Université Paris-Dauphine, PSL University, CNRS, CEREMADE
Abstract
In both arbitrage and utility **pricing** approaches, the fictitious completion appears as a powerful tool that permits to extend complete markets results to an incomplete markets framework. Does this technique permit to characterize the **equilibrium** **pricing** **interval**? This note provides a negative answer.

utility functions) and for all possible ditributions of initial endowment across the agents. In a di¤usion setting, when markets are complete but information (about prices, endowments and preferences) is incomplete or when markets are incomplete but under the assumption that there exists a possibility to complete the market by zero net supply assets without modifying the prices of the already existing ones, Jouini and Napp (2002) show that there exists a unique admissible price for any new asset and this price (as well as the **pricing** kernel) only depends on the initial assets prices that is to say it does not depend on agents utility functions or initial endowments nor on the choice of the completion. Bizid and Jouini (2005) show that the **equilibrium** **pricing** **interval** is strictly smaller than the arbitrage/utility **pricing** one. However, they only exploit partial conditions derived from the market clearing ones and the derived **pricing** **interval** is then larger than or equal to the **equilibrium** **pricing** **interval** as de…ned above.

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claim, **equilibrium** bounds delimit the set of all possible **equilibrium** prices for the asset under consideration for all possible distributions of agents preferences (within the class of vNM increasing and strictly concave utility functions) and for all possible ditributions of initial endowment across the agents. Bizid and Jouini (2005) show that this **interval** is smaller than the arbitrage/utility **pricing** one even though they do not fully characterize it. At this stage, a natural question arises: does the completion technique permit to char- acterize the **equilibrium** **pricing** **interval** (EPI ) as the set of **equilibrium** prices associated to all possible completions (that we will call …ctitious completion **equilibrium** **pricing** **interval**, FCEPI )? In other words, does the set of prices that can be reached at the **equilibrium** for at least one distribution of preferences/endowments and for at least one completion coincide with the set of prices that can be reached at the **equilibrium** for at least one distribution of preferences/endowments?

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The concept of Berge **equilibrium** is totally different from the Strong Berge **Equilibrium**. Indeed, the Berge Strong **Equilibrium** has been introduced in 1957 by Berge [6]. This **equilibrium** is a refinement 1 of the Nash **equilibrium**
[16] (see [15]), but in general, Berge **equilibrium** in the sense of Zhukovskii is not a Nash **equilibrium**. If only one player i adopts his strategy in a Strong Berge **Equilibrium**, he obliges all the other players −i to choose their strat- egy in this **equilibrium**: the adoption of other strategies by any players in the coalition −i, would provide each of them a payoff at most equal to that they get in this **equilibrium**. In other words, if any player selects his strategy in a Strong Berge **Equilibrium**, the other players have no other choice than to follow him by choosing their strategies from the same Stong Berge Equilib- rium. By contrast, if a player chooses his strategy in a Berge **equilibrium** in the sense of Zhukovskii, he cannot oblige the other players to follow him; he gets a maximum payoff if the other players are willing or obliged by some circumstances to choose their strategies in the same Berge **equilibrium**. The Berge **equilibrium** in the sense of Zhukovskii is rarely mentioned (not to say used) by game theorists. One of the most important reasons for this is that Zhukovskii published his results in Russian and within former USSR with local publishers only, so his results are not known world wide. The first

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By Proposition 4.1 , the infimum super-hedging cost of the European Call option ( S T − K ) + is given by h n t n i , St n i where h
n is defined by ( 4.14 ) with terminal
condition h n ( T, x ) = g ( x ) = ( x − K ) + . We extend the function h n on [0 , T ] in such a way that h n is constant on each **interval** [ t n i , t n i +1 [, i ∈ { 0 , . . . , n} . Such a scheme is proposed by Milstein [ 29 ] where a convergence theorem is proved when the terminal condition, i.e. the payoff function, is smooth. Precisely, the sequence of functions ( h n ( t, x )) n≥ 1 converges uniformly to h ( t, x ), solution of the Black and

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Theorem, that this transformation is actually an **interval** exchange transforma- tion on a stack, as defined in [7] (see also [19]).
The paper is organized as follows.
In Section 2, we recall some notions concerning **interval** exchange transfor- mations. We state the result of Keane [12] which proves that regularity is a sufficient condition for the minimality of such a transformation (Theorem 2.3). We study in Section 3 the relation between **interval** exchange transforma- tions and bifix codes. We prove that the transformation associated with a finite S-maximal bifix code is an **interval** exchange transformation (Proposition 3.8). We also prove a result concerning the regularity of this transformation (Theo- rem 3.12).

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Abstract: For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset’s **pricing** theory. Here, we propose a new approach based on convex duality instead of martingale measures duality : our prices will be expressed using Fenchel conjugate and bi-conjugate. This is lead naturally to a weak condition of absence of arbitrage opportunity, called Absence of Immediate Profit (AIP), which asserts that the price of the zero claim should be zero. We study the link between (AIP), (NA) and the no- free lunch condition. We show in a one step model that, under (AIP), the super-hedging cost is just the payoff’s concave envelop and that (AIP) is equivalent to the non-negativity of the super-hedging prices of some call option. In the multiple-period case, for a particular, but still general setup, we propose a recursive scheme for the computation of a the super-hedging cost of a convex option. We also give some numerical illustrations.

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Supposing that I ≤ϕ is transitive we analyse two cases: ∃(i, j) ∈ Cp ≤ϕ , i 6= j and
∀(i, j) ∈ Cp ≤ϕ , i = j and |Cp ≤ϕ | > 1. We show that these two cases are contradic-
tory with the transitivity of I ≤ϕ . ¥
This result shows that, the only structures having transitive indifference are the ones where two points of the same level are compared (intuitively it means that we do not really need intervals for the representation, points are sufficient. Let us remark that it is the case for linear orders and weak orders). Such a result is coherent with the use of **interval** representation for intransitive indifference since we need more than one point to represent intransitivity (I ≤ϕ is intransitive if and only if ∃i ∈ {1, . . . , n}, Cp ≤ϕ = {(i, j)} where

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The difference found with other **pricing** methods lies in several factors :
The relatively long maturity : it increases the errors made by the derivatives approximations thereby reducing the accuracy of the model
The spot increment chosen : it needs to be such that the multiplicative (Ψ) coefficients remain positive and between 0 and 1 (and ideally roughly equal to 1/3 each) but also such that the extreme spot values in the grid are sensible with regard to the maturity chosen, i.e. deltax must be basically proportional to the maturity chosen. However, the finite differences scheme can be quite sensitive to deltax and does not necessarily increase the option price linearly with deltax (increasing deltax can sometimes reduce the option price). To sum up, the choice of the spot increment in the grid is crucial and can be a source of imprecision in **pricing**. A usual choice is Δlog(S) = σ*Ѵ(3Δt) which we use here.

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The long distance operator Mercury pays access charges to British Telecom (BT) to reach consumers through the local loops and compete with BT in the long distance.. market.[r]

2.2. The model 47
articles in this area (Banditori and al., 2013, Table 2).
Multidimensional capacities are crucial in freight transport, such as the air cargo industry or the container shipping industry. Since dynamic **pricing** is widespread in these industries, the literature models freight transport **pricing** by extending standard revenue management models to accommodate multiple capacities. Xiao and Yang (2010) consider revenue management with two capacity dimensions. Their focus is on ocean container shipping, where container sizes are standardized (two varieties dominate the market) and maximal weights of all containers are set by on-road regulations. They derive an analytical solution and show that under some conditions the optimal policy is qualitatively different when considering the second capacity constraint. Kasilingam (1997) describes how air cargo revenue management is different from air passenger revenue management, and one of the key differences he identifies is the mul- tidimensional aspect of capacities: volume, weight and even cargo position may be constraining. Finally, the hospitality industry’s capacity management literature has also recognized the importance of dual capacities. Kimes and Thompson (2004) optimize the table mix for restaurant revenue management taking into account not only the number and distribution of seats but also the size of the service areas. Bertsimas and Romy (2003) consider both sizes of parties to be seated and expected service duration to compare several optimization-based approaches to restaurant revenue management.

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164 En savoir plus

Theorem, that this transformation is actually an **interval** exchange transforma- tion on a stack, as defined in [7] (see also [19]).
The paper is organized as follows.
In Section 2, we recall some notions concerning **interval** exchange transfor- mations. We state the result of Keane [12] which proves that regularity is a sufficient condition for the minimality of such a transformation (Theorem 2.3). We study in Section 3 the relation between **interval** exchange transforma- tions and bifix codes. We prove that the transformation associated with a finite S-maximal bifix code is an **interval** exchange transformation (Proposition 3.8). We also prove a result concerning the regularity of this transformation (Theo- rem 3.12).

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Section 3 derived the optimal regulation under the assumption that the regulator could audit the costs of the regulated monopoly well enough to separate the cost of the network from the [r]

Effect of **interval** running and drop-jumping on the time spent at 𝐕̇O2-MAX .
Our results showed that gas exchange and cardioventilatory responses to exercise during IT DJ9 was not different than IT RUN (Fig. 2 and 3). Moreover, no significant difference in T V̇ O 2-MAX , a key determinant of V̇ O 2-MAX improvement with training 12,13,38,39 , was found between IT DJ9 and IT RUN . Our findings therefore indicate that, the repetition of 192 drop jumps – a training method commonly used during physical training to improve explosive performance 3-5 -, was effective when performed at a work rate of 9 jumps per 15-s to stimulate the cardiorespiratory and oxydative systems to levels that are known to produce long term positive adaptations with training 12,13,38,39 . Our results extend previous findings from Brown et al. 9 , which showed that participants reached 83% of V̇O 2-MAX during eight series of ten drop-jumps (height: 80 cm), with three minutes of passive recovery between series. Our findings showing that T V̇O 2-MAX was greatly improved in IT DJ9 and IT DJ9-ISO WORK compared to IT DJ7 indicate that the metabolic and cardioventilatory response to repeated drop-jumps is critically dependent on the work rate. Therefore, the improved V̇O 2 response in our study compared to the abovementioned study is likely due to 1) the shorter recovery duration between work bouts (i.e. 15-s vs. 3-min) and 2) a higher number of jumps per minute, which contributed to increase the work rate and prevent a fall in V̇O 2 during recovery (Fig. 2).

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Raskin and Schobbens [8], a decidable, realtime extension of PTL.
In this short abstract, we have defined a timed extension of Dillon’s et al Graphical **Interval** Logic (TGIL) that is more expressive than previous proposals. The semantics of TGIL can be easily defined using an equivalent “textual notation” that may facilitate (such as e.g. proving the consistency of partial proof systems) We show how to apply TGIL for the definition of a realtime specification language and give an example of the synthesis of an observer. A limitation of this approach is that TGIL is essentially a linear time logic (it expresses constraints on traces), whereas some properties may require a branching time extension. In future work, we plan to enrich the logic with more expressive timing constraints and to study their interaction with a branching time variant of TGIL. We also plan to extend our compilation of diagrams into TTS observers to a larger subset of TGIL.

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interval exchange sets is closed under decoding by a maximal bifix code, that.. 62.[r]

In the next step, we consider systems of equations over intervals using least upper bound, addition, negation, multiplication with positive constants as well as intersections with constant intervals and arbitrary multiplication of intervals. We show that comput- ing the least solution of such systems can be reduced to computing the least solution of corresponding systems of integer equations. This reduction is inspired by the methods from [5] for **interval** equations with unrestricted intersections and the ideas of Leroux and Sutre [7], who first proved that **interval** equations with intersections with constant intervals as well as full multiplication can be solved in cubic time.

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