29.01.2025 12:15 Siegfried Hörmann (Graz University of Technology, AT): Measuring dependence between a scalar response and a functional covariate
We extend the scope of a recently introduced dependence coefficient between a scalar response Y and a multivariate covariate X to the case where X takes values in a general metric space. Particular attention is paid to the case where X is a curve. While on the population level, this extension is straight forward, the asymptotic behavior of the estimator we consider is delicate. It crucially depends on the nearest neighbor
structure of the infinite-dimensional covariate sample, where deterministic bounds on the degrees of the nearest neighbor graphs available in multivariate settings do no longer exist. The main contribution of this paper is to give some insight into this matter and to advise a way how to overcome the problem for our purposes. As an important application of our results, we consider an independence test.
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03.02.2025 14:15 Lorenz Schneider, EMLYON Business School: Revisiting the Gibson-Schwartz and Schwartz-Smith Commodity Models
We extend the popular Gibson and Schwartz (1990) and Schwartz and Smith (2000) two-factor models for the spot price of a commodity to include stochastic volatility and correlation. This generalization is based on the Wishart variance-covariance matrix process. For both of the extended models we present the joint characteristic functions of the two state variables. The original models are known to fit the term-structure of implied volatility in futures and options markets very well. However, the extended models are also able to match volatility smiles observed in these markets. Regarding the analysis of financial time series, the assumption of a constant correlation between the state variables is known to be too restrictive. Introducing time-varying correlation via the Wishart process allows us to study its empirical behaviour in commodity markets through the use of filtering techniques.
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03.02.2025 15:00 Juri Joussen: Two-time scale dynamics of solutions to a rimming flow equation
We consider a thin layer of an incompressible viscous Newtonian fluid coating the inner
wall of a horizontal cylinder rotating with constant speed. Assuming the film height is small
compared to the radius of the cylinder, we formally derive a closed equation for the height
h(t, θ) > 0 of the liquid film by means of a lubrication approximation:
ht + hθ + γ h3(hθθθ + hθ)
θ = g h3 cos θ
θ in (0, T ) × T
This rimming flow equation is of fourth-order, degenerate-parabolic, and quasilinear. Compet-
ing effects are observed between viscosity, the surface tension γ, and gravity g.
For g = 0 and a fixed mass m, the two-dimensional manifold
M(m) := m + a sin θ + b cos θ a2 + b2 m2
is invariant. If 0 g ∼ δ ≪ 1 is small, we show that solutions which are bounded away from
zero converge exponentially fast to a δ-neighbourhood of M(m). Here, the existence of solutions
on a large time scale t ∼ 1/δ2 can be shown. Moreover, such solutions evolve on two distinct
time scales: On the fast time scale t they only rotate around the origin with the speed of the
cylinder, while on the slow time scale τ = δ2t the dynamics are governed by an ODE in τ on
M(m).
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05.02.2025 12:15 Cecilie Recke (University of Copenhagen, DK): Identifiability and Estimation in Continuous Lyapunov Models
We study causality in systems that allow for feedback loops among the variables via models of cross-sectional data from a dynamical system. Specifically, we consider the set of distributions which appears as the steady-state distributions of a stochastic differential equation (SDE) where the drift matrix is parametrized by a directed graph. The nth-order cumulant of the steady state distribution satisfies the corresponding nth-order continuous Lyapunov equation. Under the assumption that the driving Lévy process of the SDE is not a Brownian motion (so the steady state distribution is non-Gaussian) and the coordinates are independent, we are able to prove generic identifiability for any connected graph from the second and third-order Lyapunov equations while allowing the cumulants of the driving process to be unknown diagonal. We propose a minimum distance estimator of the drift matrix, which we are able to prove is consistent and asymptotically normal by utilizing the identifiability result.
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05.02.2025 16:00 Christoph Schweigert (Hamburg): From tensor networks to Frobenius Schur indicators: some applications of state-sum models with boundaries
State-sum constructions have numerous applications in both mathematics and physics. In mathematics, they yield invariants for knots and manifolds and serve as a powerful organizing principle in representation theory. To illustrate this principle, we discuss equivariant Frobenius-Schur indicators. In the context of physics, we explain how state-sum models offer a conceptual framework for tensor network models, based on collaboration with Fuchs, Haegeman, Lootens, and Verstraete.
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