Conference Agenda

Motivated by the competitive investment behavior of hedge fund managers, competitive portfolio optimization problems are a widely studied topic in the (continuous-time) portfolio optimization literature. In this talk, we propose a new way to incorporate competition into the classical expected utility maximization problem by using a value-at-risk-based constraint. Instead of just maximizing the expected utility of their terminal wealth, agents also aim to outperform a weighted average of their competitors’ terminal wealth with some fixed probability. In the special case of logarithmic utility, we determine and discuss optimal solutions in the form of Nash equilibria. If time permits, we also discuss the influence of several model parameters on the Nash equilibria in the special case of a Black-Scholes market.

In contrast to the existing literature on mean field equilibrium models for financial markets, which focuses on differences in agents' trading needs, this research project aims to extend these models by also considering differences in agents' information flows.

We bring together the models of [Bank, Körber, 2022] and [Lacker, Zariphopoulou, 2019] and investigate equilibrium problems in continuous-time stochastic control for optimal investment. Specifically, we model a multi-agent game, where a finite number of investors receives signals about impending price shocks and interacts through relative performance concerns.

For this purpose, we introduce novel signal-driven strategies utilizing Meyer-$\sigma$-measurable controls and a utility function with interactions as in [Lacker, Zariphopoulou, 2019], where investors assess their wealth based on their individual relative risk aversion and the average wealth of their peers, mediated by a concern parameter. A key aspect is the role of a single Poisson random measure, which drives jumps in both the market and signal processes, capturing both common and idiosyncratic noise. This requires understanding of the information shared among investors to accurately define the common noise filtration and the concept of mean wealth when transitioning to the mean field setting, via randomization over type vectors representing individual investor characteristics such as signal quality or quantity.

In both the multi-agent and the mean field case, using the dynamic programming techniques from [Bank, Körber, 2022], we derive and explicitly solve the corresponding HJB-equation and prove a verification theorem for best-response controls of a single (resp. representative) investor interacting with a fixed environment of other investors.

The existence of equilibria in both the multi-agent and mean field games is established using Schauder's Fixed Point Theorem under appropriate assumptions on the investors' characteristics, particularly their signal processes.

As a final step, we provide a numerical example to illustrate equilibria from a financial-economic perspective, addressing questions such as how much investors should care about information known by their peers.

We analyze a continuous-time optimal trade execution problem in multiple assets where the price impact and the resilience can be matrix-valued stochastic processes. Our starting point is a stochastic control problem where the control process is of finite variation, possibly with jumps, and acts as an integrator both in the state dynamics and in the cost functional. We discuss how this problem can be continuously extended from finite-variation controls to progressively measurable controls and how the extended problem is linked to a linear-quadratic (LQ) stochastic control problem. We obtain a solution of the LQ stochastic control problem by using results from the theory on LQ stochastic optimal control. From this we recover a solution of the extended problem. Finally, we present an example where it is optimal to start trading also in an asset where the initial position is already zero. This is based on joint work with Thomas Kruse and Mikhail Urusov.