Das Institut für Mathematische Wirtschaftsforschung veranstaltet im Rahmen des Bielefeld Stochastic Afternoon regelmäßig Seminare zum Thema Finanzmathematik. Das Programm des aktuellen Semesters finden Sie hier.
Boualem Djehiche (KTH Royal Institute of Technology, Stockholm)
Titel: Infinite Horizon Stochastic Impulse Control with Delay
Abstract: I will highlight recent results on a class of infinite horizon impulse control problems with execution delay when the dynamics of the system is described by a general stochastic process adapted to the Brownian filtration, for which we establish the existence of an optimal strategy over all admissible strategies.
Stephan Eckstein (Hamburg University)
Titel: Calculating robust price bounds using generative adversarial nets
Abstract: To calculate robust price bounds, one has to identify risk-neutral distributions that correspond to extreme cases given the constraints on the market, leading for instance to the martingale optimal transport problem. This talk presents a method to solve such optimization problems using neural networks and a MinMax approach, which is in close relation to generative adversarial networks. We showcase how regularization methods can justify the utilization of neural networks in this framework, and further show how numerical tools from the area of generative adversarial networks can be adapted to obtain stable numerical solutions to robust pricing problems. This talk is mainly based on joint work together with Luca De Gennaro Aquino.
An Chen (University of Ulm)
Title: Success and failure of the financial regulation on a surplus-driven financial company
Abstract: This paper studies an optimal asset allocation problem for a surplus-driven financial institution facing a quantile-based constraint, i.e., under a Value-at-Risk (VaR) or an Average Value-at-Risk (AVaR) constraint or a shortfall-based constraint, i.e., an expected shortfall or an expected discounted shortfall constraint. We obtain closed-form solutions to the optimal wealth for the non-concave utility maximization problem under constraints. We find that the quantile- and shortfall-based regulation can effectively reduce the probability of default for a surplus-driven financial institution. However, the liability holders' benefits cannot be fully protected under either type of regulation. This is based on a joint work with Mitja Stadje and Fangyuan Zhang.
Felix-Benedikt Liebrich (Leibniz Universität Hannover)
Titel: Law-Invariance vs. Ambiguity: Collapse to the mean beyond convexity
Abstract: The theme of the talk is the "collapse to the mean". The term refers to the inability of law-invariant functionals to simultaneously incorporate pure risk -- understood as "linearity" or additivity -- and ambiguity. In many cases, the expectation with respect to the physical measure turns out to be the only law-invariant functional admitting an unambiguous argument in the preceding sense.
The first part of the talk discusses this phenomenon for law-invariant nonconvex Choquet integrals and consistent risk measures. While convexity is not an assumption, it is seen to be implied ex ante if an unfortunate combination of axioms is imposed on the functionals in question.
The second part deals with cooperative game theory and aims to formalise a decomposition of the worth associated with coalitions into one part explained by risk and one explained by ambiguity. To this end, we construct a large space of not necessarily convex games containing all bounded additive set functions. We thereon define the "leveling operator", an explicit rule that serves as a linear projection minimising the distance of a game to the space of signed measures, thereby providing "risk equivalents" of games. For law-invariant games, we confirm the "collapse to the mean"-based intuition showing that this risk equivalent is always a multiple of the phyiscal measure.
The talk is based on joint work with Massimiliano Amarante, Mario Ghossoub, and Cosimo Munari.
Jodi Dianetti (Universität Bielefeld)
Titel: Stochastic singular control: Existence, characterization and approximation of solutions in cost minimization problems and games
Abstract: For a class of optimal stochastic control problems with singular controls, we characterize the optimal control as the unique solution to a related Skorokhod reflection problem. We prove that the optimal control only acts when the underlying diffusion attempts to exit the so-called waiting region, and that the direction of this action is prescribed by the derivative of the value function. We next consider problems concerning existence and approximation of equilibria in N-player stochastic games and mean field games of singular control. In a not necessarily Markovian setting, we establish the existence of Nash and mean field equilibria for games with submodular costs via Tarski's fixed point theorem. This approach allows to prove that there exist minimal and maximal equilibria which can be obtained through a simple learning procedure based on the iteration of the best-response-map. Finally, we analyse stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and an ergodic performance criterion. We prove existence and uniqueness of the mean field equilibria, which are completely characterized through nonlinear equations.
This talk is based on joint works with Haoyang Cao, Giorgio Ferrari, Markus Fischer and Max Nendel.
Stéphane Villeneuve (Toulouse School of Economics)
Titel: Gaussian Agency problems and Linear Contracts
Abstract: How to explain the use of dynamic contracts which are linear in end-of-period outputs when the agent controls a process that exhibits memory? This paper addresses this question by extending the classical model of Holmstrom-Milgrom (1987) to general Gaussian settings where the output dynamics are neither semi-martingales, nor Markov processes. The class of principal-agent models we introduce is rich enough to encompass dynamic agency models with memory and also allows us to go beyond the usual continuous-time framework which generally solves for the optimal contract by the means of a Hamilton-Jacobi-Bellman equation. Our main contribution is to show that this setting allows surprisingly for optimal linear contracts in observable outcomes with a non constant optimal level of effort.
Dies ist ein gemeinsames Projekt zusammen mit Eduardo Abi Jaber from Université Pantheon-Sorbonne.