1st BYO Workshop in Probability and Finance

Internal Block Seminar, February 12, 2021

About

This internal block seminar of our Probability and Financial Mathematics Group at the Mathematical Institute of the LMU Munich is to keep us up-to-date regarding our ongoing research activities, in particular, of our doctoral and post-doctorial fellows, and to provide a regular event for our scientific exchange.

BYO stands for "bring your own cookies and coffee/tea” as there will be lots of time for our informal discussions during and between the talks.

The event will take place on Zoom. Please contact us here to obtain the link.

Program

Time	Event	Description
13:30-13:40	Welcome	Get-together
13:40-14:30	Talk	Katharina Oberpriller: Feynman-Kac formula under volatility uncertainty
14:30-14:50	Break	BYO coffee break and discussions
14:50-15:40	Talk	Leonid Kolesnikov: Cluster expansions: Sufficient and necessary conditions for absolute convergence
15:40-16:00	Break	BYO coffee break and discussions
16:00-16:50	Talk	Annika Heckel: Non-concentration of the chromatic number of a random graph
16:50-17:10	Break	BYO coffee break and discussions
17:10	Closing

Talks

Non-concentration of the chromatic number of a random graph

Speaker: Annika Heckel

Abstract: There are many impressive results asserting that the chromatic number of the random graph G(n,p) is sharply concentrated. By a (nowadays) standard argument, it takes one of at most about n^(1/2) consecutive values with high probability (whp), meaning with probability tending to 1. This can be improved considerably for sparse random graphs: if p=p(n) < n^(-1/2-epsilon), the chromatic number takes one of only two consecutive values whp. However, until recently no non-trivial lower bounds for the concentration interval length were known for any p=p(n), even though the question was raised prominently by Erdős and Bollobás since the late 80s. In this talk, I will show that the chromatic number of G(n, 1/2) is not whp contained in any sequence of intervals of length n^(1/2-o(1)), almost matching the classic upper bound. I will also discuss and give evidence for a recent conjecture on the correct concentration behaviour. Joint work with Oliver Riordan and with Konstantinos Panagiotou.

Cluster expansions: Sufficient and necessary conditions for absolute convergence

Speaker: Leonid Kolesnikov

Abstract: We consider Gibbs point processes with repulsive pair interactions. For small activities, a cluster expansion allows us to express the corresponding correlation functions by (multivariate) power series in the activity around zero. We characterize the domain of absolute convergence of these series. The characterization provides an elementary access to the classical criteria for absolute convergence and allows us to prove a new sufficient condition in the setting of abstract polymers improving the known bounds for the convergence radii.

Feynman-Kac formula under volatility uncertainty

Speaker: Katharina Oberpriller

Abstract: In this work we provide a generalization of the Feynmac-Kac formula under volatility uncertainty. We state our result under different hypothesis with respect to the derivation in [1], where the Lipschitz continuity of some functionals is assumed which is not necessarily satisfied in our setting. In particular, we obtain the G-conditional expectation of a discounted payoff as the limit of C^{1,2} solutions of some regularized PDEs, for different kinds of convergence. In applications, this permits to approximate such a sublinear expectation without resorting to Monte-Carlo methods, whose correct generalization to the G-setting is still object of research. This is a joint work with Bahar Akthari, Francesca Biagini and Andrea Mazzon.

[1] Mingshang Hu, Shaolin Ji, Shige Peng, and Yongsheng Song. Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDES driven by G-Brownian motion. Stochastic Processes and their Application, 124(2), 2014.