Veranstaltungen

There has been increasing interest in the estimation of risk capital within enterprise risk models, particularly through Monte Carlo methods. A key challenge in this area is accurately characterizing the distribution of a company’s available capital, which depends on the conditional expected value of future cash flows. Two prominent approaches are available: the “regress-now” method, which projects cash flows and regresses their discounted values on basis functions, and the “regress-later” method, which approximates cash flows using realizations of tractable processes and subsequently calculates the conditional expected value in a closed form. This paper demonstrates that the left and right singular functions of the valuation operator serve as robust approximators for both approaches, enhancing their performance. Furthermore, we describe situations where each method outperforms the other, with the regress-later method demonstrating superior performance under certain conditions, while the regress-now method generally exhibiting greater robustness.

Geometric packing problems have been investigated for centuries in mathematics and a notable example is the Kepler's conjecture, from the 1600s, about the packing of spheres in the Euclidean space. In this talk, we mainly address the knapsack problem of hyperspheres. The input is a set of hyperspheres associated with profits and a hyperrectangular container, the knapsack. The goal is to pack a maximum profit subset of the hyperspheres into the knapsack. For this problem, we present a PTAS. For a more general version, where we can have arbitrary fat objects instead of hyperspheres, we present a resource augmentation scheme (an algorithm that gives a super optimal solution in a slightly increased knapsack).

Quantum walks (QWs) can be viewed as quantum analogs of classical random walks. Mathematically, a QW is described as a unitary, local operator acting on a grid and can be written as a product of shift and coin operators. We highlight differences to classical random walks and stress their connection to quantum algorithms (see Grover’s algorithm). If the QW is assumed to be translation invariant, applying the Fourier transform yields a multiplication operator, whose bandstructure we briefly study. After equipping the underlying lattice with random phases, we turn to dynamical localization. This means that the probability to move from one lattice site to another decreases on average exponentially in the distance, independently of how many steps the QW may take. We sketch the proof of dynamical localization on the hexagonal lattice in the regime of strong disorder, which uses a finite volume method.

Decision makers desire to implement decision rules that, when applied to individuals in the population of interest, yield the best possible outcomes. For example, the current focus on precision medicine reflects the search for individualized treatment decisions, adapted to a patient's characteristics. In this presentation, I will consider how to formulate, choose and estimate effects that guide individualized treatment decisions. In particular, I will introduce a class of regimes that are guaranteed to outperform conventional optimal regimes in settings with unmeasured confounding. I will further consider how to identify or bound these "superoptimal" regimes and their values. The performance of the superoptimal regimes will be illustrated in two examples from medicine and economics.