in economies with xed costs, i.e. with transaction costs which are not proportional to the size of the transactions. In this framework, the terminal payoff of a strategy is where is a bounded non-linear function of the strategy instead of as in the classical case. Therefore, it is easy to understand that large scale transactions will the transaction cost effect and that the characterization of the no-arbitrage condition should be an asymptotic version of the classical one. Jouini, Kallal and [START_REF] Napp | [END_REF] prove that this characterization is in terms of absolutely continuous martingale measures and show that the existence of such a measure is necessary but not sufficient and that we need the existence of a family of such measures each one associated with a given date and a given event at that date in order to characterize the absence of free-lunches.

Before Merton s (1969) paper, most models of portfolio selection only considered one period. Furthermore, the investment decision by households was viewed in two parts : (a) the consumption-saving choice where the individual decides how much income and wealth to allocate into current consumption and how much to save for future consumption; and (b) the portfolio-selection choice where the investor decides how to allocate savings among the available investment opportunities. [START_REF] Merton | Lifetime portfolio selection under uncertainty : the continuous-time case[END_REF] examined . The original analysis of Merton s model is based on the Hamilton-Jacobi-Bellman equation and requires an underlying Markov state process. After the papers of [START_REF] Harrison | Martingales and arbitrage in multiperiod securities markets[END_REF] and Harrison andPliska (1981, 1983), and their characterization of the no-arbitrage assumption in terms of the existence of martingale-measures, Pliska (1986), Cox andHuang (1989, 1991) and Karatzas, Lehoczky and Shreve (1987) used this methodology in order to analyze this consumption-investment problem. This new approach is based on duality arguments and permits to transform the initial dynamic problem into a static one and to solve it without assuming any Markov condition.

Let us now introduce the main results related to this problem. Let be a xed probability space and denote the interval on which we are going to treat our problem : corresponds to the terminal date for all economic activity under consideration. All processes that we shall encounter in this section are de ned on . The real-valued interest rate process , the dimensional process , the dimensional dividend yield process as well as the volatility -matrix-valued process are supposed to be progressively measurable with respect to and bounded uniformly in in For all in the volatility matrix has full rank and the norm of is uniformly bounded.

Assumption A1 Assumption A2 ∈ { ∈ } ∞ { ∈ } { ∈ } { ∈ } { ∈ } { ∈ } ∞ { ∈ } ∈ t t
We consider a market consisting in one bond and assets. More precisely, the primitive market model is the same as in Karatzas , except that we consider here dividends paying assets. We adopt a model for the market consisting of one bond with price at time denoted by satisfying the differential equations ,

and stocks with prices at time denoted by the dimensional vector satisfying [START_REF] Ansel | Lois de martingale[END_REF] Here, is a -dimensional Brownian motion on a probability space and denotes the -augmentation of the natural ltration generated by . We assume that the sample paths of specify completely all the distinguishable events, which mathematically entail . Since standard Brownian motions start from zero with probability one, is trivial. We will denote by the set of -progressively measurable processes taking values in such that a.s.

Under this assumption , Equation admits a unique real-valued, adapted, continuous solution , satisfying .

Therefore, a -dimensional process can be de ned by : a.s.

With the above assumptions, is -progressively measurable and uniformly bounded. We shall also introduce the discounted price process de ned by for all in . Using Itô s Lemma, we easily get that is the unique solution of the following stochastic differential equation: We know that in such a model, there exists a unique equivalent probability measure de ned on that makes the full process a martingale for . It is given by We then have , where is the Brownian motion for de ned by for all in (see Girsanov s Theorem) We shall denote in the following, the martingale process by . In the context of the above market-model, consider an agent who starts out with an initial capital and can decide of the amounts that he invests at time in the different assets, and of the rate at which he withdraws funds for consumption. Assuming that at each time , sales and dividends must nance purchases and consumption, the corresponding wealth process, denoted by , satis es the following stochastic differential equation

E T T 4 5 4 5 ∈ ∈ T T T T 2 0 0 2 1 T T T T A T d t d t t t s s t s n n { ∈ } E {E ∈ } E E ∈ | ∈ ∈ ∀ ∈ 0
( Self-nancing condition) (3)
which can be rewritten

( Self-nancing condition) (4)
The set of investment-consumption strategies satisfying the previous self-nancing condition and the following no-bankruptcy condition is called the admissible strategies set and denoted by :

(No-bankruptcy condition)

This last condition amounts to saying that at each time , the investor must be able to cover his debts -see e.g. Karatzas-Lehoczky- [START_REF] Karatzas | Optimal portfolio and consumption decisions for a small investor on a nite horizon[END_REF] or [START_REF] Duffie | Dynamic asset pricing theory[END_REF] where the same assumption is made. ( ) :

T A B B T T T B ∈ ∈ F R R C m F , x R n F t, R R Assumption Proposition 2 0 ; 0 0 0; 0 + 1 + 0 1 + ( ) ( ) ; ( ) 0 0 ∈ → U → ∞ ∞ → U ∞ ∞ t s x ,c t t s u s ,c T T T t x j x j j , c u c V ,c x x ,c T X,c u t s t V T s T exp + exp ; ( ) (0) 0 0 ( ) : ( ) = ( ) + ( ) : (0 ) inf ( ) = 0 sup ( ) = + : ( ) ( ) ( ) ( ) (0 ) (0 ) ( ) ( ) ( ) ( ) sup = sup ( ) ( ) 0
( ) = 1 exp 0 = exp
As usual, a function is said to be of class if the -th derivative of and the -th derivative of exist and are continuous.

Y r ds X r du c ds t

P , c X > > X, c c X F U , U c, X u t, c dt V X , u V V R R C , V x V x u R R C R u t, u t, I t, u t, , , I V x U c, X c U c, X U c, X c , X x > c t I t, r ds M t T a.s. P X I r ds M
The function is on strictly increasing strictly concave and satis es Inada conditions ( . The function is on , strictly increasing, strictly concave and satis es Inada conditions.

A pair in is optimal for an agent with an initial endowment if and only if there exists a constant such that

Under the self-nancing condition, the process consisting in the current discounted wealth plus the total discounted consumption is a -supermartingale (Fatou s Lemma). It is then easy to see that the market excludes any arbitrage opportunity which turns out to be characterized in our context by the existence of a pair in such that ).

Let denote the set of pairs

where is an adapted nonnegative consumption rate process and is a nonnegative -measurable random variable describing the terminal wealth. An agent is represented by a utility function given by where and satisfy the following assumption. ,

Under Assumption , we shall denote by the derivative of and by the inverse function of , which is a strictly decreasing continuous function on in . We shall also denote by the inverse function of . The considered agent has an initial endowment and tries to maximize his utility on both his consumption over the time-interval and his terminal wealth. The optimal demand of the agent in the consumption commodity as well as his optimal portfolio are determined by the optimization problem .

Adapting the proofs of Duffie (1994) and [START_REF] Karatzas | Optimization problems in the theory of continuous trading[END_REF], we get that (5) (6)

E 0 0 0 V V V V T t s t T s T x x exp ( ) + exp = = r ds M c t dt r ds M X x. x x . (7)
This last proposition permits to solve explicitly the agent s optimization program. [START_REF] Huang | Optimal consumption and portfolio policies with an in nite horizon : existence and convergence[END_REF] extended this methodology to the in nite horizon framework. Karatzas, Lehoczky, Sethi and Shreve (1986) provided explicit computations in that framework assuming constant coefficients in the price evolution equations. He and Pearson (1991 a and b) and Karatzas, Lehoczky, Shreve and Xu (1991) extended the methodology to incomplete markets and proved that the optimal investment/consumption plan is given, as in the classical case, by the inverse of the marginal utility evaluated at the random variable which is optimal for a well-de ned dual problem. Cvitanic and Karatzas (1992) used the same approach in order to solve the problem when there is convex constraints on the strategies (short-sales constraints, borrowing constraints,...). [START_REF] Fleming | An optimal investment consumption model with borrowing[END_REF] solved the problem assuming different borrowing and lending rates. [START_REF] Cuoco | Optimal consumption and equilibrium prices with portfolio constraints and stochastic income[END_REF] and El Karoui and Jeanblanc-Picqué (1997) considered random endowment streams. [START_REF] Cvitanic | Hedging options for a large investor and forward-backward SDEs[END_REF] and Cuoco and Cvitanic (1998) generalized these results in the context of a large investor . In that context, the strategy of the investor has a direct nonlinear impact on the price dynamics. The main technique in all these references consists in embedding the original problem into a family of perfect (linear) ctitious markets, where security prices dynamics are modi ed and agents receive an additional stochastic endowment re ecting the nonlinearity in the market price of risk. The ctitious markets are designed in such a way that the optimal policy in one of them coincides with that in the actual, nonlinear market. Using the partial differential equations (PDE) approach, Dumas and Luciano (1989) rst formulated the problem in the presence of transaction costs. The main contributions in this context are Davis and Norman (1990), Fleming, Grossman, Vila and Zariphopoulou (1990) and [START_REF] Shreve | Optimal investment and consumption with transaction costs[END_REF].

When there are imperfections, the utility maximization approach can be used in order to provide pricing formulas for new contingent claims. There are mainly two methods. The rst one, initiated by [START_REF] Hodges | Optimal replication of contingent claims under transaction costs[END_REF] in the transaction costs setting, consists in using the marginal utility of the considered agent at his optimal consumption-investment plan as a state-price density. The second method, initiated by [START_REF] Davis | Option pricing in incomplete markets[END_REF] consists in comparing the optimal utility levels with a deterministic initial endowment and with a stochastic endowment equal to the payoff of the considered claim. The fair price is then de ned by the equation Papers along these lines include Constantinides (1986), [START_REF] Panas | Option pricing with transaction costs[END_REF], [START_REF] Davis | European option pricing with transaction costs[END_REF], [START_REF] Davis | The writting price of a European contingent claim under proportional transaction costs[END_REF], [START_REF] Davis | American options and transaction fees[END_REF], [START_REF] Cvitanic | Hedging and portfolio optimization under transaction costs : a martingale approach[END_REF], Constantinides and Zariphopoulou (1997) and Barles and Soner (1998).

. d R R

The equilibrium

In this issue, [START_REF] Cvitanic | On optima terminal wealth under transaction costs[END_REF] show that the martingale/duality approach adopted in the frictionless model works also in the transaction costs framework and prove that the optimal terminal wealth is given as the inverse of marginal utility evaluated at the random variable which is optimal for an appropriately de ned dual problem. They prove the existence of a solution for this dual problem and doing so they resolve a question left open by [START_REF] Cvitanic | Hedging and portfolio optimization under transaction costs : a martingale approach[END_REF]. A similar problem is studied by [START_REF] Deelstra | Dual formulation of the utility maximization problem under transaction costs[END_REF] where the utility functions are de ned on instead of Framstad, Øksendal and Sulem (2001) considers also the transaction costs framework but in a jump diffusion market. Using a viscosity solution approach, they show that the solution of the problem in that context has the same form as in the pure diffusion case : there is a no-transaction cone such that it is optimal to make no transactions as long as the wealth position remains in that cone and to trade on the boundary. Bellamy (2001) solves the same problem but assuming market incompleteness instead of the presence of transaction costs and using a Hamilton-Jacobi-Bellman (HJB) approach.

Using ltering techniques, Lakner (1995) considers utility maximization problems where the agent must estimate the mean rate of return of the assets. In this issue, [START_REF] Dokuchaev | Optimal investment strategies with bounded risks, general utilities and goal achieving[END_REF] considers the case where the stock appreciation rates are not observable and where the strategies depend only on the known distribution of these rates and on the current prices. Furthermore, they use general utility/loss functions (including mean-variance criteria and goal achieving problems) and they consider some lower and upper constraints on the terminal wealth. The problem is solved by means of backward stochastic differential equations as well as a dual formulation.

Finally, we want to mention another family of optimization problems related to the contingent claims pricing : the hedging problems. These problems are not represented in this special issue but are studied by many papers in the recent literature. The main problem in all these paper is to compute the hedging price of a given contingent claim with respect to a given hedging criterion. If we assume that the agents want to minimize the downside risk, then the hedging price is equal to the super-replication price and its computation leads to solve a stochastic-control-based problem as in the pionneering paper of El Karoui and Quenez (1991,1995). If we assume that the agents want to minimize the quadratic risk then we have to solve the mean-variance hedging problem and we refer to [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF] or Schweizer (1993) for a survey about related results. Models of competitive equilibrium go back to Walras (1874). The rst complete proof for the existence of an equilibrium in an economy with nitely many commodities was given by Arrow and Debreu (1954). In the chapter 7 of Debreu (1959), the author explains how this model permits to take into account dynamic markets with uncertainty. Bewley (1972) studied the competitive equi-

E 0 ∞ B E R → U = 1 : ( ) = ( ) + ( ) th j j j T j t j j j L n j , , n j x U U c, X u t, c dt V X u V
librium in an in nite-dimensional commodity space, namely and Mas-Colell (1986) generalized Bewley s results to Hausdorff locally convex, topological vector spaces under a uniform properness condition on the agent s preferences. Araujo and Monteiro (1989 a and b) and Duffie and Zame (1989) proved independently the existence of an equilibrium without Mas-Colell s uniform properness condition. Dana, Le Van and Magnien (1997) extended Mas Colell s result to topological locally solid Riesz spaces under a local non-satiation condition weaker than the uniform properness one. [START_REF] Aase | Dynamic equilibrium and the structure premium in a reinsurance market[END_REF] and Bernis (2000) applied these results to the reinsurance markets.

All the previous models does not take explicitely into account dynamic security trading. Models where the agents achieve equilibrium allocations by trading in securities like the capital asset pricing model (CAPM) or the consumption based capital asset pricing model (CCAPM) can be found in the literature going back to [START_REF] Merton | Optimum consumption and portfolio rules in a continuous-time model[END_REF], [START_REF] Cox | An intertemporal general equilibrium model of asset prices[END_REF], [START_REF] Huang | Implementing Arrow-Debreu equilibria by continuous of few long-lived securities[END_REF], [START_REF] Huang | An intertemporal general equilibrium asset pricing model : the case of diffusion information[END_REF] and Karatzas, Lehoczky and [START_REF] Karatzas | Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model[END_REF].

The link between these two approaches is made by [START_REF] Huang | Implementing Arrow-Debreu equilibria by continuous of few long-lived securities[END_REF] where the authors explain how an Arrow-Debreu equilibrium can be implemented by trading in securities. This role of securities was, in fact, already recognized by Arrow (1952). The difference between the two approaches is illustrated by [START_REF] Cuoco | Optimal consumption and equilibrium prices with portfolio constraints and stochastic income[END_REF] where the budget constraints are associated to all the possible equilibrium prices (all the risk-neutral measures) instead of a unique budget constraint associated to the equilibrium price as in the classical general equilibrium model.

In Karatzas, Lehoczky and Shreve (1990), all agents are endowed in units of the same perishable commodity, which arrives at some time-varying random rate. Agents may consume their endowment as it arrives, they may sell some portion of it to other agents, or they may buy extra endowment from other agents. The endowment, however, cannot be stored, and agents wish to hedge the variability in their endowment process by trading with one another.

In this model all the prices are in term of a unique consumption good. When the market is complete it is equivalent to assume that the agents receive their endowment initially rather than over time. In that case and in order to have a stochastic total wealth, we assume that the consumption good is produced by the rms and distributed as dividends among the shareholders. The equilibrium condition imposes then a total consumption equal to the total supply of the consumption good and a total investment in each rm equal to the total value in term of consumption good of that rm.

With the notations of Jouini and Napp (1998), the mathematical description of the model is the following. Let be an economy with agents indexed by and let us assume that the agent has an initial wealth and a utility function given by where and satisfy our assumption . As previously each agent maximizes In this framework an equilibrium consists in a dimensional price processes and trading-consumption choices which are optimal for the agents, i.e. and such that for all in , the following market clearing conditions hold almost surely:

where is the number of rm outstanding shares Note that the last condition is redundant with the two previous ones by the self-nancing condition.

In Karatzas, Lehoczky and [START_REF] Karatzas | Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model[END_REF] it is shown that under mild conditions a unique equilibrium exists. In this issue, Chiarolla and Haussmann (2001) specializes and extends the [START_REF] Karatzas | Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model[END_REF] model to a situation where the endowment streams of the agents are denominated in money, not in goods, and are not exogenous. The labor provided by the agents to a rm produces the consumable good through a production function. The agents have then to choose a consumption and a leisure levels in order to maximize their utility function. Furthermore, the rm de nes the level of employment by a pro t maximization program. The utility functions of the agents depend then on two control variables and the main contribution of this paper is to extend the classical one-dimensional approach to this framework. The authors provide rst order necessary conditions for equilibrium, and derive from there the existence of such an equilibrium. They also solve explicitly two examples.

Basak and Croitoru (2001) exploit the equilibrium conditions in order to analyze the taxation impact on the asset prices. They consider a simple two agents model and use the ctitious market techniques described in the previous section in order to solve the individual utility maximization problem. The main difficulty is due to the presence of two redundant assets but with different taxation rules. The redundancy adds an extra step in the agent s problem : once he has chosen his risk exposure, he must decide how to allocate that risk

2

  The no-arbitrage condition

  over the set of admissible strategies. More precisely, the agent problem is

5 Acknowledgements References Journal of Mathematical Economics Geneva Papers on Risk and Insurance Ann. Inst. Henri Poincaré Ann. Inst. Henri Poincaré Journal of Economic Theory Equilibrium and dynamics : Essays in honor of David Gale

between the two securities. The authors establish general necessary conditions for equilibrium and show, in particular, that arbitrage opportunities still exist at the optimum. They characterize this mispricing and they provide an analysis of its equilibrium role. Some of the papers gathered here have been presented at the International Conference on Mathematical Finance, Hammamet, Tunisia, 14-18 juin 1999 organized by Nizar Touzi and the author. We wish to thank all the participants at this conference for helpful discussions as well as the members of the scienti c committee (Gérard Debreu (President), Freddy Delbaen, Ivar Ekeland, Ioannis Karatzas, Pierre-Louis Lions, Stanley Pliska and Dieter Sondermann), the members of the organizing committee (Guillaume Bernis, Abdelhamid Bizid, Laurence Carassus, Kaïs Hamza, Clotilde Napp, Abdelhamid Trad and Amel Zenaïdi) and our sponsors (UNESCO, Tunisian authorities, French authorities, ADRES, Tunisie-Valeurs group, Amen Invest and the Tunis Stock Market). We are also most grateful to the and to its Editor, Bernard Cornet, for welcoming us in print. I have also bene tted from comments and discussions with Guillaume Bernis, Abdelhamid Bizid, Bruno Bouchard, Martino Grasselli, Clotilde Napp and Nizar Touzi in the writing of this presentation. I am also indebted to CREST for the very stimulating scienti c atmosphere during the 1997-2000 period and to the Stern Business School where this paper was written for inviting me during the 1999-2000 period.

Mathematical

Finance Mathematical Finance Journal of Mathematical Economics DP CREST n Stochastics and Finance Finance and Stochastics Theory Probab. Appl Theory Probab. App. Journal of Mathematical Economics SIAM J. Control & Optimization Math. Operations Research SIAM J. Control & Optimization Math. Operations Research SIAM J. Control & Optimization

. [8] Arrow, K. and G. Debreu (1954), Existence of equilibrium for a competitive economy. 22, 265-290.

Econométrie

Review of Economic Studies

[9] Barles G. and H. M. Soner (1998), Option pricing with transaction costs, and a nonlinear Black-Scholes equation, 2(4).

[10] Basak S. and B. Croitoru (2001), Nonlinera taxation, tax arbitrage and equilibrium asset prices, , this issue.

[11] Bellamy N. ( 2001), Wealth optimization in an incomplete market driven by a jump-diffusion process. , this issue.

[12] Bernis (2000), Equilibrium in a reinsurance market with short sale constraints. Preprint. Université de Paris I Panthéon-Sorbonne.

[13] Bewley, T. (1972), Existence of equilibria in economies with in nitely many commodities. 4, 514-540.

[14] Black, F. and M. Scholes (1973), The pricing of options and corporate liabilities, 81, 637-659.

[15] Carassus L. and E. Jouini (1997), Coûts de transaction et contraintes de vente à découvert : une approche uni ée. DP CREST n 9758

[16] Carassus L. and E. Jouini (1998), Investment and arbitrage opportunities with short sales constraints. 8(3), 169-178.

[17] Carassus L. and E. Jouini (2000), A discrete stochastic model for investment with an application to the transaction costs case. 33(1), 57-80.

[18] Chiarolla M.B.and U.G. Haussmann (2001), Equilibrium in a stochastic model with consumption, wages and investment. , this issue.

[19] Constantinides G.M. (1986), Capital market equilibrium with transaction costs, 94, 842-862.

[20] Constantinides G.M.and Th. Zariphopoulou (1997), Bounds on option prices in an intertemporal economy with proportional transaction costs and general preferences. Preprint.

[21] Cox, J.C. and C.F. Huang (1989), Optimal consumption and portfolio policies when asset prices follow a diffusion process. 49, 33-83.


Elyès Jouini

Convergence of the equilibrium prices in a family of nancial models

In this paper, we consider a family of complete or incomplete nancial models such that the price processes of the nancial assets converge in distribution to those in a limit model. Different authors pointed out that we do not have necessarily convergence of the arbitrage pricing intervals in that context. We prove here that we have very good convergence properties for the equilibrium pricing interval as de ned by Bizid, Jouini and Koehl (1998) in discrete time or Jouini and Napp (1999) in continuous time.

Introduction

It is well-known that, in a complete nancial market, there is a unique price compatible with the no-arbitrage condition for each asset. That price corresponds to the cost of any replicationg strategy. In that context, if we consider a family of complete markets converging in some sense to another complete market, it suf ces to establish good convergence properties on the gain processes in order to prove good convergence properties on the arbitrage prices. Duf e and Protter (1992) present such convergence results on the gain processes developed by [START_REF] Jakubowski | Convergence en loi des suites d'intégrales stochastiques sur l'espace GD 1 de Skorokhod[END_REF] and Kurtz and Protter (1991, a and b) and they adapt them for easy applications in nance.

Even if their results are valid in the incomplete markets framework they do not give any information about the convergence of the arbitrage interval. [START_REF] Hubalek | When does convergence of asset price processes imply convergence of option prices[END_REF] provides some results about the convergence of the arbitrage interval but they rely on the mathematical concept of contiguity, or the nancial concept of quasi-arbitrage where, losely speaking, a quasi-arbitrage is, like in the de nition of a free-lunch, a sequence of strategies converging to an arbitrage, but where the n th strategy is applied in the n th market. It is dif cult then to give a "real world" interpretation of these quasi-arbitrages since we can't observe them in anyone of the considered markets. However this concept makes sense if the (n + 1) th market can be embedded in the n th one. This is the case when the considered sequence of nancial markets is introduced in order to modelize large nancial markets. This is not the aim of this paper where the sequence of nancial markets is introduced as an approximation of the limit market.

In fact, even in very simple and classical situations (stationary trinomial or quadrinomial trees), we can construct sequences of models satisfying all the conditions of Duf e and Protter (1992) and converging in distribution to a Black and Scholes model with a given volatility , but where the arbitrage interval for a given call option does not converge to the Black and Scholes price associated to the volatility (see Examples 1 and 2).

Nevertheless as underlined by Duf e and Protter "although a large part of nancial economic theory is based on models with continuous-time security trading, it is widely felt that these models are relevant insofar as they characterize the behavior of models in which trade occur discretely in time". It seems then natural to look for pricing intervals with good convergence properties.

In this paper, we prove that the equilibrium pricing interval introduced by Bizid, Jouini and Koehl (1999) in a discrete time framework and by [START_REF] Jouini | Continuous time pricing of nonredundant assets[END_REF] in a continuous time framework has these good convergence properties.

This interval is de ned as the set of all the expected values of the terminal payoff of a given asset with respect to all the equivalent probability measures with densities "in reverse order" than the total wealth of the economy.

Our restrictions on the family of the probabilities used to price the derivative asset with respect to the classical case are similar to those introduced by [START_REF] Perrakis | Option Pricing Bounds in Discrete Time[END_REF] and [START_REF] Perrakis | Option Bounds in Discrete Time: Extensions and the Pricing of the American Put[END_REF][START_REF] Perrakis | Option pricing for multinomial stock returns in diffusion and mixed processes[END_REF]. In those papers, the authors propose an ordering principle on the probabilities used to price the derivative asset. In Bizid, Jouini and Koehl (1999) and [START_REF] Jouini | Continuous time pricing of nonredundant assets[END_REF] this ordering principle concerns the densities instead of the probabilities and is derived from the equilibrium theory, and more precisely from the market clearing conditions. If we assume that all the states of the world are equiprobable, our result permits then to establish the convergence for the [START_REF] Perrakis | Option Bounds in Discrete Time: Extensions and the Pricing of the American Put[END_REF] prices.

This result is a generalization of [START_REF] Perrakis | Option pricing for multinomial stock returns in diffusion and mixed processes[END_REF] where the author proves the convergence of the "well ordered" prices in a multinomial model where the distribution of the successive returns stay the same for every period and where the price process converges in distribution to the [START_REF] Black | The pricing of options and corporate liabilities[END_REF] one or to the Merton (1976) mixed diffusion-jump one.

The paper is organized as follows. In section 2, we introduce the main concepts and result and in section 3 we prove a Ranga [START_REF] Rao | Relations between weak and uniform convergence of measures with applications[END_REF] type Lemma for sequences of nonincreasing functions instead of equicontinuous ones and we prove our main result as a consequence of this key Lemma.

Equilibrium pricing interval

Before introducing our general framework, let us consider the simplest example of incomplete markets and let us explain the main idea of the so-called equilibrium interval. More precisely, consider a simple one-period model where the sample space and the probability are = f! 1 ; ! 2 ; ! 3 g and P = (1=3; 1=3; 1=3).

First, there are two assets in the market: a rst asset whose prices are p(0) = 25, at date 0 and p(! 1 ) = 20; p(! 2 ) = 30 or p(! 3 ) = 40 at date 1 and a risk-free asset de ned by an interest rate r = 1 19 : It is straightforward that the feasible range of the density of the martingale measures with respect to P is

(Q(! 1 ); Q(! 2 ); Q(! 3 )) = (0:368+Q(! 3 ); 0:632 2Q(! 3 ); Q(! 3 )) where Q(! 3 ) 2 [0; 0:316] :
If there is a representative agent in this economy, the maximization program of this agents leads to the following rst-order conditions:

(Q(! 1 ); Q(! 2 ); Q(! 3 )) = ( u 0 (20) u 0 (25) ; u 0 (30) u 0 (25) ; u 0 (40) u 0 (25) )
and since u is usually strictly-concave, we must have

Q(! 1 ) > Q(! 2 ) > Q(! 3 ):
After imposing these restrictions, we get the feasible range of the risk-neutral probability Q(! 3 ) 2 [0:088; 0:211] : Consequently this approach permits to obtain bounds on the price of the derivative assets, which are better than the usual arbitrage-free bounds.

Following the same idea, the option bounding approach-introduced by Perrakis and Ryan (1984) and extended by [START_REF] Perrakis | Option Bounds in Discrete Time: Extensions and the Pricing of the American Put[END_REF][START_REF] Perrakis | Option pricing for multinomial stock returns in diffusion and mixed processes[END_REF]-derives upper and lower limits on option prices by assuming that stock returns and risk-neutral probability are in reverse order. Bizid, Jouini and Koehl (1999) and [START_REF] Jouini | Continuous time pricing of nonredundant assets[END_REF] proposed to replace the risk neutral probability by the density of such a probability with respect to the historical one and called "equilibrium bounds" the bounds obtained by assuming that stock returns and risk-neutral density are in reverse order. Furthermore they proved, in different settings, that this ordering principle is a necessary condition for the derivatives prices to be compatible with an equilibrium.

From an empirical point of view, Aït-Sahalia andLo (1997,1998) propose a nonparametric estimation of the state-price de ators. Using the market information, they estimate an option-pricing formula, then differentiate twice this estimator with respect to the strike of the option. Under suitable regularity conditions, this last quantity converges to a state-price de ator. Their empirical results con rm our theoretical "ordering principle".

Let us now de ne carefully the equilibrium interval:

De nition 1 A nancial model M is de ned by: -a nite horizon date T , -a probability space ( ; F; P ), -a ltration (F t ) t2[0;T ] on ( ; F; P ) satisfying the usual assumptions of right continuity and completeness, -a set S of assets with positive càdlàg, adapted, price processes denoted by s i i=0;:::;m ;

In all this section, we will assume that one of the assets (namely s 0 ) is risk-free and that it has a constant price equal to 1. This last assumption is made only in order to normalize the price processes and has no impact on our results. Let p be in f0;

; m 1g and let us assume that assets 0; ; p are in zero net supply and that assets p + 1;

; m are in positive supply and for each one of them we will normalize its supply to one.

For a given vector x we de ne W (x) as the sum of the last (m p) coordinates. For a given date t the total wealth at t is then de ned by

W t = W ((s i (t)) i=0; ;m ) = m X i=p+1 s i (t)
and the sum is then extended over all the assets with positive (and then normalized to one) net supply.

Before to introduce our main de nitions and results let us introduce some mathematical tools.

We denote by D m the space of R m valued càdlàg sample paths on a xed interval T = [0; T ] endowed with the Skorokhod topology and by D m the -eld generated by the open sets of D m : We refer to Jacod and Shiryaev, (1987, chapter VI), for the de nitions of these concepts as well as for the de nition of the convergence in distribution of a sequence of R m valued càdlàg processes X n to another process X 1 ; denoted by

X n =) X 1 .
For a given càdlàg adapted stochastic process X and a given stopping time we will denote by X the process X de ned by

X (t; !) = X(t; !) if t < (!) lim t 0 ! (!);t 0 < (!) X(t 0 ; !) if t (!)
and, for a given positive number b by X b the stopped process X b where b is de ned by b (!) = inf ft 0; kX (t; !)k bg and where the in mum is supposed to be equal to T when the associated set is empty.

Since we assumed that X is càdlàg, we have then that W (X b ) is bounded above for b suf ciently large. In the next W (X b ) will be denoted by W b and W (X ) by W .

De nition 2 Let us consider a given positive number B: We de ne the set B as the set of equivalent martingale-measures Q for s i i=0;:::;m such that, for all stopping time ; there exists a nonincreasing, positive function g B such that

E dQ dP = g (W ):
In the next B is kept xed and we denote by the set B , by loc the set of probability measures satisfying the same conditions as those in except that the priceprocesses are local-martingales instead of martingales. We will also denote by loc;ac the set of probability measures satisfying the same conditions as those in loc except that the functions (g ) are only nonnegative instead of positive (in particular, Q is then absolutely continuous with respect to P instead of equivalent).

If we introduce now a new purely-nancial asset in the market M , which nal payoff q T is known, the equilibrium interval is de ned as follows:

EI(q) = E Q [q T ] : Q 2 :
Note that, if this "new" asset is already in S then, the previous interval is reduced to one point : the known initial price of that asset.

Assume now that we have a sequence of nancial models (M n ) n=0; ;1 on a sequence of probability spaces ( n ; F n ; P n ) n=0; ;1 and let us assume that they have the same number m of assets, the same number p + 1 of zero-net supply assets, the same number m p of normalized to one net supply assets, the same horizon T and let us denote by X n the R m -valued process de ned by X n = s i n i=1; ;m . We will also assume that X n (0) does not depend on n and we will denote it by X(0) (for instance, we can assume that the initial prices of all the initially marketed assets and in all the models are normalized to one).

We are now ready to de ne the convergence of models as follows:

De nition 3 We say that the sequence (M n ) n2N converges to a model M 1 (and we denote it by In order to prove the convergence in distribution in that case we refer to Duf e (1988, section 22).

M n ) M 1 ) if X 1 is locally bounded, if the ltration of M 1 is generated by X 1 and if X n ) X 1 : Proposition 1 (Example) If M n is
Remark that we can not apply here the results of Duf e and Protter (1992) about the convergence of nancial gain processes since we consider arbitrage prices or equilibrium prices and not replicating prices. Then our prices are not directly linked to any gain process for any strategy.

It is well known that in the Cox-Ross-Rubinstein example, the price of any derivative asset (de ned by a nal payoff equal to a given function of the nal payoff of the stock) converges to the price of the derivative asset de ned by the same function in the limit-model (see e.g. Duf e and Protter 1992).

It is well known too that even if the models (M n ) are incomplete as in a trinomial model with only one stock, the limit model can in certain cases be complete. Nevertheless, [START_REF] Prigent | Pricing of contingent claims from discrete to continuous time models : on the robustness of the Black and Scholes formula[END_REF] showed on some examples that the arbitrage price interval of a given derivative asset in (M n ) does not necessarily converge to the unique price in the limit complete market model. In fact, the arbitrage price interval converges, in general, to a given interval containing the unique price in the limit model.

Let us present here one of his examples.

Example 2 (Prigent) We consider a sequence of stationary trinomial models where the returns at each node are 1 + n + i n 1=2 with positive probabilities p i for i = 1; 2; 3: We further assume that > 0; 1 > 2 > 0; 1 + > 3 > If P 3 i=1 p i i = 0; this model converges to the Black-Scholes model with volatility q P 3 i=1 p i 2 i . The set of martingale measures in M n has at each node exactly two extremal transition martingale-measures on the form = (0; 2 ; 3 ) and = ( 1 ; 0; 3 ):

The extremal price of a call option2 is obtained using always the same extremal transition probability at each node.

For a call option the arbitrage interval converges to [BS ( p 2 3 ) ; BS ( p 1 3 )] (where BS is the Black-Scholes formula considered as a function of the volatility).

The two next examples are from [START_REF] Bizid | Pricing of nonredundant assets derivatives in a complete market[END_REF]. The rst one is a discretization of a continuous time model and we study the convergence when the time step goes to zero. The second one is parametrized by a continuous parameter and we study the convergence of the equilibrium interval when the parameter goes to in nity for a given xed size of the tree.

Example 3 (Bizid-Jouini-Koehl) We consider a sequence of stationary quadrinomial models with equiprobable returns at each node e r T n + i p T n , i = 1;

; 4 and where we have

1 = 4 def = max > 2 = 3 def = min :
The arbitrage interval for a call option converges to [BS ( min ) ; BS ( max )] as in [START_REF] Avellaneda | Pricing and hedging derivative securities in markets with uncertain volatilities[END_REF].

The model converges, when n goes to in nity, to the Black and Scholes model with volatility

q 2 min + 2 max 2 :
With min = 10%; max = 15%; T = 1 year,r = 5%; n = 500 and3 S 0 = K = 100; the arbitrage interval is [6:812824; 8:602021] ; the equilibrium interval is [7:770313; 7:779874] for K large enough and the price in the limit model is 7:764116:

In the next, a derivative asset in the model M n will be de ned as an asset with a payoff equal to a given real-valued continuous function of X n = (X n (t)) 0 t T . Our derivative asset has then a payoff described by a function of the paths of the primitive assets processes in the model.

In the next we will denote by q n (T ) = F n (X n ), the nal payoff of a given derivative asset. The equilibrium interval for this asset is the de ned by

EI n (F n ) = E Q [F n (X n )] : Q 2 n :
The interval EI loc n (F n ) (resp. EI loc;ac n (F n )) will be de ned exactly as EI n (F n ) replacing n by loc n (resp. loc;ac n

). Furthermore, we will say that a sequence of intervals I n converges in a given interval I 1 and we will denote it by lim n I n ,! I 1 in order to say that, for all convergent sequence (u n ) in I n ; we have lim u n 2 I 1 :

We can now apply these de nitions and results in order to show a convergence result on derivatives prices : Theorem 4 Assume that M n ) M 1 and let us introduce a derivative asset de ned by q n (T ) = F n (X n ) in the model M n (for n = 0;

; 1) where F n is a sequence of bounded and continuous (for the Skorokhod topology) functions converging uniformly to a given function F 1 ; then (i) we have :

lim n EI n (F n ) ,! EI loc;ac 1 (F 1 ) (ii) if EI 1 6 = ; then lim n EI n (F n ) ,! EI loc 1 (F 1 ) (iii) if EI 1 6 = ; and (X 1 ) is of class 4 (D), then lim n EI n (F n ) ,! EI 1 (F 1 )
(iv) nally, if EI n 6 = ; for n suf ciently large and if M 1 is complete and arbitragefree (more accurately, such that there exists one and only one equivalent martingalemeasure), then lim

n EI n (F n = EI 1 (F 1 )
Remark 1 Under the conditions of (iii) or (iv) and if F is not bounded but such that F (x) = x e for a given real number ; a given vector e in R m and for x suf ciently large, then it suf ces to apply our result to q P m i=1 e i s i (Put-Call Parity).

Proof of the Theorem

In order to prove the theorem we will follow the following steps : By stopping the problem is reduced to a process which is bounded from above.

It is shown that the martingale measures sequence (Q n ) is tight and thus there is a limit candidate measure.

Boundedness and monotonicity yields a Helly-type selection theorem... ... that permits to prove a Ranga [START_REF] Rao | Relations between weak and uniform convergence of measures with applications[END_REF]-type Theorem (see also Billingsley, 1968, p17) for a family of decreasing functions (instead of equicontinuous ones). This is in fact the key result of the paper.

The limit measure is then a martingale-measure.

By stopping and approximation arguments the result extends to the original sequence of processes.

Reduction to bounded processes

Since X 1 is locally bounded there exists a sequence b k converging to 1 such that b k converges to T and X b k 1 converges to X 1 in probability. Furthermore, (b k ) can be chosen such that X b k n ) X b k 1 for all k: Indeed, when the set of paths with a jump or a "plateau" when they reach b have a zero measure with respect to P then we have X b n ) X b 1 : Furthermore, we refer to Jacod and Shiryaev (1987, chapter IX) in order to see that the set of real numbers b for which the set of paths with a jump or a "plateau" when they reach b does not have a zero measure with respect to P 1 is at most countable.

In the next we will omit the subscript k and we consider a sequence of martingalemeasures Q n such that, for all n; Q n belongs to n . Therefore, there exists a nonincreasing function

g b n such that E Pn b h dQn dPn i = g b n (W n ( b ))
where W n stands for W (X n ). We will denote by b n the distribution probability of X b n . In the next, b will be, unless speci ed, kept xed and we will denote by the stopping time b , by X n the process X b n , by g n the function g b n and by n the distribution probability n : The density of the restriction of Q n to ( ; F ) with respect to P n is then given by g n :

A tightness result

By construction, we have kg n k 1 B:

Let us denote by Qn the probability measure with density g n (W (x(T ))) with respect to n . Let us prove that Qn is tight. By de nition (see [START_REF] Billingsley | Convergence of probability measures[END_REF]), it suf ces to prove that for all " > 0; there exists a compact set K in D m (endowed with the Skorokhod topology) such that Qn (K) > 1 " for all n.

But we know that the family ( n ) converges weakly and then is tight (Billingsley, 1968, Theorem 29.3). There exists then K such that n (K) > 1 " 0 for all n with " 0 < " B . We have then,

Qn (K c ) = Z K c g n (W (x(T ))) d n (x) B Z K c d n (x) B" 0 < "
The family ( Qn ) is then tight and we can extract from it a subsequence ( Q'(n) ) converging weakly to some measure Q1 : In the next we will denote by ( Qn ) that subsequence.

We want now to prove that Q1 has a density with respect to 1 de ned by a nonincreasing function of the wealth. For that we need rst a Helly-type selection Theorem.

A Helly-type selection theorem

Lemma 5 If (h n ) is a sequence of uniformly bounded real-valued nonincreasing nonnegative functions de ned on [d; b] then we can extract a subsequence also denoted by (h n ) converging pointwise on [d; b] to a given real-valued nonnegative nonincreasing function h 1 such that for all u except on a countable set we have 8"; 9n u " ; 9 u " : 8n n u " ; jv uj u " =) jh n (v) h n (u)j < " [START_REF] Helly | über lineare Funktionaloperationen[END_REF] theorem permits us to construct a subsequence h (n) converging on [d; b] to some nonincreasing function h 1 .

Now, for all continuity point u of h 1 we have

h (n) (u) ! h 1 (u):
and we can nd u " such that, 8v 2 [d; b] ; ju vj < u " ) jh 1 (u) h 1 (v)j < " furthermore we can chose u " such that u u " and u + u " are both continuity points for h 1 :

Then for n suf ciently large we have

h (n) (u + u " ) h 1 (u + u " )
< ", = 1; 0; 1: and since all the functions (h n ) are nonincreasing we have, for all v such that u v u + u " ;

h (n) (u) h (n) (v) h (n) (u) h (n) (u + u " ) h (n) (u) h 1 (u) + (h 1 (u) h 1 (u + u " )) + h 1 (u + u " ) h (n) (u + u " ) < 3"
and we have exactly the same result for v such that u u " v u:

We have then proved that, there exists u " such that, for n suf ciently large and for all v such that ju vj < u " we have converging uniformly to a function f 1 , there exists a subsequence also denoted (h n ) and a real-valued nonnegative nonincreasing function h 1 de ned on [d; b] such that :

h (n) (u) h (n) (v) < 3" 3.4 A Ranga Rao-type Theorem Lemma 6 If M n ) M 1 and if (h n ) n2N is
Z f n (x)h n (W (x(T )))d n ! Z f 1 (x)h 1 (W (x(T )))d 1 :
This is in fact the key Lemma of this paper. In order to prove this result we rst reduce the problem by considering that f does not depend on n: Then, thanks to previous Lemma we provide an equicontinuous-like decomposition of R + : The following step follows the same lines as the original Ranga Rao Theorem (1962) proof and leads to an approximation of our measures by coutable support measures. The last step concludes.

Let " be a given positive real number and let us continue to use the same notations as in the previous proof.

Step 1 : Reduction of the problem

First remark that Z f n (x)h n (W (x(T )))d n Z f 1 (x)h 1 (W (x(T )))d 1 Z f n (x)h n (W (x(T )))d n Z f 1 (x)h n (W (x(T )))d n + Z f 1 (x)h n (W (x(T )))d n Z f 1 (x)h 1 (W (x(T )))d 1
and that h n is uniformly bounded, the measures n are probability measures and (f n ) converges uniformly to f 1 : The rst term on the right of the inequality symbol is then arbitrarily small for n suf ciently large. We will work then on the second term and we will simply denote by f the function f 1 :

Step 2 : Equicontinuous like decomposition For all continuity points u for h 1 , let us denote by N u the set de ned by N u = fv 2 R : ju vj < u " g we have then that 8v 2 N u ; 8n n u " ; jh n (u) h n (v)j < " and without any loss of generality, we can assume that

1 T W 1 (N u ) is a continuity set for 1 .
Let us consider N = [N u where the union is extended over all the continuity points for h 1 : The set N is open and has a countable complement in R + def = R + n f0g. It has then a countable number of connected components. Let us consider (w; w 0 ) one of these components. The set w + 1 n ; w 0 1 n is compact and can be covered by a nite union of sets N u : There exists then a sequence (u j ) such that

R + = 1 [ j=0 N uj [ D
where D is a subset of the set of discontinuity points for h 1 and is then nite or countable. We will denote by (a k ) k2N the elements of D:

If we denote by C 0 the set N u0 ; C 1 the set N u1 \ C c 0 ; and so on, we will obtain a countable partition of R + nD: We have then

1 X j=0 1 1 T W 1 (C j ) + 1 X k=0 1 1 T W 1 (fa k g) = 1
and it exists then N " suf ciently large such that

1 X j N " 1 1 T W 1 (C j ) + 1 X k N " 1 1 T W 1 (fa k g) ":
Furthermore for all k, there exists N k suf ciently large such that jh n (a k ) h 1 (a k )j < ", for all n N k : In the next, we will denote by n " the integer de ned by

n " = sup sup k N " N k ; sup j N " n uj " :
We have then for all j; k N " ; all n n " , all pair (u; v) in C j ,

jh n (u) h n (v)j < "; jh n (a k ) h 1 (a k )j < ":
Step 3 : Approximation by countable support measures

Let us now consider a family (x j ) in D m and a family (y k ) in D m such that for all j 1 and all k 0; we have W ( T (x j )) 2 C j and W ( T (y k )) = a k , and let us construct, for n = 1;

; 1 a measure n de ned on its support fx j : j 0g [ fy k : k 0g by n (x j ) = ~ n (C j ); and n (y k ) =~ n (a k ) where ~ n = n 1 T W 1 : If we omit the x's and if we denote by W T the quantity W (x(T )), we have for n n " and for n = 1;

Z f h n (W T ) d n Z f h n (W T ) d n = N" X j=0 Z Cj f h n (W T ) d n Z Cj f h n (W T ) d n + 1 X j=N"+1 Z Cj f h n (W T ) d n Z Cj f h n (W T ) d n + 1 X k=0 jf (y k )h n (a k ) n (fy k g) f (y k )h n (a k )~ n (fa k g)j
The last term is, by de nition, equal to zero and using the monotonicity of (h n ) we have,

Z f h n (W T ) d n Z f h (W T ) d n M N" X j=0 Z Cj jh n (u j ) h n (u)j d~ n (u) + 2M B 1 X j=N " +1 ~ n (C j ) "M ~ n ( 1 [ j=1 C j ) + 2"M B "M (1 + 2B)
where M is a bound on f and B is the uniform bound on the family (h n ):

Furthermore, it is easy to see that for n n 0

Z f h n (W T ) d n Z f h n (W T ) d 1 M B 1 X j=0 j~ n (C j ) ~ 1 (C j )j + M B 1 X k=0 j~ n (fa k g) ~ 1 (fa k g)j
We have ~ n (C j ) ! ~ 1 (C j ) and ~ n (fa k g) ! ~ 1 (fa k g) for all j 1 and all k 0 with

1 X j=0 ~ n (C j ) + 1 X k=0 ~ n (fa k g) = 1 X j=0 ~ 1 (C j ) + 1 X k=0 ~ 1 (fa k g) = 1
then by Scheffe's Theorem (Billingsley, 1968, p224), P 1 j=1 j~ n (C j ) ~ 1 (C j )j and P 1 k=0 j~ n (fa k g) ~ 1 (fa k g)j converge both to zero. We have then, for n sufciently large

Z f h n d n Z f h n d 1 M B" Finally, we have Z f h n (W T )d 1 Z f h 1 (W T )d 1 N " X j=0 Z Cj f h n (W T )d 1 Z Cj f h 1 (W T )d 1 + 1 X j=N " +1 Z Cj f h n (W T )d 1 Z Cj f h 1 (W T )d 1 + M N " X k=0 jh n (a k ) h 1 (a k )j + 1 X k=N " +1 jh n (a k ) h 1 (a k )j ! 1 (fy k g) M N " X j=0 jh n (u j ) h 1 (u j )j ~ 1 (C j ) + 2M B 1 X j=N " +1 ~ 1 (C j ) + 2M B 1 X k=N " +1 ~ 1 (fa k g) + M N " X k=0 jh n (a k ) h 1 (a k )j ~ 1 (fa k g) :
The sum of the second and the third term is lower than 2M B", furthermore, for all j and all k;

jh n (u j ) h 1 (u j )j ~ 1 (C j ) 2B ~ 1 (C j ) ; jh n (u j ) h 1 (u j )j ~ 1 (C j ) ! 0 jh n (a k ) h 1 (a k )j ~ 1 (fa k g) 2B ~ 1 (fa k g) ; jh n (a k ) h 1 (a k )j ~ 1 (fa k g) ! 0 1 = X j ~ 1 (C j ) + X k ~ 1 (fa k g)
We have then, by the Weierstrass M-test (see Billingsley (1995, Theorem A28, p. 543

) that N " X j=1 jh n (u j ) h 1 (u j )j ~ 1 (C j ) + N " X k=1 jh n (a k ) h 1 (a k )j ~ 1 (fa k g) ! 0
and, for n suf ciently large, we have then

Z f h n (W T )d 1 Z f h 1 (W T )d 1 < 2M "(1 + B):
Step 4 : Back to the original measures

In conclusion, we have

Z f h n (W T )d n Z f h 1 (W T )d 1 Z f h n (W T )d n Z f h n (W T )d n + Z f h n (W T )d 1 Z f h n (W T )d n + Z f h n (W T )d 1 Z f h 1 (W T )d 1 + Z f h 1 (W T )d 1 Z f h 1 (W T )d 1 < "M (4 + 7B)
We have then proved that

Z f n h n (W T )d n ! Z f 1 h 1 (W T )d 1 : 3.5 Q 1 is a martingale-measure for X 1
Applying last Lemma to our family of probability measures (Q n ) or more precisely to their densities (g n ) with respect to the probability measures n and to a constant sequence (f ); we obtain that for or a given subsequence that we will continue to denote by (g n ) and for any bounded random variable

f on D m , Z f g n (W T )d n ! Z f g 1 (W T )d 1 Or, in other words, Z f (u)d Qn ! Z f (u)g 1 (u)d 1 :
It is easy then to conclude that g 1 is the density of Q1 with respect to 1 : Furthermore, we know that, for n 2 N, X n is a martingale with respect to Q n and for n = 1;

; 1, it admits Qn as a distribution probability with respect to Q n : The sequence (X n ; Q n ) converges then in distribution to (X 1 ; Q 1 ) where Q 1 is the probability measure with density g 1 (X 1 (T )) with respect to P 1 :

The probability-measure Q 1 is then a martingale-measure for X 1 with respect to the ltration generated by X 1 (see Jacod and Shiryaev, Proposition 1.1, Chapter IX, p481).

Back to the original sequence of processes

If we reintroduce our notations with b, we proved that the stopped process X b 1 is a martingale with respect to the ltration that it generates and the probability Q b 1 with density g b 1 (W (X b 1 (T ))) with respect to P 1 . Therefore X b 1 is a local martingale for all the probability-measures with density g b 0 1 (W (X b 0 1 (T ))) for all b 0 > b: Furthermore X b 1 is bounded and X 1 is locally bounded. Therefore, for b 0 suf ciently large P 1 (g b 0 1 (W (X b 0 1 (T ))) = g 1 (W (X 1 (T )))) is arbitrarily close to 1: The process X b 1 is then a local-martingale for the probabilitymeasure with density g 1 (W (X 1 (T )))) and so is the process X 1 :

If we apply again the result of our Lemma to the sequence (F n ) which de nes our derivative assets, we have

Z F n g n (W T )d n ! Z F 1 g 1 (W T )d 1 or equivalently Z F n (X n )dQ n ! Z F 1 (X 1 )dQ 1 where R F (X n )dQ n is in EI n (F n ) and R F (X 1
)dQ 1 is in EI loc;ac 1 (F 1 ). We proved then that EI n (F n ) ,! EI loc;ac 1 (F 1 ):

If the set EI 1 (F ) is nonempty, i.e. if it exists at least one absolutely-continuous, martingale measure with a density de ned by a non-increasing function of the total wealth in the model M 1 , it is easy then to see that we have EI n (F ) ,! EI loc 1 (F ): If, nally X 1 is of class(D) with respect to P 1 and if all the assets are in positive supply, with respect to P 1 then, by de nition, for all "; there exists k; such that for all nite valued stopping time ; Z kX1( )k b X 1 ( )dP 1 ":

Since g 1 is bounded, we have then

Z kX1( )k b X 1 ( )g 1 (W ) dP 1 "B or equivalently Z kX1( )k b X 1 ( )dQ 1 = Z kX1( )k b X 1 ( )g 1 (W ) dP 1 "B:
The process X 1 is then of class (D) with respect to Q 1 and consequently a martingale with respect to Q 1 (Jacod and Shiryaev, 1987, Proposition 1-47, p11): We have then EI n (F ) ,! EI 1 (F ):

Assume now that we only have EI n (F n ) ,! EI loc;ac 1 (F 1 ):

but that there exists only one equivalent martingale measure, then there exists only one equivalent local-martingale measure and consequently there exists only one absolutelycontinuous martingale measure (Jacod, 1979, Theorem 11-4). We have then EI loc;ac 1 (F 1 ) = EI 1 (F 1 ) and is reduced to a single element, therefore lim n EI n (F n ) = EI 1 (F 1 ):

Remark 2 If we want the equality in (i), we don't really need market completeness nor the uniqueness of the equivalent martingale measure. It suf ces to impose that there is only one element in EI loc;ac 1 (F 1 ): This is, in particular, the case when there is only one absolutely-continuous local-martingale measure in "reverse order" than the total wealth. Such a situation may occur even in incomplete markets as in the following example.

Example 7 Consider a one step trinomial model with equiprobable returns 1 + " with " > 0 and = 1; 0; 1: This model is obviously incomplete. If we assume that there is a zero-interest rate, it is easy to check that there exists one and only one martingale-measure in reverse order of the stock prices, namely (1=3; 1=3; 1=3):

  a sequence of uniformly bounded realvalued nonincreasing nonnegative functions de ned on [d; b], then for all sequence of bounded real-valued, continuous (for the Skorokhod topology) functions f n on D m

The author wants to thank an anonymous referee for careful reading, valuable comments and improvements. Part of this paper was written when the author was visiting the Stern School of Business at New-York University.

Or any other derivative asset de ned by a convex function of the terminal payoff of the primitive asset

Where, as usual, K is the option strike and S 0 the initial price of the underlying asset.

i.e. the family (X 1 ) 2 ; where is the set of all nite value stopping times, is uniformly integrable, seeJacod and Shiryaev, 1987, p11.


Stochastic approaches to out-of-equilibrium dynamics

P.