## CRC Workshop Information and Enforcement

## Asymptotic Analysis of Deep Residual Networks

## CANCEELLED

## Open Markets in Stochastic Portfolio Theory and Rank Jacobi Processes

Stochastic portfolio theory is a framework to study large equity markets over long time horizons. In such settings investors are often confined to trading in an “open market” setup consisting of only assets with high capitalizations. In this work we relax previously studied notions of open markets and develop a tractable framework for them under mild structural conditions on the market. Within this framework we also introduce a large parametric class of processes, which we call rank Jacobi processes. They produce a stable capital distribution curve consistent with empirical observations. Moreover, there are explicit expressions for the growth-optimal portfolio, and they are also shown to serve as worst-case models for a robust asymptotic growth problem under model ambiguity. Lastly, the rank Jacobi models are shown to be stable with respect to the total number of stocks in the market. Time permitting, we will show that, under suitable assumptions on the parameters, the capital distribution curves converge to a limiting quantity as the size of the market tends to infinity. This convergence result provides a theoretical explanation for an important empirically observed phenomenon.

This talk is based on joint work with Martin Larsson.

## Stochastic Optimal Control of Heating Systems with a Geothermal Energy Storage

Thermal storage facilities help to mitigate and to manage temporal fluctuations of heat supply and demand for heating and cooling systems of single buildings as well as for district heating systems. We focus on a heating system equipped with several heat production units using also renewable energies and an underground thermal storage. The thermal energy is stored by raising the temperature of the soil inside that storage. It is charged and discharged via heat exchanger pipes filled with a moving fluid.

Besides the numerous technical challenges and the computation of the spatio-temporal temperature dis- tribution in the storage also economic issues such as the cost-optimal control and management of such systems play a central role. The latter leads to challenging mathematical optimization problems. There we incorporate uncertainties about randomly fluctuating renewable heat production, environmental conditions driving the heat demand and supply.

The dynamics of the controlled state process is governed by a PDE, a random ODE, and SDEs modeling energy prices and the difference between supply and demand. Model reduction techniques are adopted to cope with the PDE describing the spatio-temporal temperature distribution in the geothermal storage. Finally, time- discretization leads to a Markov decision process for which we apply numerical methods to determine a cost-optimal control.

This is joint work with Paul Honore Takam (BTU Cottbus-Senftenberg) and Olivier Menoukeu Pamen (AIMS Ghana, University of Liverpool).

## Detecting asset price bubbles using deep learning

In this talk we present a novel deep learning methodology to detect financial asset bubbles by using observed call option prices. We start with an introduction to deep learning and asset price bubbles. We then examine the pitfalls of a naive approach for deep learning-based bubble detection and subsequently introduce our method. We provide theoretical foundations for the method in the context of local volatility models and show numerical results from experiments both on simulated and market data.

The talk is based on joint work with Francesca Biagini, Andrea Mazzon and Thilo Meyer-Brandis.

## Large-scale principal-agent problems

In this talk, we introduce two problems of contract theory, in continuous-time, with a multitude of agents. First, we will study a model of optimal contracting in a hierarchy, which generalises the one-period framework of Sung (2015). The hierarchy is modelled by a series of interlinked principal-agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal (she) can contract with a manager (he), to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts the agents below him. We will see through a simple example that, while the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Therefore, even this relatively simple introductory example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory on 2BSDEs.

This will lead us to introduce the second problem, namely optimal contracting for demand-response management, which consists in extending the model by Aid, Possamai, and Touzi (2022) to a mean-field of consumers. More precisely, the principal (an electricity producer, or provider) contracts with a continuum of agents (the consumers), to incentivise them to decrease the mean and the volatility of their energy consumption during high peak demand. In addition, we introduce a common noise, impacting all consumption processes, to take into account the impact of weather conditions on the agents’ electricity consumption. This mean-field framework with common noise leads us to consider a more extensive class of contracts. In particular, we prove that the results of [1] can be improved by indexing the contracts on the consumption of one agent and aggregate consumption statistics from the distribution of the entire population of consumers.

*Talk based on Hubert (2020) and Elie, Hubert, Mastrolia, and Possamai (2021).*

## Equilibria in non-Markovian zero-sum stopping games with asymmetric information

I will show that a zero-sum stopping game in continuous time with partial and/or asymmetric information admits a saddle point (and, consequently, a value) in randomised stopping times when stopping payoffs of players are general càdlàg adapted processes. We do not assume a Markovian nature of the game nor a particular structure of the information available to the players. I will discuss links with classical results by Baxter, Chacon (1977) and Meyer (1978) derived for optimal stopping problems. Based on a joint work with Tiziano De Angelis, Nikita Merkulov and Jacob Smith.

## A Tikhonov Theorem for McKean Vlasov SDEs and an application to mean-field control problems.

We present a stochastic Tikhonov theorem for two-scales systems of SDEs, which cover the case of McKean-Vlasov SDEs. Our approach extends and generalizes previous results on two-scales systems of SDEs without mean-field interaction. As an application we provide a novel method for approximating the solution of certain systems of FBSDEs, related to the Pontryagin maximum principle, which is new even for the case without mean-field interaction. This is a joint work with A. Cosso.

## Utility indifference pricing with high risk aversion and small linear price impact

We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options and we compute their non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, we find explicitly a family of portfolios which are asymptotically optimal.