@article{14451,
  abstract     = {We investigate the potential of Multi-Objective, Deep Reinforcement Learning for stock and cryptocurrency single-asset trading: in particular, we consider a Multi-Objective algorithm which generalizes the reward functions and discount factor (i.e., these components are not specified a priori, but incorporated in the learning process). Firstly, using several important assets (BTCUSD, ETHUSDT, XRPUSDT, AAPL, SPY, NIFTY50), we verify the reward generalization property of the proposed Multi-Objective algorithm, and provide preliminary statistical evidence showing increased predictive stability over the corresponding Single-Objective strategy. Secondly, we show that the Multi-Objective algorithm has a clear edge over the corresponding Single-Objective strategy when the reward mechanism is sparse (i.e., when non-null feedback is infrequent over time). Finally, we discuss the generalization properties with respect to the discount factor. The entirety of our code is provided in open-source format.},
  author       = {Cornalba, Federico and Disselkamp, Constantin and Scassola, Davide and Helf, Christopher},
  issn         = {1433-3058},
  journal      = {Neural Computing and Applications},
  publisher    = {Springer Nature},
  title        = {{Multi-objective reward generalization: improving performance of Deep Reinforcement Learning for applications in single-asset trading}},
  doi          = {10.1007/s00521-023-09033-7},
  year         = {2023},
}

@article{14554,
  abstract     = {The Regularised Inertial Dean–Kawasaki model (RIDK) – introduced by the authors and J. Zimmer in earlier works – is a nonlinear stochastic PDE capturing fluctuations around the meanfield limit for large-scale particle systems in both particle density and momentum density. We focus on the following two aspects. Firstly, we set up a Discontinuous Galerkin (DG) discretisation scheme for the RIDK model: we provide suitable definitions of numerical fluxes at the interface of the mesh elements which are consistent with the wave-type nature of the RIDK model and grant stability of the simulations, and we quantify the rate of convergence in mean square to the continuous RIDK model. Secondly, we introduce modifications of the RIDK model in order to preserve positivity of the density (such a feature only holds in a “high-probability sense” for the original RIDK model). By means of numerical simulations, we show that the modifications lead to physically realistic and positive density profiles. In one case, subject to additional regularity constraints, we also prove positivity. Finally, we present an application of our methodology to a system of diffusing and reacting particles. Our Python code is available in open-source format.},
  author       = {Cornalba, Federico and Shardlow, Tony},
  issn         = {2804-7214},
  journal      = {ESAIM: Mathematical Modelling and Numerical Analysis},
  number       = {5},
  pages        = {3061--3090},
  publisher    = {EDP Sciences},
  title        = {{The regularised inertial Dean' Kawasaki equation: Discontinuous Galerkin approximation and modelling for low-density regime}},
  doi          = {10.1051/m2an/2023077},
  volume       = {57},
  year         = {2023},
}

@article{10551,
  abstract     = {The Dean–Kawasaki equation—a strongly singular SPDE—is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N≫1. The singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean–Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean–Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in N−1  (in suitable weak metrics). In other words, the Dean–Kawasaki equation may be interpreted as a “recipe” for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.},
  author       = {Cornalba, Federico and Fischer, Julian L},
  issn         = {1432-0673},
  journal      = {Archive for Rational Mechanics and Analysis},
  number       = {5},
  publisher    = {Springer Nature},
  title        = {{The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles}},
  doi          = {10.1007/s00205-023-01903-7},
  volume       = {247},
  year         = {2023},
}

@article{9240,
  abstract     = {A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension . It is a regularised and inertial version of the Dean–Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in d-dimensions when the particle-width ϵ is proportional to  for  and N is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the d-spatial dimensions, and use of the Faà di Bruno's formula.},
  author       = {Cornalba, Federico and Shardlow, Tony and Zimmer, Johannes},
  issn         = {1090-2732},
  journal      = {Journal of Differential Equations},
  number       = {5},
  pages        = {253--283},
  publisher    = {Elsevier},
  title        = {{Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions}},
  doi          = {10.1016/j.jde.2021.02.048},
  volume       = {284},
  year         = {2021},
}

@article{7637,
  abstract     = {The evolution of finitely many particles obeying Langevin dynamics is described by Dean–Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a regularised Dean–Kawasaki model based on second order Langevin dynamics by analysing a system of particles interacting via a pairwise potential. Key tools of our analysis are the propagation of chaos and Simon's compactness criterion. The model we obtain is a small-noise stochastic perturbation of the undamped McKean–Vlasov equation. We also provide a high-probability result for existence and uniqueness for our model.},
  author       = {Cornalba, Federico and Shardlow, Tony and Zimmer, Johannes},
  issn         = {13616544},
  journal      = {Nonlinearity},
  number       = {2},
  pages        = {864--891},
  publisher    = {IOP Publishing},
  title        = {{From weakly interacting particles to a regularised Dean-Kawasaki model}},
  doi          = {10.1088/1361-6544/ab5174},
  volume       = {33},
  year         = {2020},
}