A spectral method for an Optimal Investment problem with Transaction Costs under Potential Utility

TL;DR

This paper introduces a spectral numerical method for solving a finite-horizon optimal investment problem with transaction costs under potential utility, reformulating it as a parabolic double obstacle problem for improved efficiency.

Contribution

The paper develops a spectral method tailored for a reformulated obstacle problem, enhancing numerical accuracy and efficiency over traditional finite difference approaches.

Findings

Spectral method outperforms finite differences in precision and efficiency.

Reformulation in polar coordinates simplifies the problem domain.

Explicit properties and formulas aid in numerical solution accuracy.

Abstract

This paper concerns the numerical solution of the finite-horizon Optimal Investment problem with transaction costs under Potential Utility. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints. In Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obstacle Problem, Day-Yi, the problem is reformulated as a non-linear parabolic double obstacle problem posed in one spatial variable and defined in an unbounded domain where several explicit properties and formulas are obtained. The restatement of the problem in polar coordinates allows to pose the problem in one spatial variable in a finite domain, avoiding some of the technical difficulties of the numerical solution of the previous statement of the problem. If high precision is required, the spectral numerical method proposed becomes more efficient than simpler methods as finite differences for example.