Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations

TL;DR

This paper demonstrates that higher-order interpolation can be used in piecewise constant policy timestepping for HJB equations without losing convergence, supported by theoretical analysis and numerical tests.

Contribution

It extends standard convergence results by allowing higher-order interpolation in mesh transfer for HJB equations, enhancing accuracy.

Findings

Higher-order interpolation guarantees convergence.

Numerical tests confirm improved accuracy.

Applicable to control and finance problems.

Abstract

An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation must be used to transfer data between meshes. Using the equivalence to a switching system and an adaptation of the usual arguments based on consistency, stability and monotonicity, we show that if limited, potentially higher order interpolation is used for the mesh transfer, convergence is guaranteed. We provide numerical tests for the mean-variance optimal investment problem and the uncertain volatility option pricing model, and compare the results to published test cases.