It is a by now classical observation that in a (realistic) financial market (model) simple portfolio strategies can outperform more sophisticated optimized portfolio strategies. For example, in a one period setting, the equal weight or 1/N-strategy often provides more stable results than mean-variance- optimal strategies. This is due to the fact that a good estimation of the mean returns is not possible for volatile financial assets. Pflug, Pichler and Wozabel (2012) gave a rigorous explanation of this observation by showing that for increasing uncertainty on the means the equal weight strategy becomes optimal in a mean-variance setting which is due to its robustness. We aim at extending this result to continuous-time strategies in a multivariate Black-Scholes type market. To this end we investigate how optimal trading strategies for maximizing expected utility of terminal wealth under CRRA utility behave when we have Knightian uncertainty on the drift, meaning that the only information is that the drift parameter lies in a so-called uncertainty set. The investor takes into account that the true drift may be the worst possible drift within this set. In this setting we can show that a minimax theorem holds which enables us to find the worst- case drift and the optimal robust strategy quite explicitly. This again allows us to derive the limits when uncertainty increases and hence to show that a uniform strategy is asymptotically optimal. We also discuss the extension to a financial market with a stochastic drift process, combining the worst-case approach with filtering techniques. This leads to local optimization problems, and the resulting optimal strategy needs to be updated continuously in time. We carry over the minimax theorem for the local optimization problems and derive the optimal strategy. In this setting we show how an ellipsoidal uncertainty set can be defined based on filtering techniques and we demonstrate that investors need to choose a robust strategy to be able to profit from additional information.

## Utility Maximization in a Multivariate Black Scholes Type Market with Model Uncertainty on the Drift

## Mean field games with Wright-Fisher common noise

Motivated by restoration of uniqueness in finite state mean field games, we introduce a common noise which is inspired by Wright-Fisher models in population genetics. Thus we analyze the master equation of this mean field game, which is a degenerate parabolic second-order partial differential equation set on the simplex whose characteristics solve the stochastic forward-backward system associated with the mean field game. We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo (AMS, 2013) has a unique smooth solution whenever the normal component of the drift at the boundary is strong enough. This is enough to conclude that the mean field game with such type of common noise is uniquely solvable. Then we introduce the finite player version of the game and show that N-player Nash equilibria converge towards the solution of such a kind of Wright-Fisher mean field game. The analysis is more subtle than in the standard setting because the mean field interaction between the players now occurs through a weighted empirical measure. In other words, each player carries its own weight, which hence may differ from 1/N and which, most of all, evolves with the common noise. Finally, we give an idea on how the randomly forced and uniquely solvable mean field game is used to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the intensity of a the Wright-Fisher common noise. Based on joint works with Erhan Byraktar, Asaf Cohen, and François Delarue.

## Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature and show how liquidation problems with portfolio processes of unbounded variation can be obtained as limiting cases in models with small instantaneous impact as well as a microscopic foundation for the use of semimartingale liquidation strategies. Our convergence results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes.

## Trading with the Crowd

We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact, along with taking into account a common general price predicting signal. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal to the aggregated transient price impact induced by all other agents. This unfolds a quantitative relation between trading signals and the order flow in crowded markets. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents. We prove that the equilibrium trading speed and the value function of an agent in the finite $N$-player game converges to the corresponding trading speed and value function in the mean field game at rate $O(N^{-2})$. In addition, we prove that the mean field optimal strategy provides an approximate Nash-equilibrium for the finite-player game. This is a joint work with Moritz Voss.

## Optimal transport and risk aversion i Kyle's model of informed trading

We establish connections between optimal transport theory and the dynamic version of the Kyle model, including new characterizations of informed trading profits via conjugate duality and Monge-Kantorovich duality. We use these connections to extend the model to multiple assets, general distributions, and risk-averse market makers. With risk-averse mar- ket makers, liquidity is lower, assets exhibit short-term reversals, and risk premia depend on market maker inventories, which are mean re- verting. We illustrate the model by showing that implied volatilities predict stock returns when there is informed trading in stocks and options and market makers are risk averse.

## The Microstructure of Stochastic Volatility Models with Self-Exciting Jump Dynamics

We provide a general probabilistic framework within which we establish scaling limits for a class of continuous-time stochastic volatility models with self-exciting jump dynamics. In the scaling limit, the joint dynamics of asset returns and volatility is driven by independent Gaussian white noises and two independent Poisson random measures that capture the arrival of exogenous shocks and the arrival of self-excited shocks, respectively. Various well-studied stochastic volatility models with and without self-exciting price/volatility co-jumps are obtained as special cases under different scaling regimes. We analyze the impact of external shocks on the market dynamics, especially their impact on jump cascades and show in a mathematically rigorous manner that many small external shocks may trigger endogenous jump cascades in asset returns and stock price volatility.

## Junior female researchers in probability

The workshop will take place **online** and, **if possible**, as a **hybrid event in Berlin**.

We warmly **invite those who identify as female to submit abstracts** for contributed talks, and **apply for financial support** for travelling to Berlin in case we can have a hybrid event.

However, please be aware that being unable to travel should not restrain from submitting an abstract. There are **special travel grants for female master students** interested in gaining some insight into research and get in touch with researchers.

Of course **everybody** **is** very **welcome to participate**, but presentations and financial support are reserved for female participants.

**Deadline for submission** of abstracts and funding requests: **June 30, 2021 **

**Conference webpage: **https://www.wias-berlin.de/workshops/JFRP21/** **

## 6th Berlin Workshop for Young Researchers in Mathematical Finance

The 6th Berlin Workshop for Young Researchers in Mathematical Finance takes place August 23-26. For more information, please visit

https://t1p.de/YoungResearchersBerlin2021

or contact the organizer Dirk Becherer.

## Deep Order Flow Imbalance : Extracting Alpha From the Limit Order Book

In this talk I will describe how deep learning methods are being applied to forecast stock returns from high frequency order book states. I will review the literature in this area and describe a working paper where we evaluate return forecasts for several deep learning models for a large subset of symbols traded on the Nasdaq exchange. We investigate whether transformation of the order book states is necessary and we relate the performance of deep learning models for a symbol to its microstructural properties. This is joint work with Petter Kolm (NYU), Jeremy Turiel (UCL) and Antonio Briola (UCL).

## Reinforcement learning for linear-convex models with jumps

We study finite-time horizon continuous-time linear-convex reinforcement learning problems in an episodic setting. In these problems, an unknown linear jump-diffusion process is controlled subject to nonsmooth convex costs. We start with the pure diffusion case with quadratic costs, and propose a least-squares algorithm which achieves a logarithmic regret bound of order O((lnM)(lnlnM)), with M being the number of learning episodes; the proof relies on the robustness of the associated Riccati differential equation and sub-exponential properties of the least-squares estimators. We then extend the least-squares algorithm to linear-convex learning problems with jumps, and establish a regret of the order O((MlnM)1/2); the analysis leverages the Lipschitz stability of the associated forward-backward stochastic differential equation and concentration properties of sub-Weibull random variables.

This is joint work with Matteo Basei, Xin Guo and Anran Hu.