Studierende und Gäste sind jederzeit willkommen.

Wenn nicht explizit anders unten angegeben, finden alle Vorträge im Seminarraum 2.58 im Mathebau (Gebäude 20.30) statt.

# AG Stochastik (Sommersemester 2017)

Dozent: | Prof. Dr. Günter Last, Prof. Dr. Nicole Bäuerle, Prof. Dr. Vicky Fasen-Hartmann, Prof. Dr. Norbert Henze, Prof. Dr. Daniel Hug |
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Veranstaltungen: | Seminar (0175900) |

Semesterwochenstunden: | 2 |

Seminar: | Dienstag 15:45-17:15 | SR 2.58 |
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Seminarleitung | Prof. Dr. Günter Last |
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Sprechstunde: Nach Vereinbarung. | |

Zimmer 2.001, Sekretariat 2.056 Kollegiengebäude Mathematik (20.30) | |

Email: guenter.last@kit.edu | |

Seminarleitung | Prof. Dr. Nicole Bäuerle |

Sprechstunde: In den Semesterferien nach Vereinbarung | |

Zimmer 2.016 Kollegiengebäude Mathematik (20.30) | |

Email: nicole.baeuerle@kit.edu | |

Seminarleitung | Prof. Dr. Vicky Fasen-Hartmann |

Sprechstunde: Im WS 2017/2018 im Forschungsfreisemester. | |

Zimmer 2.053 Kollegiengebäude Mathematik (20.30) | |

Email: vicky.fasen@kit.edu | |

Seminarleitung | Prof. Dr. Norbert Henze |

Sprechstunde: Nach Vereinbarung. | |

Zimmer 2.020, Sekretariat 2.002 Kollegiengebäude Mathematik (20.30) | |

Email: henze@kit.edu | |

Seminarleitung | Prof. Dr. Daniel Hug |

Sprechstunde: Nach Vereinbarung. | |

Zimmer 2.051 Kollegiengebäude Mathematik (20.30) | |

Email: daniel.hug@kit.edu |

# Vorträge

### Dienstag, 18.07.2017

15.45 Uhr **M.Sc. Daniel Schmithals (Institut für Stochastik, KIT)**

## Model-independent derivative pricing via martingale optimal transport

**Abstract:**The talk is threefold.

In the first part we motivate the concept of model-independent derivative pricing and explain the basic idea. We use a lemma of Breeden & Litzenberger (1978) and adapt the theory of classical optimal transport to find the set of so called martingale transport plans. Martingale transport plans are martingale measures with prespecified marginals. Based on this we introduce the pricing problem and its dual. The solutions may be interpreted as an upper bound for consistent derivative prices and as a super-replicating hedging strategy for the derivative payoff respectively.

In the second part we present a strong duality result for the two problems in a very general setting showing that the problems correspond to best-possible bounds for prices and hedging strategies.

In the third part in a less general setting we impose certain assumptions on the derivative payoff under which it is possible to characterize solutions to the problems. We then explicitly solve the pricing and hedging prob-lems for different types of marginals.

### Dienstag, 27.06.2017

15.45 Uhr **Dr. Fabian Schaller (Universität Erlangen-Nürnberg)**

## Die Struktur zufälliger granularer Packungen

**Abstract:**In granular packings and other particulate systems with short-range interactions, the local structure has cru-cial influence on the macroscopic physical properties such as stability or response to external forces. Packings of monodisperse spherical particles are a common simple model for granular matter and packing problems, but particles in nature and industries are rarely spherical. This talk focuses on packings of ellipsoidal particles, a system which offers the possibility to study the influ-ence of particle shape on packing properties. Results of a large scale experimental study of jammed packings of oblate ellipsoids as well as results of sim-ulations of frictional and frictionless particles with and without gravity are presented. The structure of the packings is analyzed by Set Voronoi diagrams, an extension of the conventional Voronoi diagram to aspherical particles, and the average number of contacts between the particles. Furthermore, a common model for the Voronoi Volume distribution of spheres is extended towards aspherical particles. Finally, the effect of particle polydispersity to the packing structure is addressed. In the broader context of the physics of particulate systems, our analysis emphasizes the need for simple toy models to understand packing properties of complex shaped particles. (Vortrag wird auf Deutsch gehalten)

### Dienstag, 13.06.2017

15.45 Uhr **M.Sc. Jan Weis(Institut für Stochastik, KIT)**

## Integral Geometric Formulae for Tensorial Curvature Measures

**Abstract:**The tensorial curvature measures are the natural tensor-valued generalizations of the curvature measures of convex bodies in Euclidean space. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex bodies. The talk provides two complete sets of integral geometric formulae, so called kinematic formulae and Crofton formulae, for such (generalized) tensorial curvature measures. These formulae treat the intersection of a convex body with a second geometric object (in the kinematic formulae this is another convex body; in the Crofton formulae this is an affine subspace) which is uniformly moved by a proper rigid motion. The proofs, which are also sketched in this talk, proceed in a more direct way than the classical proofs of the corresponding integral formulae for curvature measures.

### Dienstag, 30.05.2017

15.45 Uhr **Dr. Tobias Fissler(Universität Bern)**

## The Elicitation Problem

**Abstract:**A statistical functional, such as the mean, the median or a certain risk measure, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to com- pare competing forecasts and to rank them in terms of their realized scores. Acknowledging the relevance of elicitability for forecast ranking and comparison, but also for M-estimation and regression, this motivates the following three-fold elicitation problem for a fixed functional T: (i) Is T elicitable, i.e., is there a strictly consistent scoring function for T? (ii) Characterize the class of strictly consistent scoring functions for T. (iii) Are there particularly distinguished instances of strictly consistent scoring functions for T satisfying secondary quality criteria? While most parts of the literature have concentrated on real- valued functionals, the talk lies particular emphasis on studying the elicitation problem for higher dimensional functionals. It turns out that a functional, albeit not being elicitable, can be a component of an elicitable vector-valued functional. In the ca- se of the variance, this is a known result. In the same direction, it was recently shown in Fissler and Ziegel (2016, AOS) that the pair of the two practically relevant risk measures (Value at Risk, Expected Shortfall) is elicitable, even though Expected Shortfall itself is not. This result has direct consequences for the practice of backtesting in quantitative risk management.

The talk is based on joint work with Johanna F. Ziegel and Tilmann Gneiting.

### Dienstag, 16.05.2017

15.45 Uhr **Prof. Dr. Johannes Ruf (London School of Economics)**

## Some remarks on functionally generated portfolios In

**Abstract:**In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities. This is joint work with Bob Fernholz and Ioannis Karatzas.

### Dienstag, 9.05.2017

15.45 Uhr **Prof. Dr. Richard Gardner (Western Washington University)**

## Symmetrization

**Abstract:** The idea of symmetrization---taking a subset of Euclidean space (for example) and replacing it by one which preserves some quantitative aspect of the set but which is symmetric in some sense---is both prevalent and important in mathematics. The most famous example is Steiner symmetrization, introduced by Jakob Steiner around 1838 in his attempt to prove the isoperimetric inequality (the inequality which essentially explains why soap bubbles are spheres rather than some other shape). Steiner symmetrization is still a very widely used tool in geometry, but it and other types of symmetrization are of vital significance in analysis, PDE's, and mathematical physics as well.

The talk focuses on symmetrization processes that associate to a given set one that is symmetric with respect to a subspace. In the first phase of an ongoing joint project with Gabriele Bianchi and Paolo Gronchi of the University of Florence, we consider various properties of an arbitrary symmetrization, the relations between these properties, and which properties characterize Steiner symmetrization. Several other well-known symmetrizations, such as Minkowski symmetrization and central symmetrization, will also be discussed. After summarizing these results, we shall discuss the second phase, in which we attempt to understand the convergence of iterated symmetrals.