# Probability Day

**(in connection with the celebration of Sergio Albeverio's 75th birthday)**

**Date**: April 11, 2014**Venue:** Mathematik-Zentrum, Lipschitz Lecture Hall (Room 1.016), Endenicher Allee 60, Bonn

## Program:

14:00-15:00 |
Eva Lütkebohmert-Holtz (Universität Freiburg): A Structural Model for Credit and Liquidity Risk |

15:00-15:45 |
Coffee break |

15:45-16:45 |
Massimiliano Gubinelli (Université Paris Dauphine): The regularising effects of irregular functions |

17:00-18:00 |
Yuri Kondratiev (Universität Bielefeld): Statistical dynamics of continuous systems |

18:00-19:30 |
Get-together (Hausdorff Room, Endenicher Allee 60) Sergio Albeverio's 75th birthday |

## Abstracts

#### Eva Lütkebohmert-Holtz : A Structural Model for Credit and Liquidity Risk

We present a structural credit risk model that allows a firm to effectively monitor and manage its exposure to both insolvency and illiquidity risks inherent in its financing structure. Besides insolvency risk, the firmis exposed to funding liquidity risk (rollover risk) through possible runs by short-term creditors.

Credit spreads are derived endogenously such that rollover risk is minimized while equity holders have to bear the gains and losses from rolling over short-term debt. Moreover, market liquidity risk enters our model as asset price volatilities are subject to macro-economic shocks and influence creditors' risk attitudes and margin requirements.

#### Massimiliano Gubinelli: The regularising effects of irregular functions

We will discuss some examples of the "good" effects of "very bad", "irregular" functions. In particular we will look at non-linear differential (partial or ordinary) equations perturbed by some kind of noise. By defining a suitable notion of "irregular" noise we are able to show, in a quantitative way, that the more the noise is irregular the more the properties of the equation are better. Some examples includes: ODE perturbed by additive noise, linear stochastic transport equations and non-linear modulated dispersive PDEs. It is possible to show that the sample paths of Brownian motion or fractional Brownian motion and related processes have almost surely this kind of irregularity.

#### Yuri Kondratiev: Statistical dynamics of continuous systems

We describe a general concept of the statistical dynamics for Markov stochastic evolutions of interacting particle systems in the continuum.

A mesoscopic scaling limit algorithm for statistical dynamics leads to the kinetic description of considered systems. The latter is formulated in terms of nonlocal evolution equations for the densities. These equations define nonlinear Markov propagators and related nonlinear Markov processes.

We apply this approach to a number of models of interacting particle systems. In particular, certain models of spatial ecology describing notions of growth, expansion and aggregation in populations will be discussed.