A Deep Neural Network Algorithm for Linear-Quadratic Portfolio Optimization with MGARCH and Small Transaction Costs

We analyze a fixed-point algorithm for reinforcement learning (RL) of optimal portfolio mean-variance preferences in the setting of multivariate generalized autoregressive conditional-heteroskedasticity (MGARCH) with a small penalty on trading. A numerical solution is obtained using a neural network (NN) architecture within a recursive RL loop. A fixed-point theorem proves that NN approximation error has a big-oh bound that we can reduce by increasing the number of NN parameters. The functional form of the trading penalty has a parameter $\epsilon>0$ that controls the magnitude of transaction costs. When $\epsilon$ is small, we can implement an NN algorithm based on the expansion of the solution in powers of $\epsilon$. This expansion has a base term equal to a myopic solution with an explicit form, and a first-order correction term that we compute in the RL loop. Our expansion-based algorithm is stable, allows for fast computation, and outputs a solution that shows positive testing performance

A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier-Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad-div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at $Re_{\tau}=180$ as well as $590$.Comment: 28 pages, in preparation for submission to Journal of Computational Physic

An Institutional Frame to Compare Alternative Market Designs in EU Electricity Balancing

The so-called â electricity wholesale marketâ is, in fact, a sequence of several markets. The chain is closed with a provision for â balancing,â in which energy from all wholesale markets is balanced under the authority of the Transmission Grid Manager (TSO in Europe, ISO in the United States). In selecting the market design, engineers in the European Union have traditionally preferred the technical role of balancing mechanisms as â security mechanisms.â They favour using penalties to restrict the use of balancing energy by market actors. While our paper in no way disputes the importance of grid security, nor the competency of engineers to elaborate the technical rules, we wish to attract attention to the real economic consequences of alternative balancing designs. We propose a numerical simulation in the framework of a two-stage equilibrium model. This simulation allows us to compare the economic properties of designs currently existing within the European Union and to measure their fallout. It reveals that balancing designs, which are typically presented as simple variants on technical security, are in actuality alternative institutional frameworks having at least four potential economic consequences: a distortion of the forward price; an asymmetric shift in the participantsâ profits; an increase in the System Operatorâ s revenues; and inefficiencies