Elisa Perrone (TU Eindhoven)

Geometric structure in dependence models for testing positive dependence

The growing availability of data makes it challenging yet crucial to model and detect dependence traits such as tail dependence, non-exchangeability, or positive/negative dependence. Copulas are fascinating mathematical objects that serve as tools for capturing these complex traits and constructing accurate dependence models. In this talk, we explore the geometric properties of copulas to address challenges in applications. First, we study the class of discrete copulas, i.e., restrictions of copulas to grid domains that admit representations as convex polytopes. We draw connections to the popular Birkhoff and transportation polytopes, thereby unifying and extending results from the statistics and the discrete geometry literature. Then, we exploit the geometric representation of discrete copulas as convex polytopes to define a hypothesis test for positive quadrant dependence. Finally, we compare the proposed approach with existing tests in various simulated scenarios and discuss its advantages and limitations. 

Odysseas Kanavetas (Leiden University)

Optimal data driven policies under constrained multi-armed bandit observations

After a brief review of the multi-armed bandit (MAB) problem and its online machine learning applications, we present our work on the model with side constraints. The constraints represent circumstances in which bandit activations are restricted by the availability of certain resources that are replenished at a constant rate.

We consider the class of feasible uniformly fast (f-UF) convergent policies, that satisfy sample path wise the constraints. We first establish a necessary asymptotic lower bound for the rate of increase of the regret (i.e., loss due to the need to estimate unknown parameters) function of f-UF policies. Then, under pertinent conditions, we establish the existence of asymptotically optimal policies by constructing a class of f-UF policies that achieve this lower bound.

We provide the explicit form of such policies for the case in which the unknown distributions are Normal with unknown means and known variances.

Antonis Papapantoleon (TU Delft)

Model-free bounds in finance: a journey through probability, statistics and optimization

Academics, practitioners and regulators have understood that the classical paradigm in mathematical finance, where all computations are based on a single “correct” model, is flawed. Model-free methods, were computations are based on a variety of models, offer an alternative. In this talk, we will discuss model-free methods and bounds, starting from the improved Fréchet-Hoeffding bounds and their applications in option pricing and risk management, and will present how ideas from probability, statistics, optimal transport and optimization can be applied in this field.

Annika Betken (U Twente)

Time series analysis based on ordinal patterns

In time series analysis,  ordinal patterns describe the relative position of consecutive realizations generated by the underlying data-generating stochastic process. Among other things, we consider estimators for the probabilities of occurrence of ordinal patterns (ordinal pattern probabilities) in time series. We investigate statistical properties of  these estimators in discrete-time Gaussian processes with stationary increments. By means of Rao-Blackwellization, we further improve the estimation of ordinal pattern probabilities. Moreover, limit theorems that describe the asymptotic distribution of the considered estimators are established. Depending on the behavior of the data-generating processes’ autocorrelation function, these limit distributions differ. As an application, we discuss the Zero-Crossing  estimator for the Hurst parameter which characterizes fractional Brownian motion processes.