Optimal bounds for numerical approximations of infinite horizon problems based on dynamic programming approach

TL;DR

This paper establishes that the error bounds for fully discrete approximations of infinite horizon problems via dynamic programming are actually first order in both time and space, improving upon previous bounds and aligning with observed numerical results.

Contribution

The paper proves a new first-order error bound of O(h + k) for fully discrete dynamic programming approximations of infinite horizon problems, refining previous bounds.

Findings

Error bound of O(h + k) for fully discrete methods

First-order accuracy in both time and space

Numerical results align with the new theoretical bounds

Abstract

In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering a time discretization with a positive step size $h$ an error bound of size $h$ can be proved for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size $k$ an error bound of size $O(k/h)$ can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under similar assumptions to those of the time discrete case, that the error of the fully discrete case is in fact $O(h+k)$ which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behaviour $1/h$ from the bound $O(k/h)$ have not been observed.