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.

## Dynamic trading under integer constraints

## 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.

## MOT Duality and Robust Finance

Without assuming any probabilistic price dynamics, we consider a frictionless financial market given by the Skorokhod space, on which some financial options are liquidly traded. In this model-free setting we show various pricing-hedging dualities and the analogue of the fundamental theorem of asset pricing. For this purpose we study the corresponding martingale optimal transport (MOT) problem: We obtain a dual representation of the Kantorovich functional (super-replication functional) defined for functions (financial derivatives) on the Skorokhod space using quotient sets (hedging sets). Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. The talk is based on a joint work with Patrick Cheridito, Matti Kiiski and H. Mete Soner.

## From systemic risk to supercooling and back

I will explain how structural models of default cascades in the systemic risk literature naturally lead to the supercooled Stefan problem of mathematical physics. On the one hand, this connection allows us to uncover a notion of global solutions to the supercooled Stefan problem, which we analyze in detail. On the other hand, the supercooled Stefan problem formulation allows to provide a truly intrinsic definition of systemic crises and to characterize the fragile states of the economy. Time permitting, I will also explain the network and game extensions of the problem. Based on a series of works with Francois Delarue and Sergey Nadtochiy.

## XXVIII European Workshop on Economic Theory

The XXVIII. European Workshop on Economic Theory (EWET 2019) is hosted by the Finance Group @ Humboldt. It will take place at the School of Business and Economics, Humboldt University, in the city center of Berlin from June 13 to June 15. The workshop is a forum for researchers interested in the latest developments in economic theory and mathematical economics. Participants present and discuss recent results in areas such as general equilibrium theory, decision theory, information economics, game theory, bargaining and matching, financial markets, and social choice. Here is a link to the official webpage.

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## N-player games and mean-field games with smooth dependence on past absorptions

Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, a direct dependence on past absorptions has been introduced in the drift of players' state dynamics. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth. Therefore, the mean-field interaction among the players takes place in two ways: via the empirical sub-probability measure of the surviving players and through a process representing the fraction of past absorptions over time. Moreover, relaxing the boundedness of the coefficients allows for more realistic dynamics for players' private states. We prove existence of solutions of the mean-field game in strict as well as relaxed feedback form. Finally, we show that such solutions induce approximate Nash equilibria for the N-player game with vanishing error in the mean-field limit as $N \to \infty$. This talk is based on a joint work with M. Ghio and G. Livieri (SNS Pisa).