## Risk Quantification, Optimal Stopping and Reflected BSDEs for a Class of Informational Markets

## Mean-field liquidation games with market drop-out

## Equilibrium in Infinite-Dimensional Stochastic Games with Mean-Field Interaction

We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic objective functional includes linear operators acting on square-integrable controls. We propose a novel approach for deriving explicitly the Nash equilibrium of the game by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their closed-form solution. Furthermore, by proving stability results for the system of Fredholm equations, we derive the convergence of the equilibrium of the N-player game to the corresponding mean- field equilibrium. As a by-product of our results we also derive epsilon-Nash equilibrium for the mean- field game and we show that the conditions for existence of an equilibrium in the mean-field limit are significantly less restrictive than in the finite-player game. Finally we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay and optimal liquidation games with transient price impact.

The talk is based on a joint work with Eduardo Abi-Jaber and Moritz Voss.

## Reinforcement Learning for Convex MDPs with application to hedging and pricing

Convex MDPs generalize the standard reinforcement learning (RL) problem formulation to a larger framework that includes many supervised and unsupervised RL problems, such as apprenticeship learning, constrained MDPs, and so-called ‘pure exploration’. We consider the reformulation of the convex MDP problem as a min-max game involving policy and cost (negative reward) ‘players’, using duality. Then we study the application of this strategy to pricing and hedging in Pricing/Hedging under optimized certainty equivalents (OCEs) which is a family of risk measures widely used by practitioners and academics. This class of risk measures includes many important examples, e.g. entropic risk measures and average value at risk.

## Stability and approximation of projection filters

Nonlinear filtering is a central mathematical tool in understanding how we process information. Sadly, the equations involved are often very high dimensional, which may lead to difficulties in applications. One possible resolution (due to D. Brigo and collaborators) is to replace the filter by a low-dimensional approximation, with hopefully small error. In this talk we will see how, in the case where the underlying process is a finite-state Markov Chain, results on the stability of filters can be strengthened to show that this introduces a well-controlled error, leveraging tools from information geometry. (Based on joint work with Eliana Fausti)

## Statistical inference for rough volatility

In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter H < 0.5. In this paper, we derive a consistent and asymptotically mixed normal estimator of H based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

## Practical Applications of Machine Learning in Risk and Pricing

Machine Learning (ML) provides techniques for universal function approximation. In this talk, we apply such techniques to the acceleration of complex derivatives pricing, focusing on Value-at- Risk computations for Bermudan interest rate options. We introduce different applicable ML methods, and we present results from our client projects. Moreover, we propose ways to address regulatory requirements via the model lifecycle process.

## When to efficiently rebalance a portfolio

A constant weight asset allocation is a popular investment strategy and is optimal under a suitable continuous model. We study the tracking error for the target continuous rebalancing by a feasible finite-time rebalanc- ing under a general multi-dimensional Brownian semimartingale model of asset prices. In a high-frequency asymptotic framework, we derive an asymptotically efficient strategy among simple predictable processes.

## Mean-field molecular dynamics derived from quantum mechanics

Quantum observables determine for instance the stress tensor and heat flux in fluid dynamics. In the talk I will show which properties are used for approximating such quantum observables by classical molecular dynamics and what is new when mean-field molecular dynamics improves the classical setting, using features of the Gibbs distribution.

## Optimal Investment and Consumption with Epstein-Zin Stochastic Differential Utility and Proportional Transaction Costs

We study the optimal investment-consumption problem for an agent whose preferences are governed by Epstein–Zin stochastic differential utility and who invests in a constant-parameter Black–Scholes– Merton market. We assume that purchases and sales of the risky asset are subject to proportional transaction costs. We fully characterise all parameter combinations for which the problem is well posed (which may depend on the level of transaction costs). We also provide a full verification argument that relies on no additional technical assumptions and uses primary methods only. Even in the special case of power utility, our arguments are significantly simpler and more elegant than the results in the extant literature. A novel key idea is to parametrise consumption and the value function in terms of the shadow fraction of wealth. The talk is based on joint work with David Hobson and Alex Tse.