Events

This talk is concerned with the following epidemic process: A set of nodes is partitioned into three states: susceptible, infectious, and recovered. We start with a single infectious node. Proceeding in rounds whose length is antiproportional to the population size, a fixed amount of nodes are drawn independently at random. If at least one of the selected nodes is infectious, every susceptible node in the sample becomes infected. Moreover, any infectious vertex recovers independently at a constant rate. If the expected amount of infections caused by single node is less than one, the epidemic dies out quickly and leaves almost the entire population untouched. If it is above one, either the infection dies out quickly or a large outbreak occurs, during which a non-vanishing fraction of the population is affected. Moreover, if enough nodes are infectious at the same time, the system’s behaviour is essentially deterministic.

We study Blaschke-Santaló diagrams for the inradius, circumradius and diameter in general Minkowski spaces. Independent of the gauge, they can be described by at most five parts of boundaries. We analyse which bodies fill these bounds and for which gauges they become redundant. Furthermore, we give a complete description of the union over all these diagrams with respect to planar symmetric gauges (solving an open problem stated by Brandenberg and Gonzáles Merino in a recent paper) by providing a new inequality that tightens Bohnenblust's bound. This union is equal to the union over all diagrams with respect to intersections of triangles with their origin reflection. This is joint work with René Brandenberg and Bernardo Gonzáles Merino.

We propose a continuous-time stochastic model to analyze the dynamics of impermanent loss in liquidity pools in decentralized finance (DeFi) protocols. We replicate the impermanent loss using option portfolios for the individual tokens. We estimate the risk-neutral joint distribution of the tokens by minimizing the Hansen–Jagannathan bound, which we then use for the valuation of options on relative prices and for the calculation of implied correlations. In our analyses, we investigate implied volatilities and implied correlations as possible drivers of the impermanent loss and show that they explain the cross-sectional returns of liquidity pools. We test our hypothesis on options data from a major centralized derivative exchange.

In this talk, we introduce a novel mesh-free and direct method for computing the shape derivative in PDE-constrained shape optimization problems. Our approach is based on a probabilistic representation of the shape derivative and is applicable for second-order semilinear elliptic PDEs with Dirichlet boundary conditions and a general class of target functions. The probabilistic representation derives from a boundary sensitivity result for diffusion processes due to Costantini, Gobet and El Karoui. Via so-called Taylor tests we verify the numerical accuracy of our methodology.