The seminar of the “Order” team will be held on Thursday of each week from 13h to 15h at the Faculty of Sciences of Tunis. The novelty of this year is to encourage the invitation of teachers-researchers from other laboratories than ours.

The sessions will begin on Thursday, September 12, 2019.

The unfinished program is below. It should be completed as participants give their confirmations.

Thursday, October 3, 2019

This session will be presential.

Pedro Tradacete Pérez : RECENT PROGRESS ON FREE BANACH LATTICES

Abstract:

The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.

Thursday, October 10, 2019

This session will be presential.

Foivos Xanthos : Some applications of vector lattices to the axiomatic theory of risk measures.

Abstarct:

The axiomatic theory of risk measures goes back to the seminal paper of Artzner et al. (1999). The classical literature assumes that the underlying model space $\mathcal{X}$ is $L^\infty(\mathbb{P})$. This puts the foundational aspects of the theory on safe mathematical grounds. However, the assumption $\mathcal{X}=L^\infty(\mathbb{P})$ is unduly strong for most applications. In this talk, we will explore the case where $\mathcal{X}$ is an abstract vector lattice. Recent advances have shown that this framework can give new insights in the theory. In this set-up we will discuss dual representations of risk measures and characterizations of risk aversion.

The slides of the talks are available here:

This is the video of the talk:

Thursday, October 24, 2019

This session will be by videoconference.

Till Hauser: Order continuity from a topological perspective

Abstract:

We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups and partially ordered vector spaces, respectively.
An order topology is introduced such that
in the latter two settings under mild conditions order continuity is a topological property.
We present a generalisation of the Ogasawara theorem on the structure of the set of order continuous operators.

The slides of the talks are available here:

Thursday, October 31, 2019

This session will be presential.

Emmanuel Lepinette : Diffusion equations: convergence of the functional scheme derived from the binomial tree with local volatility for non smooth payoff functions.

Abstract:

The function solution to the functional scheme derived from the Binomial tree financial model with local volatility converges to the solution of a diffusion equation of parabolic type as the number of discrete dates $n \to \infty$. Contrarily to classical numerical methods, in particular finite difference methods, the principle is only based on a discretization in time. We establish the uniform convergence in time of the scheme and provide the rate of convergence when the payoff function is not necessarily smooth as in finance. We illustrate the convergence result and compare its performance to the finite difference and finite element methods by numerical examples.

Thursday, November 14, 2019

This session will be presential.

Karim Boulabiar: Gleason-Kahane-Zelasko type theorems for complex lattice-ordred algebras

Abstract: Paragraphe

Let $\mathfrak{A}$ be a complex lattice-ordered algebra with positive identity $e$. The principal band $\mathfrak{B}{e}$ generated by $e$ is a projection band. It is shown that a linear functional $f$ on $\mathfrak{A}$ with $f\left( e\right) =1$ is a lattice homomorphism if and only if $f\left( \mathfrak{a}\right) \in \sigma\left( P{e}\left( \mathfrak{a}\right) \right)$ for all $\mathfrak{a}\in\mathfrak{A}$, where $P_{e}$ denotes the band projection of $\mathfrak{A}$ on $\mathfrak{B}{e}$. It follows that if $E$ is a Dedekind complete complex vector lattice and $f$ is an identity preserving linear functional on $\mathcal{L}^{r}\left( E\right)$, then $f$ is a lattice homomorphism if and only if for every $T\in\mathcal{L}{r}\left( E\right)$ the scalar $f\left( T\right)$ is a spectral value in $\mathcal{L}\left( E\right)$ of the diagonal component of $T$.