This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by Carmona and Lacker, we consider a fully non-Markovian setting allowing for drift control, and interactions through the joint distribution of players’ states and controls. We provide first a new characterisation of mean-field equilibria as arising from solutions to a novel kind of McKean–Vlasov back- ward stochastic differential equations, for which we provide a well-posedness theory. We incidentally obtain there unusual existence and uniqueness results for mean-field equilibria, which do not require short-time horizon, separability assumptions on the coefficients, nor Lasry and Lions’s monotonicity conditions, but rather smallness conditions on the terminal reward. We then take advantage of this characterisation to provide non-asymptotic rates of convergence for the value functions and the Nash-equilibria of the N-player version to their mean-field counterparts, for general open-loop equilibria. This relies on new back- ward propagation of chaos results, which are of independent interest. This is a joint work with Ludovic Tangpi.

## Non-asymptotic convergence rates for mean-field games: weak formulation and McKean–Vlasov BSDEs.

## Learning in continuous time mean-field control problems

The theory and applications of mean-field games/control have stimulated a growing interest and generated important literature over the last decade since the seminal papers by Lasry/Lions and Caines, Huang, Malhamé. This talk will address some learning methods for numerically solving continuous time mean-field control problems, also called McKean-Vlasov control (MKV). In a first part, we consider a model-based setting, and present numerical approximation methods for the Master Bellman equation that characterises the solution to MKV, based on the one hand on particles approximation of the Master equation, and on the other hand on cylindrical neural networks approximation of functions defined on the Wasserstein space. The second part of the lecture is devoted to a model-free setting, a.k.a. reinforcement learning. We develop a policy gradient approach under entropy regularisation based on a suitable representation of the gradient value function with respect to parametrised randomised policies. This study leads to actor-critic algorithms for learning simultaneously and alternately value function and optimal policies. Numerical examples in a linear-quadratic mean-field setting illustrate our results.

## Moral hazard for time-inconsistent agents and BSVIEs

*We address the problem of Moral Hazard in continuous time between a Principal and an Agent that has time-inconsistent preferences. Building upon previous results on non-Markovian time-inconsistent control for sophisticated agents, we are able to reduce the problem of the principal to a novel class of control problems, whose structure is intimately linked to the representation of the problem of the Agent via a so-called extended Backward Stochastic Volterra Integral equation. We will present some results on the characterization of the solution to problem for different specifications of preferences for both the Principal and the Agent.*

## The Dark Side of Circuit Breakers

Market-wide trading halts, also called circuit breakers, have been widely adopted as part of the stock market architecture, in the hope of stabilizing the market during dramatic price declines. We develop an intertemporal equilibrium model to examine how circuit breakers impact market behavior and welfare. We show that a circuit breaker tends to lower the level of price and significantly alters its dynamics. In particular, as the price approaches the circuit breaker, its volatility rises drastically, accelerating the chance of triggering the circuit breaker -- the so-called ``magnet effect''. In addition, returns exhibit increasing negative skewness and positive drift, while trading activity spikes up. Our empirical analysis finds supportive evidence for the model's predictions. Moreover, we show that a circuit breaker can affect the overall welfare either negatively or positively, depending on the relative significance of investors' trading motives for risk sharing vs. irrational speculation. This is a joint work with Hui Chen, Anton Petukhov, and Jiang Wang.

## Exploration vs Exploitation: From stochastic control theory to Reinforcement Learning

## Conservation laws arising in the study of forward-forward Mean Field Games

In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). These models arise, for example, in the study of numerical schemes to approximate stationary states of MFGs. We establish a link between the forward-forward Mean-Field Games and a class of hyper- bolic conservation laws. Furthermore, we show how these models are connected to certain nonlinear wave equations. Finally, we investigate the existence of solutions and examine their long-time limit properties. Joint work with Diogo Gomes and Levon Nurbekyan.

## Smoothing Techniques Combined with Hierarchical Approximations for Efficient Option Pricing

When approximating the expectation of a functional of a stochastic process, in particular for option pricing purposes, the performance of numerical integration methods based on deterministic quadra- ture, quasi-Monte Carlo (QMC), or multilevel Monte Carlo (MLMC) techniques may critically depend on the regularity of the integrated. To overcome this issue, we introduce in [1,2,3] different smoothing tech- niques. In the first part of the talk, we will discuss our novel numerical smoothing approach [1,2] in which we combine root-finding methods with one-dimensional integration with respect to a single well-selected variable, focusing on cases where the discretization of the asset price dynamics is necessary. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). Our analysis in [1] demonstrates the advantages of combining numerical smoothing with AS- GQ and QMC methods over ASGQ and QMC methods without smoothing, and the Monte Carlo approach. Moreover, our analysis in [2] shows that our numerical smoothing improves the robustness (the kurtosis at deep levels becomes bounded) and complexity of the MLMC method. In particular, we recover the optimal MLMC complexities obtained for Lipschitz functionals.In the second part of the talk, we will discuss our efficient Fourier-based method in [3] for pricing European multi-asset options under L ́evy models. Given that the integrand in the frequency space often has higher regularity than in the physical space, we extend the one-dimensional Fourier valuation formula to the multivariate case and employ two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed heuristic optimization rule. These parameters ensure integrability and control the regularity class of the integrand. Second, we use sparsification and dimension-adaptivity techniques to accelerate the convergence of the numerical quadrature in high dimensions. We demonstrate the advantages of adaptivity and our damping parameter rule on the numerical complexity of the quadrature methods. Moreover, we reveal that our approach achieves substantial computational gains compared to the Monte Carlo method for different dimensions and parameter constellations.

## Many Player Games and Applications

We are organizing a workshop on Many Player Games and Applications in Berlin from August 29-31. This workshop brings together leadings experts from mathematics, economics, operations research and engineering departments to discuss recent developments in the theory of many player games and their applications to finance and engineering. The event follows up on a series of previous events, held at the Center for Interdisciplinary research (ZiF). It is sponsored by the CRC TRR 190 (Berlin-Munich), the IRTG 2544 (Berlin-Oxford) and the SFB 1238 (Bielefeld). Confirmed speakers include

- Peter Cains (McGill),
- Diogo Gomes (KAUST),
- Johannes Muhle-Karbe (Emperial),
- Mathias Blonski (Frankfurt)
- Julio Backhoff-Veraguas (Vienna)
- Martin Herdegen (Warwick)
- Sujoy Mukerji (Queen Mary U)
- Chao Zhou (Singapore)

and many others. More information is available on the conference webpage.

## Stochastic and rough analysis

Meeting held in Berlin, August 22 - August 26.

Scientific Committee: P Bank, P Friz, U Horst, S Paycha, N Perkowski, W Stannat

Organizing Institutions: FU, HU and TU Berlin, U Potsdam and WAS Berlin

Support: DFG FOR 2402, ERC GPSART GPSART 683164, Berlin-Oxford IRTG 2544

This meeting is devoted to recent progress on stochastic & rough analysis and its applications. It is also a

welcome opportunity to congratulate our colleaque Michael Scheutzow (FOR 2402) on his retirement.

## On the martingale projection of a Brownian motion given initial and terminal marginals

In one of its dynamic formulations, the optimal transport problem asks to determine the stochastic process that interpolates between given initial and terminal marginals and is as close as possible to the constant-speed particle. Typically, the answer to this question is a stochastic process with constant-speed trajectories. We explore the analogue problem in the setting of martingales, and ask: what is the martingale that interpolates between given initial and terminal marginals and is as close as possible to the constant-volatility particle? The answer this time is a process called ’stretched Brownian motion’, a generalization of the well-known Bass martingale. After introducing this process and discussing some of its properties, I will present current work in progress (with Mathias Beiglbo ̈ck, Walter Schachermayer and Bertram Tschiderer) concerning the fine structure of stretched Brownian motions.