Optimal decision rules for marked point process models

TL;DR

This paper analyzes an optimal control problem for marked point processes with applications to thinning strategies, providing explicit thresholds and bounds for rewards under specific stochastic dynamics.

Contribution

It introduces explicit optimal thinning rules and reward calculations for marked point processes with birth-death-growth dynamics, including cases with hard core constraints.

Findings

Thinning points with large marks is optimal under Poisson births and logistic growth.

Explicit thresholds for thinning and expected reward calculations are derived.

Bounds on rewards are established when points must respect a hard core distance.

Abstract

We study a Markov decision problem in which the state space is the set of finite marked point configurations in the plane, the actions represent thinnings, the reward is proportional to the mark sum which is discounted over time, and the transitions are governed by a birth-death-growth process. We show that thinning points with large marks is optimal when births follow a Poisson process and marks grow logistically. Explicit values for the thinning threshold and the discounted total expected reward over finite and infinite horizons are also provided. When the points are required to respect a hard core distance, upper and lower bounds on the discounted total expected reward are derived.