University of Freiburg

A conjecture due to Zilber predicts that the complex exponential field is quasiminimal: that is, that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the complex exponential function are countable or cocountable.
Zilber showed that this conjecture would follow from Schanuel's Conjecture and an existential closedness type property asserting that certain systems of exponential-polynomial equations can be solved in the complex numbers; later on, Bays and Kirby were able to remove the dependence on Schanuel's Conjecture, shifting all the focus to the existence of solutions. In this talk, I will discuss recent work about the quasiminimality of a reduct of the complex exponential field, that is, the complex numbers expanded by multivalued power functions. This is joint work with Jonathan Kirby.

We present a dynamic bank run model for liquidity risk where a financial institution finances its risky assets by a mixture of short- and long-term debt. The financial institution is exposed to liquidity risk as its short-term creditors have the possibility not to renew their funding at a finite number of rollover dates. Besides, the financial institution can default due to insolvency at any time until maturity. We compute both insolvency and illiquidity default probabilities in this multi-period setting. We show that liquidity risk is increasing in the volatility of the risky assets and in the ratio of the return that can be earned on the outside market over the return for short-term debt promised by the financial institution. Moreover, we study the influence of the capital structure on the illiquidity probability and derive that illiquidity risk is increasing with the ratio of short-term funding.