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.

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

## 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.

## Understanding the Time-Space Evolution of Economic Activities: Recent Mathematical Models and their application

The goal of this talk is to present some recent models on the time evolution of most important economic variables (e.g. consumption and capital) across different locations, taking into account space heterogeneity. In particular we focus on two recent papers looking at the macro level where there is one planner which, in a spatial Ramsey setting, maximizes utility across space with heterogeneous productivity in a deterministic (paper with R. Boucekkine, G. Fabbri, S. Federico) or in a stochastic setting (paper with M. Leocata). If time allows we will also introduce a mean field game model looking at the micro level where the agents move across space maximizing their own utility which also depends on the choices of the other agents.

## Mean-field Langevin dynamics and neural networks

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## A synthetic model for ALM in life insurance and numerical methods for SCR computation

We introduce a synthetic ALM model that catches the main specificity of life insurance contracts. First, it keeps track of both market and book values to apply the regulatory profit sharing rule. Second, it introduces a determination of the crediting rate to policyholders that is close to the practice and is a trade-off between the regulatory rate, a competitor rate and the available profits. Third, it considers an investment in bonds that enables to match a part of the cash outflow due to surrenders, while avoiding to store the trading history. We use this model to evaluate the Solvency Capital Requirement (SCR) with the standard formula, and illustrate the importance of matching cash-flows.

Then, we focus on the problem of evaluating the SCR at future dates. For this purpose, we study the multilevel Monte-Carlo estimator for the expectation of a maximum of conditional expectations. We obtain theoretical convergence results that complements the recent work of Giles and Goda. We then apply the MLMC estimator to the calculation of the SCR at future dates and compare it with estimators obtained with Least Squares Monte-Carlo or Neural Networks. Last, we discuss the effect of the portfolio allocation on the SCR at future dates.