On asymptotic relations between singular and constrained control problems of one-dimensional diffusions

TL;DR

This paper investigates the asymptotic relationships between singular and constrained control problems for one-dimensional diffusions, showing convergence of solutions under various limits and illustrating with specific stochastic processes.

Contribution

It establishes the convergence relationships between constrained, singular, and ergodic control problems for diffusions, extending understanding of their asymptotic behavior.

Findings

Solutions of discounted problems converge to ergodic solutions for recurrent diffusions.

Constrained control solutions approach singular control solutions as Poisson rate increases.

Results are demonstrated with drifted Brownian motion and Ornstein-Uhlenbeck processes.

Abstract

We study the asymptotic relations between certain singular and constrained control problems for one-dimensional diffusions with both discounted and ergodic objectives. By constrained control problems we mean that controlling is allowed only at independent Poisson arrival times. We show that when the underlying diffusion is recurrent, the solutions of the discounted problems converge in Abelian sense to those of their ergodic counterparts. Moreover, we show that the solutions of the constrained problems converge to those of their singular counterparts when the Poisson rate tends to infinity. We illustrate the results with drifted Brownian motion and Ornstein-Uhlenbeck process.