Ergodic control of diffusions with random intervention times

TL;DR

This paper investigates an ergodic control problem for a one-dimensional diffusion where control actions are only possible at random jump times, characterizing optimal impulse controls and their connection to ergodic singular control.

Contribution

It introduces a novel framework for ergodic control with random intervention times and characterizes the optimal impulse control policy under weak assumptions.

Findings

Optimal control is of impulse type, pushing the process to a threshold.

The control policy is optimal at specific jump times of a Poisson process.

Results are illustrated with various cost and diffusion models.

Abstract

We study an ergodic singular control problem with constraint of a regular one-dimensional linear diffusion. The constraint allows the agent to control the diffusion only at jump times of independent Poisson process. Under relatively weak assumptions, we characterize the optimal solution as impulse type control policy, where it is optimal to exert the exact amount of control needed to push the process to a unique threshold. Moreover, we discuss the connection of the present problem to ergodic singular control problems, and finally, illustrate the results with different well-known cost and diffusion structures.