Comparison Principle and Semi-Lagrangian Methods for a Hamilton--Jacobi--Bellman System with Recursive Utility

Abstract

We consider a system of coupled first-order Hamilton--Jacobi--Bellman equations arising in continuous-time heterogeneous agent models with recursive utility, a class of models that has attracted increasing attention in macroeconomics. The first part of the paper establishes a comparison principle for the system when preferences exhibit early resolution of uncertainty. A key ingredient in the proof is an ordinally equivalent transformation of the recursive utility representation. In the second part, we develop semi-Lagrangian schemes and iterative algorithms for solving the associated nonlinear discrete systems. For preferences exhibiting late resolution of uncertainty, we establish a first-order error estimate for the semi-Lagrangian approximation. Numerical experiments demonstrate that the proposed methods yield results consistent with those obtained using the previously developed finite difference schemes.