## Deep Optimal Stopping

## The entry and exit game in the electricity markets: a mean-field game approach

## Mean Field Utility Maximization Game with Partial Information

We study a mean field utility maximization game, where each player manages a stock whose return process depends on a hidden factor, which cannot be observed by the manager. The manager needs to infer the return process and rewrite the dynamics of the stock price based on the information available to her. Moreover, each manager is concerned not only with her own terminal wealth but also with the relative performance of her competitiors. We use the probabilistic approach to consider exponential and power utilities. Due to the mean field interaction and the nature of the game, the FBSDE systems characterizing the equilibria of our problem become coupled mean-field FBSDEs with possibly quadratic growth. We establish the well-posedness result of the mean-field FBSDEs in some suitable BMO space; firstly we work on a short time interval and secondly the local solution is extended to an arbitrary interval by considering the corresponding variational FBSDEs.

## Understanding the dual formulation for the hedging of path-dependent options with price impact

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in a path-dependent setting and under weak regularity conditions, that any solution to this dual problem actually allows one to construct explicitly a perfect hedging portfolio. From a pure probabilistic point of view, our approach also allows one to exhibit solutions to a specific class of second order forward backward stochastic differential equations, in the sense of Cheridito et al. (2007). Existence of a solution to the dual optimal control problem is also addressed in particular settings. As a by-product of our arguments, we prove a version of Itô’s Lemma for path-dependent functionals that are only Cˆ{0,1} in the sense of Dupire.

## Some Explicit Results on Dynkin Games with Incomplete and Asymmetric Information

In this talk I will consider two types of Dynkin game with non-standard information structures. The first one is a zero-sum game between two players who observe a geometric Brownian motion but in which the minimiser knows the drift of the process whereas the maximiser doesn't know it. We construct an explicit Nash equilibrium in which the uninformed player uses a pure strategy and the informed player uses a randomised strategy. The second game is a non-zero sum game between two agents interested in the purchase of the same asset. Neither of the two players knows with certainty whether their competitor is `active' and in that sense that they have uncertain competition. Also in this case we construct explicitly a Nash equilibrium in which both players randomise their strategy.

## Dynamic trading under integer constraints

We first review results on arbitrage theory for some notions of "simple" strategies, which do not allow continuous portfolio rebalancing by arbitrary amounts. Then, the focus of the talk is on trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer arbitrage free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. The set of prices of a contingent claim is not necessarily an interval, but is either empty or dense in an interval. We also discuss superhedging with integral portfolios. Joint work with Paul Eisenberg.

## The Mean Field Schrödinger Problem

I will introduce the mean field Schrödinger problem, concerned with finding the most likely evolution of a cloud of interacting Brownian particles conditionally on their initial and final configurations. New energy dissipation estimates are shown, yielding exponential convergence to equilibrium as the time between initial and final observations grows to infinity. The method reveals novel functional inequalities involving the mean field entropic cost, as well as an interesting connection with the theory of PDEs. (Joint work with Giovani Conforti, Ivan Gentil and Christian Léonard.)

## Polynomial processes - a universal modeling class

We introduce polynomial jump-diffusions taking values in an arbitrary Banach space via their infinitesimal generator. We obtain two representations of the (conditional) moments in terms of solution of systems of ODEs. We illustrate the wide applicability of these formulas by analyzing several state spaces. We start by the finite dimensional setting, where we recover the well-known moment formulas. We then study the probability-measure valued setting, where we also obtain an existence result for the corresponding martingale problems. Moving to more recent results, we consider (potentially rough) forward variance polynomial models and we illustrate how to use the moment formulas to compute prices of VIX options. Finally, we show that the signature process of a d-dimensional Brownian motion is polynomial and derive its expected value via the polynomial approach. This is in fact just an illustrative example: the same applies to solutions of every SDE with analytic coefficients.

## Should I invest in the market portfolio?

Guided by stylised facts and inspired by Robert Fernholz' stochastic portfolio theory, we present a parsimonious stationary diffusion model for the entire stock market. Its ultimate purpose is to decide whether there is a simple more efficient alternative to the market portfolio. At this stage we discuss the qualitative implications of the model. A crucial role is played by the speed measure and by the local time at the boundary of the support of the diffusion process under consideration.

## Optimal Installation of Solar Panels with Price Impact: a Solvable Singular Stochastic Control Problem

We consider a price-maker company which generates electricity and sells it in the spot market. The company can increase its level of installed power by irreversible installations of solar panels. In absence of any actions of the company, the electricity's spot price evolves as an Ornstein-Uhlenbeck process, and therefore it has a mean-reverting behavior. The current level of the company's installed power has a permanent impact on the electricity's price and affects its mean-reversion level. The company aims at maximizing the total expected profits from selling electricity in the market, net of the total expected proportional costs of installation. This problem is modeled as a two-dimensional degenerate singular stochastic control problem in which the installation strategy is identified as the company's control variable. We follow a guess-and-verify approach to solve the problem. We find that the optimal installation strategy is triggered by a curve which separates the waiting region, where it is not optimal to install additional panels, and the installation region, where it is. Such a curve depends on the current level of the company's installed power, and is the unique strictly increasing function which solves a first-order ODE. While studying the ODE, we obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions.