Time Discretization

Pathwise analysis of log-optimal portfolios ArXiv ID: 2507.18232 “View on arXiv” Authors: Andrew L. Allan, Anna P. Kwossek, Chong Liu, David J. Prömel Abstract Based on the theory of càdlàg rough paths, we develop a pathwise approach to analyze stability and approximation properties of portfolios along individual price trajectories generated by standard models of financial markets. As a prototypical example from portfolio theory, we study the log-optimal portfolio in a classical investment-consumption optimization problem on a frictionless financial market modelled by an Itô diffusion process. We identify a fully deterministic framework that enables a pathwise construction of the log-optimal portfolio, for which we then establish pathwise stability estimates with respect to the underlying model parameters. We also derive pathwise error estimates arising from the time-discretization of the log-optimal portfolio and its associated capital process. ...

Boundary treatment for high-order IMEX Runge-Kutta local discontinuous Galerkin schemes for multidimensional nonlinear parabolic PDEs ArXiv ID: 2410.02927 “View on arXiv” Authors: Unknown Abstract In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di-richlet boundary conditions. We consider Cartesian meshes and PDEs with stiff terms coming from the diffusive parts of the PDE. The algorithms treat boundary values at the implicit-explicit internal stages in the same way as the interior points. The boundary treatment strategy is designed to work with multidimensional problems with possible nonlinear advection and source terms. The proposed methods recover the designed order of convergence by numerical verification. For the spatial discretization, in this work, we consider Local Discontinuous Galerkin methods, although the developed boundary treatment algorithms can operate with other discretization schemes in space, such as Finite Differences, Finite Elements or Finite Volumes. ...

Commodities Trading through Deep Policy Gradient Methods ArXiv ID: 2309.00630 “View on arXiv” Authors: Unknown Abstract Algorithmic trading has gained attention due to its potential for generating superior returns. This paper investigates the effectiveness of deep reinforcement learning (DRL) methods in algorithmic commodities trading. It formulates the commodities trading problem as a continuous, discrete-time stochastic dynamical system. The proposed system employs a novel time-discretization scheme that adapts to market volatility, enhancing the statistical properties of subsampled financial time series. To optimize transaction-cost- and risk-sensitive trading agents, two policy gradient algorithms, namely actor-based and actor-critic-based approaches, are introduced. These agents utilize CNNs and LSTMs as parametric function approximators to map historical price observations to market positions.Backtesting on front-month natural gas futures demonstrates that DRL models increase the Sharpe ratio by $83%$ compared to the buy-and-hold baseline. Additionally, the risk profile of the agents can be customized through a hyperparameter that regulates risk sensitivity in the reward function during the optimization process. The actor-based models outperform the actor-critic-based models, while the CNN-based models show a slight performance advantage over the LSTM-based models. ...