TU Wien

Busenberg and Travis suggested in 1983 a population system that exhibits complete segregation of the species. This system can be rigorously derived from interacting particle systems in a mean-field-type limit. It consists of parabolic cross-diffusion equations with an indefinite diffusion matrix. It is known that this system can be formulated in terms of so-called entropy variables such that the transformed equations possess a positive semidefinite diffusion matrix. We consider in this talk the case of incomplete diffusion, which means that the diffusion matrix has zero eigenvalues, and the problem is not parabolic in the sense of Petrovskii. 

We show that the cross-diffusion equations can be written as a normal form of symmetric hyperbolic-parabolic type beyond the Kawashima-Shizuta theory. Using results for symmetric hyperbolic systems, we prove the existence of a unique local classical solution. As solutions may become discontinuous in finite time, only global solutions with very low regularity can be expected. We prove the existence of global dissipative measure-valued solutions satisfying a weak-strong uniqueness property. The proof is based on entropy methods and a finite-volume approximation with a mesh-dependent artificial diffusion. 

A defect-based a posteriori error estimate for Krylov subspace approximations to the matrix exponential is introduced. This error estimate constitutes an upper norm bound on the error and can be computed during the construction of the Krylov subspace with nearly no computational effort. The matrix exponential function itself can be understood as a time propagation with restarts. In practice, we are interested in finding time steps for which the error of the Krylov subspace approximation is smaller than a given tolerance. Finding correct time steps is a simple task with our error estimate. Apart from step size control, the upper error bound can be used on the fly to test if the dimension of the Krylov subspace is already sufficiently large to solve the problem in a single time step with the required accuracy.

We show how to adapt methods originally developed in
model-independent finance / martingale optimal transport to give a
geometric description of optimal stopping times tau of Brownian Motion
subject to the constraint that the distribution of tau is a given
distribution. The methods work for a large class of cost processes.
(At a minimum we need the cost process to be adapted. Continuity
assumptions can be used to guarantee existence of solutions.) We find
that for many of the cost processes one can come up with, the solution
is given by the first hitting time of a barrier in a suitable phase
space. As a by-product we thus recover Anulova's classical solution of
the inverse first passage time problem.

In a market with one safe and one risky asset, an investor with a long

horizon and constant relative risk aversion trades with constant

investment opportunities and proportional transaction costs. We derive

the optimal investment policy, its welfare, and the resulting trading

volume, explicitly as functions of the market and preference parameters,

and of the implied liquidity premium, which is identified as the

solution of a scalar equation. For small transaction costs, all these

quantities admit asymptotic expansions of arbitrary order. The results

exploit the equivalence of the transaction cost market to another

frictionless market, with a shadow risky asset, in which investment

opportunities are stochastic. The shadow price is also derived

explicitly. (Joint work with Paolo Guasoni, Johannes Muhle-Karbe, and

Walter Schachermayer)