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Abstract Many modern computational approaches to classical problems in quantitative finance are formulated as empirical loss minimization (ERM), allowing direct applications of classical results from statistical machine learning. These methods, designed to directly construct the optimal feedback representation of hedging or investment decisions, are analyzed in this framework demonstrating their effectiveness as well as their susceptibility to generalization error. Use of classical techniques shows that over‐training renders trained investment decisions to become anticipative, and proves overlearning for large hypothesis spaces. On the other hand, nonasymptotic estimates based on Rademacher complexity show the convergence for sufficiently large training sets. These results emphasize the importance of synthetic data generation and the appropriate calibration of complex models to market data. A numerically studied stylized example illustrates these possibilities, including the importance of problem dimension in the degree of overlearning, and the effectiveness of this approach. 

Dynamic programming equations for mean field control problems with a separable structure are Eikonal type equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to prove a comparison result among semi-continuous sub and super solutions, obtaining a unique characterization of the value function. 

An optimal control problem in the space of probability measures, and the viscosity solu- tions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier-Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution. 

The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both syn- chronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uni- form distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial dif- ferential equations, viscosity solutions, stochastic optimal control and stochastic processes. ARTICLE HISTORY Received 23 October 2022 Accepted 17 June 2023 KEYWORDS Mean field games; Kuramoto model; synchronization; viscosity solutions 2020 MATHEMATICS SUBJECT CLASSIFICATION 35Q89; 35D40; 39N80; 91A16; 92B25 1. Introduction Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model [1] has found an amazing range of applications from neuroscience to Josephson junctions in superconductors, and has become a key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long time behavior of the system. While the system is unsynchronized when this term is not sufficiently strong, fascinatingly they exhibit an abrupt transition to self organization above a critical value of the interaction parameter. Synchronization is an emergent property that occurs in a broad range of complex systems such as neural signals, heart beats, fire-fly lights and circadian rhythms, and the Kuramoto dynamical system is widely used as the main phenomenological model. Expository papers [2, 3] and the references therein provide an excellent introduction to the model and its applications. The analysis of the coupled Kuramoto oscillators through a mean field game formalism is first explored by [4, 5] proving bifurcation from incoherence to coordination by a formal linearization and a spectral argument. [6] further develops this analysis in their application to a jet-lag recovery model. We follow these pioneering studies and analyze the Kuramoto model as a discounted infinite horizon stochastic game in the limit when the number of oscillators goes to infinity. We treat the system of oscillators as an infinite particle system, but instead of positing the dynamics of the particles, we let the individual particles endogenously determine their behaviors by minimizing a cost functional and hopefully, settling in a Nash equilibrium. Once the search for equilibrium is recast in this way, equilibria are given by solutions of nonlinear systems. Analytically, they are characterized by a backward dynamic CONTACT H. Mete Soner soner@princeton.edu Department of Operations Research and Financial Engineering, Prince- ton University, Princeton, NJ, 08540, USA. © 2023 Taylor & Francis Group, LLC 

Although leveraged exchange-traded funds (ETFs) are popular products for retail investors, how to hedge them poses a great challenge to financial institutions. We develop an optimal rebalancing (hedging) model for leveraged ETFs in a comprehensive setting, including overnight market closure and market frictions. The model allows for an analytical optimal rebalancing strategy. The result extends the principle of “aiming in front of target” introduced by Gârleanu and Pedersen (2013) from a constant weight between current and future positions to a time-varying weight because the rebalancing performance is monitored only at discrete time points, but the rebalancing takes place continuously. Empirical findings and implications for the weekend effect and the intraday trading volume are also presented. This paper was accepted by Agostino Capponi, finance. Funding: M. Dai acknowledges support from the National Natural Science Foundation of China [Grant 12071333], the Hong Kong Polytechnic University [Grant P0039114], and the Singapore Ministry of Education [Grants R-146-000-243/306/311-114 and R-703-000-032-112]. H. M. Soner acknowledges partial support from the National Science Foundation [Grant DMS 2106462]. C. Yang acknowledges support from the Chinese University of Hong Kong [Grant 4055132 and a University Startup Grant]. Supplemental Material: Data and the online supplement are available at https://doi.org/10.1287/mnsc.2022.4407 . 

This paper outlines, and through stylized examples evaluates a novel and highly effective computational technique in quantitative finance. Empirical Risk Minimi- zation (ERM) and neural networks are key to this approach. Powerful open source optimization libraries allow for efficient implementations of this algorithm making it viable in high-dimensional structures. The free-boundary problems related to Amer- ican and Bermudan options showcase both the power and the potential difficulties that specific applications may face. The impact of the size of the training data is studied in a simplified Merton type problem. The classical option hedging problem exemplifies the need of market generators or large number of simulations.