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.

In an investment contest with incomplete information, a finite number of agents dynamically trade assets with idiosyncratic risk and are rewarded based on the relative ranking of their terminal portfolio values. We explicitly characterize a symmetric Nash equilibrium of the contest and rigorously verify its uniqueness. The connection between the reward structure and the agents’ portfolio strategies is examined. A top-heavy payout rule results in an equilibrium portfolio return distribution with high positive skewness, which suffers from a large likelihood of poor performance. Risky asset holding increases when competition intensifies in a winner-takes all contest. This is joint work with Yumin Lu.

We study finite player stochastic differential games on possibly bounded spatial domains. The equilibrium problem is formulated through the dynamic programming principle, leading to a coupled Nash system of HJB equations and, in probabilistic form, to a corresponding Nash FBSDE with stopping at the first exit from the parabolic domain (covering both boundary and terminal conditions). The main focus of the talk is the analysis of a fictitious-play procedure applied at the level of FBSDEs. At each iteration, a player solves a best-response FBSDE against fixed opponent strategies, giving rise to a sequence of fictitious-play FBSDEs. We show that this sequence converges exponentially fast to the Nash FBSDE. In unbounded domains, this holds under a small-time assumption; in bounded domains, exponential convergence is obtained for arbitrary horizons under additional regularity conditions. For completeness, we also discuss how the fictitious-play FBSDE is approximated by a numerically tractable surrogate FBSDE, which itself converges exponentially to the fictitious-play equation. Since the surrogate FBSDE admits a standard time-discrete approximation of order 1/2, this provides a transparent overall error structure for the numerical approximation of the Nash FBSDE. We conclude with representative numerical illustrations of the full approximation scheme.