Long-run impulse control with generalised discounting

TL;DR

This paper explores how generalised discounting affects long-run impulse control problems, revealing that optimal values remain consistent across discounted and undiscounted cases, and proposing simpler solution methods.

Contribution

It demonstrates the equivalence of optimal values between discounted and undiscounted problems and introduces a time-consistent approach for complex discounting scenarios.

Findings

Optimal value of discounted problem equals that of the undiscounted version.

Optimal strategies for undiscounted problems are nearly optimal in discounted continuous-time settings.

Time-inconsistency issues can be addressed with simpler, time-independent equations.

Abstract

In this paper, we investigate the effects of applying generalised (non-exponential) discounting on a long-run impulse control problem for a Feller-Markov process. We show that the optimal value of the discounted problem is the same as the optimal value of its undiscounted version. Next, we prove that an optimal strategy for the undiscounted discrete time functional is also optimal for the discrete-time discounted criterion and nearly optimal for the continuous-time discounted one. This shows that the discounted problem, being time-inconsistent in nature, admits a time-consistent solution. Also, instead of a complex time-dependent Bellman equation one may consider its simpler time-independent version.