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.

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

## A probabilistic approach to the convergence of large population games to mean field games

## Optimal Investment and Equilibrium Pricing under Ambiguity

## Convergence of Optimal Strategies in a Multivariate Black Scholes Type Market with Model Uncertainty on the Drift

## Multidimensional singular control and related Skorokhod problem: sufficient conditions for the characterization of optimal controls

We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The considered optimization problems concern the minimization of a discounted cost functional over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. In a multidimensional setting, we prove that the optimal control acts only 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. Our approach is based on the study of a suitable monotonicity property of the derivative of the value function through its interpretation as the value of an optimal stopping game. Such a monotonicity allows to construct nearly optimal policies which reflect the underlying diffusion at the boundary of approximating waiting regions. The limit of this approximation scheme then provides the desired characterization. Our result applies to a relevant class of linear-quadratic models, among others. Furthermore, it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature, where this important aspect was only partially addressed. This talk is based on a joint work with Giorgio Ferrari.

## Control and stopping mean-field games: the linear programming approach

In this talk, we present recent results on the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via other approaches used in the previous literature. We then present a fictious play algorithm to approximate the mean-field game population dynamics in the context of the linear programming approach. Finally, we give an application of the theoretical and numerical contributions introduced in the first part of the talk to an entry-exit game in electricity markets. The talk is based on several works, joint with R. A ̈ıd, G. Bouveret, M. Leutscher and P. Tankov.

## Multilevel Picard approximations for high-dimensional semilinear parabolic PDEs and further applications

We present the multilevel Picard approximation method for high-dimensional semilinear parabolic PDEs which in particular appear in the pricing of financial derivatives. A key idea of our method is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of the proposed method grows polynomially both in the dimension and in the reciprocal of the required accuracy. Moreover, we present further applications of the multilevel Picard approximation method and illustrate its efficiency by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe Von Wurstemberger.

## Well-posedness of path-dependent semilinear parabolic master equations

Master equations are partial differential equations for measure-dependent unknowns, and are introduced to describe asymptotic equilibrium of large scale mean-field interacting systems, especially in stochastic games and control. In this paper we introduce new semilinear master equations whose unknowns are functionals of both paths and path measures. They include state-dependent master equations, path-dependent partial differential equations (PPDEs), history information dependent equations and time inconsistent (e.g. time-delayed) equations, which naturally arise in applications (e.g. option pricing, risk control). We give a classical solution to the master equation by introducing a new notation called strong vertical derivative (SVD) for path-dependent functionals, inspired by Dupire’s vertical derivative, and applying forward-backward stochastic system argument. This talk is based on a joint work with Shanjian Tang.

## Quantum Computing in Machine Learning

Im Rahmen des Forschungsseminars findet ein d-fine day statt. Neben einer allgemeinen Vorstellung des Unternehmens mit anschliessendem get-together wird es einen Vortrag zum Thema "Quantum Machine Learning – Approaches, Applications and Results" geben. Interessierte Studierende sind herzlich eingeladen.

## Efficient Allocations under Ambiguous Model Uncertainty

We investigate consequences of model uncertainty (or ambiguity) on ex ante efficient allocations in an exchange economy. The ambiguity we consider is embodied in the model uncertainty perceived by the decision maker: they are unsure what would be the appropriate probability measure to apply to evaluate contingent consumption contingent plans and keep in consideration a set of alternative probabilistic laws. We study the case where the typical consumer in the economy is ambiguity-averse with smooth ambiguity preferences and the set of priors $\mathcal{P}$ is point identified, i.e., the true law $p\in \mathcal{P}$ can be recovered empirically from observed events. Differently from the literature, we allow for the case where the aggregate risk is ambiguous and agents are heterogeneously ambiguity averse. Our analysis addresses, in particular, the full range of set-ups where under expected utility the Pareto efficient consumption sharing rule is a linear function of the aggregate endowment. We identify systematic differences ambiguity aversion introduces to optimal sharing arrangements in these environments and also characterize the representative consumer. Furthermore, we investigate the implications for the state-price function, in particular, the effect of heterogeneity in ambiguity aversion.