Restrictions on the geometry of the periodic vorticity equation

We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, the axisymmetric Euler flow in higher space dimensions, and De Gregorio's vorticity model equation.Comment: 14 pages, 1 tabl

Global Existence for the Generalized Two-Component Hunter-Saxton System

We study the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting. We first prove a persistence result for the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter-Saxton system under proper assumptions on the initial data. This significantly improves recent result

Hedging goals

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Goal-based investing is concerned with reaching a monetary investment goal by a given finite deadline, which differs from mean-variance optimization in modern portfolio theory. In this article, we expand the close connection between goalbased investing and option hedging that was originally discovered in Browne (Adv Appl Probab 31(2):551–577, 1999) by allowing for varying degrees of investor risk aversion using lower partial moments of different orders. Moreover, we show that maximizing the probability of reaching the goal (quantile hedging, cf. Föllmer and Leukert in Finance Stoch 3:251–273, 1999) and minimizing the expected shortfall (efficient hedging, cf. Föllmer and Leukert in Finance Stoch 4:117–146, 2000) yield, in fact, the same optimal investment policy. We furthermore present an innovative and model-free approach to goal-based investing using methods of reinforcement learning. To the best of our knowledge, we offer the first algorithmic approach to goal-based investing that can find optimal solutions in the presence of transaction costs

Asymptotics of nonlinear diffusion and fluid synamics equations

Die Dissertation zerfällt in drei Teile: Der erste behandelt Themen des Optimalen Transports, der zweite das Verhalten von Lösungen von Modellgleichungen der Strömungsmechanik, und der letzte stetige monoton steigende singuläre Funktionen. Der erste Teil beginnt mit einem historischen Abriss der Theorie des Optimalen Transports seit den Arbeiten von Gaspard Monge und führt über eine eingehende Behandlung der Eigenschaften der Wasserstein-Distanzen zur Präsentation des ersten Resultats des Autors über das Langzeitverhalten von Lösungen des gekoppelten Drift-Diffusion-Poisson -- Modells aus der Halbleitertechnik. Weiters wird bewiesen, dass ein Optimierungsproblem mit Nebenbedingungen auf dem quadratischen Wassersteinraum eindeutig lösbar ist und das entsprechende zeitdiskrete Schema Lösungen einer nichtlokalen Fokker-Planck´schen Gleichung approximiert. Der zweite Teil behandelt zwei Modellgleichungen, welche den Euler´schen Gleichungen für inkompressible Flüssigkeiten abgeleitet sind. Für erstere, die verallgemeinerte Proudman-Johnson´schen Gleichung, werden für Spezialfälle einfache Beweise für den Verlust der Regularität von Lösungen nach endlicher Zeit gegeben. Für die verallgemeinerte Constantin-Lax-Majda´schen Gleichung wird gezeigt, dass es ein dem dreidimensionalen Falle entsprechendes Fortsetzungskriterium gibt. Im letzten Teil stellen wir ein Verfahren zur Erzeugung stetiger monoton steigender singulärer Funktionen vor.This thesis consists of three parts. The first is concerned with topics from optimal transport, the second deals with the behavior of solutions to model equations in fluid dynamics, and the third part treats continuous monotonously increasing singular functions. The first part commences with a historical introduction to the theory of optimal transport since the seminal contributions of Monge, continues with a detailed description of the properties of Wasserstein distances and displays the first result of the author on the large-time asymptotics of solutions to the coupled drift-diffusion-Poisson system. Moreover, it is proven that a constrained minimization problem on the quadratic Wasserstein space is uniquely solvable, yielding a time-discrete scheme approximating the solution to a nonlocal Fokker-Planck equation. The second part is concerned with two model equations in one space dimension derived from the famous Euler equations. We demonstrate, for some special cases, simple proofs of the loss of regularity in finite time of solutions to the so-called generalized Proudman-Johnson equation. Moreover, this part presents a continuation result for solutions to the generalized Constantin-Lax-Majda equation reminiscent of the three-dimensional case. In the last part, we introduce a constructive method for obtaining continuous monotonously increasing singular functions

Weighted variance swaps hedge against impermanent loss

Decentralized Exchanges (DEXes) allow users to trade in a fully noncustodial manner. Traders can directly swap their digital currencies using a smart contract, a program running on the blockchain, rather than trusting a central counterparty with their funds. In the early stages, the low throughput of blockchains required another trading model than the traditional order book approach, which gave rise to Automated Market Makers (AMMs). An AMM is a smart contract that determines the price for which traders can swap their digital currency against another digital currency. For the trade to happen, liquidity providers lock digital currencies into a smart contract, the liquidity pool. The AMM deposits the trader's digital currency into the liquidity pool and pays the trader with the other digital currency from the liquidity pool according to the price provided by the AMM. This alters the amounts owned by liquidity providers. In turn, liquidity providers earn trading fees, cf. Mohan (Citation2022). In a Constant Function Market, the AMM determines the price via a so-called trading function – a function of the liquidity pool's reserves – so that the value of the trading function given the post-trade reserves equals its value given the pre-trade reserves

The geometry of a vorticity model equation

We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point, with respect to right-invariant metric induced by the homogeneous Sobolev norm $H^{1/2}$ and show the local existence of the geodesics in the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$.Comment: 24 page