We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal robability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a L²-set of Wasserstein space-valued functions. This talk is based on a joint work with A. De Crescenzo, M. Fuhrman and H. Pham.

Hawkes processes are a popular model for self-exciting phenomena, from earthquakes to finance. In this talk, I will first present them in a simple way, using a Poisson imbedding construction. I will then review what is known about their long-time behavior, through limit theorems for both linear and non-linear cases. The focus will be on three regimes that appear when the process has a long memory and the branching ratio gets close to or above one: the Nearly Unstable, the Weakly Critical, and the Supercritical Nearly Unstable Hawkes processes. These regimes have been studied qualitatively, but quantitative convergence results have been missing. I will explain how we obtain explicit convergence rates, relying on a coupling with a Brownian sheet, Fourier analysis, and a careful approximation of the absolute value function.
 

We develop a continuous-time stochastic model for optimal cybersecurity investment under the threat of cyberattacks. The arrival of attacks is modeled using a Hawkes process, capturing the empirically relevant feature of clustering in cyberattacks. Extending the Gordon-Loeb model, each attack may result in a breach, with breach probability depending on the system's vulnerability. We aim at determining the optimal cybersecurity investment to reduce vulnerability. The problem is cast as a two-dimensional Markovian stochastic optimal control problem and solved using dynamic programming methods. We perform a numerical study of the value function and the associated optimal investment strategy in cyber-security, highlighting the impact of randomly arriving clustered cyber-attacks. Based on a joint work with G. Callegaro, C. Fontana and C. Hillairet.
 

When returns are partially predictable and trading is costly, utility maximizing investors track a target portfolio at a constant trading speed. The target portfolio is optimal for a frictionless market, where asset returns are scaled back to account for trading costs and volatilities are adjusted to proxy the “execution risk” of holding assets that are costly to trade and exposed to volatile states. The trading speed solves an optimal execution problem, which describes how the legacy portfolio inherited from the past is traded towards the target portfolio in an optimal manner. Unlike for period-by-period mean-variance preferences as in Garleanu and Pedersen (2013), the target portfolio hedges changes in investment opportunities, and both it and the trading speed are linked and depend on execution risk. We set the problem out first in an “absolute” framework – price shocks independent of the price level and investors have CARA preferences – and then in a “relative” framework, with price shocks scaled by price levels and CRRA preferences.

We analyze the effect of regulatory capital constraints on financial stability in a large homogeneous banking system using a mean-field game (MFG) model. Each bank holds cash and a tradable risky asset. Banks choose absolutely continuous trading rates in order to maximize expected terminal equity, with trades subject to transaction costs. Capital regulation requires equity to exceed a fixed multiple of the position in the tradable asset; breaches trigger forced liquidation. The asset drift depends on changes in average asset holdings across banks, so aggregate deleveraging creates contagion effects, leading to an MFG. We discuss the coupled forward-backward PDE system characterizing equilibria of the MFG, and we solve the constrained MFG numerically. Experiments demonstrate that capital constraints accelerate deleveraging and limit risk-bearing capacity. In some regimes, simultaneous breaches trigger liquidation cascades. The last part of the presentation is devoted to the mathematical analysis of a related model with time-smoothed contagion as in, e.g., Hambly, Ledger and Sojmark (2019) or Campi and Burzoni (2024). We characterize optimal strategies for a given evolution of the system, establish the existence of an MFG equilibrium and discuss limit results for a finite but large homogeneous banking system

Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma has paved the way for a unified theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise. As an application, we will then investigate the existence of a robust representation of the conditional distribution in a stochastic filtering model for multidimensional correlated jump-diffusions.