Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, a direct dependence on past absorptions has been introduced in the drift of players' state dynamics. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth. Therefore, the mean-field interaction among the players takes place in two ways: via the empirical sub-probability measure of the surviving players and through a process representing the fraction of past absorptions over time. Moreover, relaxing the boundedness of the coefficients allows for more realistic dynamics for players' private states. We prove existence of solutions of the mean-field game in strict as well as relaxed feedback form. Finally, we show that such solutions induce approximate Nash equilibria for the N-player game with vanishing error in the mean-field limit as $N \to \infty$. This talk is based on a joint work with M. Ghio and G. Livieri (SNS Pisa).

## N-player games and mean-field games with smooth dependence on past absorptions

## 4th Berlin-Princeton-Singapore Workshop on Quantitative Finance

The 4th 4th Berlin-Princeton-Singapore Workshop on Quantitative Finance takes place March 18-20, 2019 at NUS. The workshop is supported through the HU's profile partnership program with NUS and Princeton University. More information will be made available soon.

## Dynamic Noisy Rational Expectations Equilibrium with Insider Information

In this talk, we study equilibria in multi-asset and multi-agent continuous-time economies with asymmetric information. We establish existence of two equilibria. First, a full communication one where the informed agents' signal is disclosed to the market, and static policies are optimal. Second, a partial communication one where the signal disclosed is ane in the informed and noise traders' signals. Here, information asymmetry creates demand for a dark pool with endogenous participation where private information trades can be implemented. Markets are endogenously complete and equilibrium prices have a three factor structure. Results are valid for multiple dimensions; constant absolute risk averse investors; fundamental processes following a general diffusion; non-linear terminal payoffs, and non-Gaussian noise trading. Asset price dynamics and public information flows are endogenous, and are established using multiple filtration enlargements, in conjunction with predictable representation theorems for random analytic maps. Rational expectations equilibria are special cases of the general results.

## Beratung bei d-fine – analytisch. technologisch. quantitativ.

Die d-fine GmbH ist seit über 15 Jahre mit ihrem Konzept, Naturwissenschaftler und Mathematiker (m/w/d) in der Beratung für den Finanzdienstleistungssektor in Deutschland und Europa einzusetzen, sehr erfolgreich. Bekannt u.a. durch Werbung im Physik Journal, Fachliteratur und Stipendienförderungen für Nachwuchswissenschaftler, erfreut sich d-fine einem regen Mitarbeiterwachstum, das in naher Zukunft die 1.000-Personen-Marke überschreiten wird. Dieses Wachstum spiegelt den Erfolg von d-fine sowohl in seinen etablierten Geschäftsfeldern als auch in neuen Themenbereichen wie Autonomes Fahren, Health Care, Machine Learning, Künstliche Intelligenz und Blockchain-Technologie wider. In diesem Vortrag stehen der berufliche Werdegang und die praktische Erfahrung des Vortragenden Dr. rer. nat. Patrick Mack, Alumnus des Karlsruher Instituts für Technologie, als Manager bei d-fine im Vordergrund. Herr Dr. Mack hat unter seinem inzwischen an der HU Berlin forschenden Doktorvater Prof. Dr. Kurt Busch in der Photonics Group an der Physikfakultät des KIT in 2011 promoviert. Der Vortrag ist eine kurzweilige first-hand Schilderung der Projekt- und Reiseerfahrungen von Herrn Dr. Mack bei d-fine und steht beispielhaft für die vielen attraktiven Beschäftigungsmöglichkeiten, die d-fine Berufseinsteigern bietet.

## A mean-field game approach to price formation

Here, we introduce a price-formation model where a large number of small players can store and trade an asset. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition. We establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well-defined and it is a Lipschitz function of time. Then, we study linear-quadratic models that can be solved explicitly.

## Large Tournament Games

We consider a stochastic tournament game in which each player is rewarded based on her rank in terms of the completion time of her own task and is subject to cost of effort. When players are homogeneous and the rewards are purely rank dependent, the equilibrium has a surprisingly explicit characterization, which allows us to conduct comparative statics and obtain explicit solution to several optimal reward design problems. In the general case when the players are heterogenous and payoffs are not purely rank dependent, we prove the existence, uniqueness and stability of the Nash equilibrium of the associated mean field game, and the existence of an approximate Nash equilibrium of the finite-player game.

Joint work with Jakša Cvitanić and Yuchong Zhang.

## Mean field games with a major player

Mean field games with a major agent study optimal control problems with infinitely many small controllers facing a major controller. The "value function" of the agents then satisfy a nonlinear nonlocal system of partial differential equations stated in the space of measures. In this joint work with Marco Cirant (U. Padova) and A. Porretta (U. Rome Tor Vergata) we explain how to build short time a classical solution for this system and use the solution to prove the mean field limit of the associated N player game as the number N of the players tends to infinity.

## Probabilistic numerical methods for MFC and MFG based on deep learning

We propose two probabilistic numerical methods for mean field type problems based on deep learning. The first method amounts to solve mean field control problems (i.e., problems of optimal control of McKean-Vlasov dynamics) by learning the optimal control using Monte-Carlo samples and stochastic gradient descent. This can be done in a somewhat brute force fashion thanks to deep learning. The second method deals with forward-backward stochastic differential equation (FBSDE) systems of mean field type. As such, this method can be applied to both mean field control problems and mean field games. We rephrase the problem of finding a solution to a generic mean field FBSDE system as a certain mean field control problem, and we then apply a variant of the first method proposed. Several numerical examples will be provided. This is joint work with René Carmona (Princeton University).

## Financial Contagion in a Generalized Stochastic Block Model

One of the most defining features of modern financial networks is their inherent complex and intertwined structure. In particular the often observed core-periphery structure plays a prominent role. Here we study and quantify the impact that the complexity of networks has on contagion effects and system stability, and our focus is on the channel of default contagion that describes the spread of initial distress via direct balance sheet exposures. We present a general approach describing the financial network by a random graph, where we distinguish vertices (institutions) of different types – for example core/periphery – and let edge proba- bilities and weights (exposures) depend on the types of both the receiving and the sending vertex. Our main result allows to compute explicitly the systemic damage caused by some initial local shock event, and we derive a complete characterization of resilient respectively non-resilient financial systems. Due to the random graphs approach these results bear a considerable robustness to local uncertainties and small changes of the network structure over time. In particular, it is possible to condense the precise micro-structure of the network to macroscopic statistics. Applications of our theory demonstrate that indeed the features captured by our model can have significant impact on system stability; we derive resilience conditions for the global network based on subnetwork conditions only. (Joint with Thilo Meyer-Brandis, Konstantinos Panagiotou and Daniel Ritter (LMU)

## Epstein-Zin utility and its utility maximization

Epstein-Zin utility is widely used in many macro-economics and asset pricing models because it decouples risk aversion and elasticity of intertemporal substitution. In the first part of the talk, we will review results on existence and uniqueness of finite horizon Epstein-Zin utilities. Then we will present new results on infinite horizon Epstein-Zin utilities. In the second part of the talk, we will consider an optimal consumption and investment problem for Epstein-Zin utilities from two approaches: control of BSDEs and convex duality.