In this presentation based on on-line demonstrations of algorithms and on the examination of several practical examples, I will reflect on the problem of modeling a detection task in images. I will place myself in the (very frequent) case where the detection task can not be formulated in a Bayesian framework or, rather equivalently that can not be solved by simultaneous learning of the model of the object and that of the background. (In the case where there are plenty of examples of the background and of the object to be detected, the neural networks provide a practical answer, but without explanatory power). Nevertheless for the detection without "learning", I will show that we can not avoid building a background model, or possibly learn it. But this will not require many examples.

Joint works with Axel Davy, Tristan Dagobert, Agnes Desolneux, Thibaud Ehret.[-]

Random mosaics generated by stationary Poisson hyperplane processes in Euclidean space are a much studied object of Stochastic Geometry, and their typical cells or zero cells belong to the most prominent models of random polytopes. After a brief review, we turn to analogues in spherical space or, roughly equivalently, in a conic setting. A given number of i.i.d. random hyperplanes through the origin in $\mathbb{R}^d$ generate a tessellation of $\mathbb{R}^d$ into polyhedral cones. The typical cone of this tessellation, called a 'random Schläfli cone', is the object of our study. We provide first moments and mixed second moments of some geometric functionals, and compute probabilities of non-trivial intersection of a random Schläfli cone with a fixed polyhedral cone, or of two independent random Schläfli cones.

Parts are joint work with Matthias Reitzner, others with Daniel Hug.[-]

Gibbs spatial point processes are important models in theoretical physics and in spatial statistics. After a brief survey of Gibbs point processes, we will present a method for approximating their most important characteristic, the intensity of the process. The method has some affinity with the classical saddlepoint approximations of probability densities. For pairwise-interaction processes the approximation can be computed directly : it performs very well in many cases, but not in all cases. For higher-order interactions, we invoke limit results from stochastic geometry due to Roger Miles and the late Peter Hall, in order to compute the approximation.

Joint work with Gopalan Nair.[-]

I will discuss second order results for the length of nodal sets and the number of phase singularities associated with Gaussian random Laplace eigenfunctions, both on compact manifolds (the flat torus) and on subset of the plane. I will mainly focus on 'cancellation phenomena' for nodal variances in the high-frequency limit, with specific emphasis on central and non-central second order results.

Based on joint works with F. Dalmao, D. Marinucci, I. Nourdin, M. Rossi and I. Wigman.[-]

We study the behaviour of random walk on dynamical percolation. In this model, the edges of a graph are either open or closed and refresh their status at rate $\mu$, while at the same time a random walker moves on $G$ at rate 1, but only along edges which are open. On the d-dimensional torus with side length $n$, when the bond parameter is subcritical, the mixing times for both the full system and the random walker were determined by Peres, Stauffer and Steif. I will talk about the supercritical case, which was left open, but can be analysed using evolving sets.

Joint work with Y. Peres and J. Steif.[-]

In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.
This new distance $\widetilde{W}^{2}$ is defined similarly to the classical Wasserstein distance $\widetilde{W}^{2}$ but the set of couplings is restricted to the set of laws of solutions of 2$d$-dimensional stochastic differential equations. We prove that this new distance $\widetilde{W}^{2}$ metrizes the weak topology. Furthermore this distance $\widetilde{W}^{2}$ is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.[-]

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 biconjugate.
This naturally leads 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.[-]

Some questions in mathematics are not answered for quite some time, but just sidestepped. One of those questions is the following: What is the quantile of a multi-dimensional random variable? The "sidestepping" in this case produced so-called depth functions and depth regions, and the most prominent among them is the halfspace depth invented by Tukey in 1975, a very popular tool in statistics. When it comes to the definition of multivariate quantiles, depth functions replace cummulative distribution functions, and depth regions provide potential candidates for quantile vectors. However, Tukey depth functions, for example, do not share all features with (univariate) cdf's and do not even generalize them.
On the other hand, the naive definition of quantiles via the joint distribution function turned out to be not very helpful for statistical purposes, although it is still in use to define multivariate V@Rs (Embrechts and others) as well as stochastic dominance orders (Muller/Stoyan and others).
The crucial point and an obstacle for substantial progress for a long time is the missing (total) order for the values of a multi-dimensional random variable. On the other hand, (non-total) orders appear quite natural in financial models with proportional transaction costs (a.k.a. the Kabanov market) in form of solvency cones.
We propose new concepts for multivariate ranking functions with features very close to univariate cdf's and for set-valued quantile functions which, at the same time, generalize univariate quantiles as well as Tukey's halfspace depth regions. Our constructions are designed to deal with general vector orders for the values of random variables, and they produce unambigious lower and upper multivariate quantiles, multivariate V@Rs as well as a multivariate first order stochastic dominance relation. Financial applications to markets with frictions are discussed as well as many other examples and pictures which show the interesting geometric features of the new quantile sets.
The talk is based on: AH Hamel, D Kostner, Cone distribution functions and quantiles for multivariate random variables , J. Multivariate Analysis 167, 2018[-]

For large financial markets as introduced in Kramkov and Kabanov 94, there are several existing absence-of-arbitrage conditions in the literature. They all have in common that they depend in a crucial way on the discounting factor. We introduce a new concept, generalizing NAA1 (K&K 94) and NAA (Rokhlin 08), which is invariant with respect to discounting. We derive a dual characterization by a contiguity property (FTAP).We investigate connections to the in finite time horizon framework (as for example in Karatzas and Kardaras 07) and illustrate negative result by counterexamples. Based on joint work with M. Schweizer.[-]