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Math. Finance Seminar (Summer Semester 2024)

The Center for Mathematical Economics organizes regularly seminars on Mathematical Finance / Financial Economics within the Bielefeld Stochastic Afternoon. This term's program is below

April 24, 2024 (Time: 2-4 pm, Location: V10-122)

First Speaker: Patrizia Semeraro (Polytechnic University of Torino)

Title: Factor-based subordination for multivariate asset return models

Abstract: We build a theoretical framework for multivariate time-changed Brownian motions. The change of time is constructed to incorporate both a time transform common to all assets and an idiosyncratic one. We first consider the case in which the change of time is a subordinator.

The resulting processes, named factor-based-models, belong to the Lèvy class and extend some well known subordinated Lèvy processes, as the normal inverse Gaussian process. We then use Sato subordinators to extend time-changed Brownian motions to  additive processes with inhomogeneous increments. The construction is designed to obtain a multivariate process with the same unit time distribution as the factor-based-model and with time varying correlations.

We show the importance to model time-inhomogeneity in multi-FX option pricing, by calibrating the NIG specification of factor-based-models on currency triangles. This work is partially joint with Elisa Luciano and partially joint with Giovanni Amici and Laura Ballotta.

Second Speaker: Federico Cannerozzi (University of Milano)

Title: Coarse correlated equilibria in mean field games

Abstract: Despite their popularity, Nash equilibria (NE) are not the only notion of equilibria between competitive rational players. A good alternative is given by coarse correlated equilibria (CCE), as they may lead to higher expected payoffs, thus being more efficient. CCEs feature a moderator who randomly selects a strategy profile for the players, correlating their strategies without requiring them to cooperate.

We extend this concept to finite horizon continuous-time stochastic mean field games (MFGs), introducing coarse correlated solutions to the MFG; we prove existence and show that approximate N-player CCEs can be constructed starting from a coarse correlated solution. Then, motivated by a simple model of irreversible investment, we consider a stationary MFG with singular controls and ergodic payoff, in which the representative player interacts with the average of the long-time distribution of the population. We provide simple classes of CCEs in the stationary MFG, and draw a comparison between CCEs, NEs and social optima; to compute the latter, we solve the associated stationary mean field control problem. Finally, we address the backward approximation problem.

This talk is based on joint works Luciano Campi (University of Milan "La Statale"), Markus Fischer (University of Padua) and Giorgio Ferrari (University of Bielefeld).


June 5, 2024 (Time: 2 - 4 pm, Location: V10-122)

First Speaker: Yufei Zhang (Imperial College London)

Title: Alpha potential games: A new paradigm for N-player games


Static potential games, pioneered by Monderer and Shapley (1996), are non-cooperative games in which there exists an auxiliary function called static potential function, so that any player's change in utility function upon unilaterally deviating from her policy can be evaluated through the change in the value of this potential function. The introduction of the potential function is powerful as it simplifies the otherwise challenging task of finding Nash equilibria for non-cooperative games: maximizers of potential functions lead to the game's Nash equilibria.
In this talk, we propose an analogous and new framework called $\alpha$-potential game for dynamic $N$-player games, with the potential function in the static setting replaced by an $\alpha$-potential function. We present an analytical characterization of $\alpha$-potential functions for any dynamic game. For stochastic differential games in which the state dynamic is a controlled diffusion, $\alpha$ is explicitly identified in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. 

We provide detailed analysis for games with mean-field interactions, distributed games, and crowd aversion games, for which $\alpha$ is shown to decay to zero as the number of players goes to infinity, even with heterogeneity in state dynamics, cost functions, and admissible strategy classes. We also show $\alpha$ is capable of capturing the subtle difference between the open-loop and closed-loop strategies.

Second Speaker: David Criens (Universität Freiburg)

Title: TBA

Abstract: TBA

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