Theoretical solutions to stochastic optimal control problems are well understood in many scenarios, however their practicability suffers from the assumption of known dynamics of the underlying stochastic process. This raises the challenge of developing purely data-driven strategies, which we explore for ergodic singular control problems associated to continuous diffusions and Lévy processes. In case of diffusion processes, the primary challenge consists of solving an exploration/exploitation tradeoff based on a minimax optimal estimation procedure of the optimal reflection boundaries with data collected in the exploration periods. Even though for Lévy processes such exploration/exploitation problem does not occur due to spatial homogeneity of the process, in this scenario we face the statistical challenge of estimating a generator functional of a subordinator associated to the Lévy process which a) cannot be observed directly from the data and b) is non-ergodic in time. We solve this problem by considering a space/time transformation of the process in form of its overshoots such that we can work with a spatially ergodic process that allows the construction of an unbiased estimator of the generator functional determining the optimal reflection boundary. We compare the results of our statistical procedure with those from deep learning approaches.

## Learning to reflect: data-driven stochastic optimal control strategies for diffusions and Lévy processes

## Stochastic Stability for the Utility Maximization Problem

## Itô’s formula for semimartingales on flows of probability measures

## Neural Network based Approximation Algorithm for nonlinear PDEs with Application to Pricing

## Stochastic Volterra equations: theory, numerics and control

## Mean-field reflected backward stochastic differential equations

In this talk, we study a class of reflected backward stochastic differential equations (BSDEs) of mean-field type, where the mean-field interaction in terms of the expected value $\E[Y]$ of the $Y$-component of the solution enters both the driver and the lower obstacle. Under mild Lipschitz and integrability conditions on the coefficients, we obtain the well-posedness of such a class of equations. Under further monotonicity conditions we show convergence of the standard penalization scheme to the solution of the equation. This class of models is motivated by applications in pricing life insurance contracts with surrender options.

## Viscosity Solutions of Stochastic Hamilton-Jacobi-Bellman Equations and Applications

Fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equations will be discussed for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and the value function of the optimal stochastic control problem is the unique viscosity solution to the associated stochastic HJB equation. Applications in mathematical finance and some recent developments will be reported as well.

## Looking at the smile from Roger Lee's shoulders

## Submodular mean field games: Existence and approximation of solutions

We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. The mean field game is first defined over ordinary stochastic controls, then extended to relaxed controls. Our approach also allows to treat a class of submodular mean field games with common noise in which the representative player at equilibrium interacts with the (conditional) mean of its state's distribution.

This talks is based on a joint work together with Giorgio Ferrari, Markus Fischer and Max Nendel