On the link between infinite horizon control and quasi-stationary distributions

TL;DR

This paper explores the connection between infinite horizon control of non-linear branching processes and their quasi-stationary distributions, providing new insights into optimal control and extinction behavior.

Contribution

It establishes a link between control problems and quasi-stationary distributions, offering a new proof of the dynamic programming principle and characterizing optimal controls near critical discount thresholds.

Findings

Derived an equivalent of the value function near the critical discount threshold.

Characterized the optimal Markov control in the limit as the discount approaches the extinction threshold.

Proved convergence to a unique quasi-stationary distribution under controlled non-linear branching processes.

Abstract

We study infinite horizon control of continuous-time non-linear branching processes with almost sure extinction for general (positive or negative) discount. Our main goal is to study the link between infinite horizon control of these processes and an optimization problem involving their quasi-stationary distributions and the corresponding extinction rates. More precisely, we obtain an equivalent of the value function when the discount parameter is close to the threshold where the value function becomes infinite, and we characterize the optimal Markov control in this limit. To achieve this, we present a new proof of the dynamic programming principle based upon a pseudo-Markov property for controlled jump processes. We also prove the convergence to a unique quasi-stationary distribution of non-linear branching processes controlled by a Markov control conditioned on non-extinction.