## 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

## Hedging with market and limit orders

## Asset-pricing puzzles and price-impact

We solve in closed-form a continuous-time Nash equilibrium model in which a finite number of investors with exponential utilities continuously consume and trade strategically with price-impact. Compared to the analogous Pareto-efficient equilibrium model, price-impact has an amplification effect on risk-sharing distortions that helps resolve the interest rate puzzle. However, price impact has little quantitative effect on the equity premium and stock-return volatility puzzles. Joint work with Xiao Chen (Rutgers), Jin Hyuk Choi (UNIST), and Duane J. Seppi (CMU).

## Portfolio Optimisation within a Wasserstein Ball

We consider the problem of active portfolio management where a loss-averse and/or gain-seeking investor aims to outperform a benchmark strategy's risk profile while not deviating too much from it. Specifically, an investor considers alternative strategies that have a specified copula with the benchmark and whose terminal wealth lies within a Wasserstein ball surrounding it. The investor then chooses the alternative strategy that minimises a distortion risk measure. We prove that an optimal dynamic strategy exists and is unique, and provide its characterisation through the notion of isotonic projections. Finally, we illustrate how investors with different risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped distortion risk measures, and lower- and upper-tail risk measures as examples. We find that investors' optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving some aspects of the benchmark. This is joint work with Silvana Pesenti, U. Toronto.