## Januar – Februar 2019

Estimating covariances between financial assets plays an important role in risk management and portfolio allocation. Here, we show that from a Bayesian perspective market factor models, such as the famous CAPM, can be understood as linear latent space embeddings. Based on this insight, we consider extensions allowing for non-linear embeddings via Gaussian processes and discuss some applications.

In general, all these models are unidentified as the choice of coordinate frame for the latent space is arbitrary. To remove this symmetry we reparameterize the factor loadings in terms of an orthogonal frame and its singular values and provide an efficient implementation based on Householder transformations. Finally, relying on results from random matrix theory we derive the parameter distribution corresponding to a Gaussian prior on the factor loadings.

According to the Buddhist doctrine all things have no essence, but only shape. In this talk I will discuss the shape of the entropic cone.

Entropic cone is the closure of the set of values of entropies of n finitely-valued random variables and their joints. For n=0,1,2,3 the entropic cone is easy to evaluate. Starting with n=4 things become more complicated and interesting. I will discuss what is (un)known and present some new results.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

We study a system of cross-diffusion equations that results, as a

formal limit, from an interacting particle system with multiple

species. In the first part of the talk we exploit the (formal)

gradient flow structure of the system to prove the existence of weak

solutions. This is based on a priori bounds obtained from the

dissipation of the corresponding entropy and the use of dual

variables. In the second part, we discuss strong solutions and a

weak-strong stability result under certain conditions on the diffusion

constants.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2019/eco-2019/index.html

For a smooth projective variety X with a C^*-action, every orbit compactifies to a map from P^1 and one can split the points of X according to where infinity is mapped. The obtained division is called the negative Bialynicki-Birula decomposition.

In the talk I will explain how to generalize the BB decomposition and use it to find new components and prove non-reducedness and other pathologies for the Hilbert scheme of points. Surprisingly, the key role is played by the tangent space to the Hilbert scheme, which is easily computable, hence the search can be effectively conducted. I will also list some open questions.

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2019/eco-2019/index.html

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

Abstract zu dieser Veranstaltung (PDF-Datei)

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

Diffusion processes on the L2-Wasserstein space were introduced by Sturm and von Renesse in 2009 and have been subject of much interest in recent years. In particular, Konarovskyi proposed in 2017 a construction of a diffusion based on a system of massive coalescing particles. The aim of my talk is to present a Girsanov Theorem and a Bismut-Elworthy formula for a diffusion process inspired by Konarovskyi’s model.

We will first introduce a diffusion on the Wasserstein space which is a regularized version of Konarovskyi’s model. It can be viewed as a continuum of particles evolving on the real line according to a Gaussian interaction kernel weighted by the mass associated to each particle.

Second, we will prove a Girsanov-type Theorem on this process, using an appropriate Fourier inversion of the Gaussian kernel. As a Corollary, we obtain weak existence and weak uniqueness of the solution to a Fokker-Planck equation on the L2-Wasserstein space with very irregular drift.

Third, we will present a regularization result on the semi-group associated to this process with non-smooth drift and prove a Bismut-Elworthy formula which provides an upper bound for the gradient of the semi-group in smooth directions of differentiation.

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2019/eco-2019/index.html

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander’s bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris’s lemma to work directly on the Malliavin matrix, instead of the “reduced Malliavin matrix” which is not available in this context.

On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild Itô formula for rough PDEs.

This is a joint work with Martin Hairer.