Partial Differential Equations

Position-building in competition with real-world constraints ArXiv ID: 2409.15459 “View on arXiv” Authors: Unknown Abstract This paper extends the optimal-trading framework developed in arXiv:2409.03586v1 to compute optimal strategies with real-world constraints. The aim of the current paper, as with the previous, is to study trading in the context of multi-player non-cooperative games. While the former paper relies on methods from the calculus of variations and optimal strategies arise as the solution of partial differential equations, the current paper demonstrates that the entire framework may be re-framed as a quadratic programming problem and cast in this light constraints are readily incorporated into the calculation of optimal strategies. An added benefit is that two-trader equilibria may be calculated as the end-points of a dynamic process of traders forming repeated adjustments to each other’s strategy. ...

Deep-MacroFin: Informed Equilibrium Neural Network for Continuous Time Economic Models ArXiv ID: 2408.10368 “View on arXiv” Authors: Unknown Abstract In this paper, we present Deep-MacroFin, a comprehensive framework designed to solve partial differential equations, with a particular focus on models in continuous time economics. This framework leverages deep learning methodologies, including Multi-Layer Perceptrons and the newly developed Kolmogorov-Arnold Networks. It is optimized using economic information encapsulated by Hamilton-Jacobi-Bellman (HJB) equations and coupled algebraic equations. The application of neural networks holds the promise of accurately resolving high-dimensional problems with fewer computational demands and limitations compared to other numerical methods. This framework can be readily adapted for systems of partial differential equations in high dimensions. Importantly, it offers a more efficient (5$\times$ less CUDA memory and 40$\times$ fewer FLOPs in 100D problems) and user-friendly implementation than existing libraries. We also incorporate a time-stepping scheme to enhance training stability for nonlinear HJB equations, enabling the solution of 50D economic models. ...

A monotone piecewise constant control integration approach for the two-factor uncertain volatility model ArXiv ID: 2402.06840 “View on arXiv” Authors: Unknown Abstract Option contracts on two underlying assets within uncertain volatility models have their worst-case and best-case prices determined by a two-dimensional (2D) Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) with cross-derivative terms. This paper introduces a novel ``decompose and integrate, then optimize’’ approach to tackle this HJB PDE. Within each timestep, our method applies piecewise constant control, yielding a set of independent linear 2D PDEs, each corresponding to a discretized control value. Leveraging closed-form Green’s functions, these PDEs are efficiently solved via 2D convolution integrals using a monotone numerical integration method. The value function and optimal control are then obtained by synthesizing the solutions of the individual PDEs. For enhanced efficiency, we implement the integration via Fast Fourier Transforms, exploiting the Toeplitz matrix structure. The proposed method is $\ell_{"\infty"}$-stable, consistent in the viscosity sense, and converges to the viscosity solution of the HJB equation. Numerical results show excellent agreement with benchmark solutions obtained by finite differences, tree methods, and Monte Carlo simulation, highlighting its robustness and effectiveness. ...

Quantum-inspired nonlinear Galerkin ansatz for high-dimensional HJB equations ArXiv ID: 2311.12239 “View on arXiv” Authors: Unknown Abstract Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of time-dependent PDEs, the dominant paradigm involves casting the approximate solution in terms of stochastic minimization of an objective function given by the norm of the PDE residual, viewed as a function of the neural network parameters. Recently, advancements have been made in the direction of an alternative approach which shares aspects of nonlinearly parametrized Galerkin methods and variational quantum Monte Carlo, especially for high-dimensional, time-dependent PDEs that extend beyond the usual scope of quantum physics. This paper is inspired by the potential of solving Hamilton-Jacobi-Bellman (HJB) PDEs using Neural Galerkin methods and commences the exploration of nonlinearly parametrized trial functions for which the evolution equations are analytically tractable. As a precursor to the Neural Galerkin scheme, we present trial functions with evolution equations that admit closed-form solutions, focusing on time-dependent HJB equations relevant to finance. ...