University of Vienna

A plethora of stochastic models used in particular in mathematical finance, but also population genetics and physics, stems from the class of affine and polynomial processes. The history of these processes is on the one hand closely connected with the important concept of tractability, that is a substantial reduction of computational efforts due to special structural features, and on the other hand with a unifying framework for a large number of probabilistic models. One early instance in the literature where this unifying affine and polynomial point of view can be applied is Lévy's stochastic area formula. Starting from this example,  we present a guided tour through the main properties and recent results, which lead to signature stochastic differential equations (SDEs). They constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are entire or real-analytic functions of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in subsets of group-like elements of the extended tensor algebra. Signature SDEs are thus a class of stochastic processes, which is universal within Itô processes with path-dependent characteristics and which allows - due to the affine theory - for a relatively explicit characterization of the Fourier-Laplace transform and hence the full law on path space.

In this talk, we discuss continuous in time dynamics for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V . Such problems are motivated by minimax convexconcave and, in particular, by convex optimization problems with linear constraints. The central role is played by a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. We show that these methods exhibit fast convergence rates for kV (z(t))k as t ! +1, where z( ) denotes the generated trajectory, and for the restricted gap function, and that z( ) converges to a zero of the operator V . For the corresponding implicit and explicit discrete time models with Nesterov’s momentum, we prove that they share the asymptotic features of the continuous dynamics.

Extensions to variational inequalities and fixed-point problems are also addressed. The theoretical results are illustrated by numerical experiments on bilinear games and the training of generative adversarial networks.

R. Schoen has conjectured that an asymptotically flat Riemannian n-manifold (M,g) with non-negative scalar curvature is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface. This has been confirmed by O. Chodosh and M. Eichmair in the case where n=3. In this talk, I will present recent work with M. Eichmair where we confirm this conjecture in the case where 3<n<8 and the area-minimizing hypersurface arises as the limit of large isoperimetric hypersurfaces. By contrast, we show that a large part of spatial Schwarzschild of dimension 3<n<8 is foliated by non-compact area-minimizing hypersurfaces.

Abstract: A recent paradigm views deep neural networks as discretizations of certain controlled ordinary differential equations, sometimes called neural ordinary differential equations. We make use of this perspective to link expressiveness of deep networks to the notion of controllability of dynamical systems. Using this connection, we study an expressiveness property that we call universal interpolation, and show that it is generic in a certain sense. The universal interpolation property is slightly weaker than universal approximation, and disentangles supervised learning on finite training sets from generalization properties. We also show that universal interpolation holds for certain deep neural networks even if large numbers of parameters are left untrained, and are instead chosen randomly. This lends theoretical support to the observation that training with random initialization can be successful even when most parameters are largely unchanged through the training. Our results also explore what a minimal amount of trainable parameters in neural ordinary differential equations could be without giving up on expressiveness.

Joint work with Martin Larsson, Josef Teichmann.

We reconsider the approximations of the Black-Scholes model by discrete time models such as the binominal or the trinominal model.

We show that for continuous and bounded claims one may approximate the replication in the Black-Scholes model by trading in the discrete time models. The approximations holds true in measure as well as "with bounded risk", the latter assertion being the delicate issue. The remarkable aspect is that this result does not apply to the well-known binominal model, but to a much wider class of discrete approximating models, including, eg.,the trinominal model. by an example we show that we cannot do the approximation with "vanishing risk".

We apply this result to portfolio optimization and show that, for utility functions with "reasonable asymptotic elasticity" the solution to the discrete time portfolio optimization converge to their continuous limit, again in a wide class of discretizations including the trinominal model. In the absence of "reasonable asymptotic elasticity", however, surprising pathologies may occur.

Joint work with David Kreps (Stanford University)

In this talk we will discuss a new approach to non rationality of projective varieties based on HMS. Examples will be discussed.

The problem of model uncertainty in financial mathematics has received considerable attention in the last years. In this talk I will follow a non-parametric point of view, and argue that an insightful approach to model uncertainty should not be based on the familiar Wasserstein distances. I will then provide evidence supporting the better suitability of the recent notion of adapted Wasserstein distances (also known as Nested Distances in the literature). Unlike their more familiar counterparts, these transport metrics take the role of information/filtrations explicitly into account. Based on joint work with M. Beiglböck, D. Bartl and M. Eder.

I will present a construction of large families of singularity-free stationary solutions of Einstein equations, for a large class of matter models including vacuum, with a negative cosmological constant. The solutions, which are of course real-valued Lorentzian metrics, are determined by a set of free data at conformal infinity, and the construction proceeds through elliptic equations for complex-valued tensor fields. One thus obtains infinite dimensional families of both strictly stationary spacetimes and black hole spacetimes.

Abstract: We study nonuniform sampling in shift-invariant spaces whose generator is a totally positive function. For a subclass of such generators the sampling theorems can be formulated in analogy to the theorems of Beurling and Landau for bandlimited functions. These results are  optimal and validate  the  heuristic reasonings in the engineering literature. In contrast to the cardinal series, the reconstruction procedures for sampling in a shift-invariant space with a totally positive generator  are local and thus accessible to numerical linear algebra.

A subtle  connection between sampling in shift-invariant spaces and the theory of Gabor frames leads to new and optimal  results for Gabor frames.  We show that the set of phase-space shifts of  $g$ (totally positive with a Gaussian part) with respect to a rectangular lattice forms a frame, if and only if the density of the lattice  is strictly larger than 1. This solves an open problem going backto Daubechies in 1990 for the class of totally positive functions of Gaussian type.
 

Hawking radiation and particle creation by an expanding Universe
are paradigmatic predictions of quantum field theory in curved spacetime.
Although the theory is a few decades old, it still awaits experimental
demonstration. At first sight, the effects predicted by the theory are too
small to be measured in the laboratory. Therefore, current experimental
efforts have been directed towards siumlating Hawking radiation and
studying quantum particle creation in analogue spacetimes.
In this talk, I will present a proposal to test directly effects of
quantum field theory in the Earth's spacetime using quantum technologies.
Under certain circumstances, real spacetime distortions (such as
gravitational waves) can produce observable effects in the state of
phonons of a Bose-Einstein condensate. The sensitivity of the phononic
field to the underlying spacetime can also be used to measure spacetime
parameters such as the Schwarzschild radius of the Earth.