Two-sided Poisson control of linear diffusions

TL;DR

This paper addresses a class of two-sided optimal control problems for linear diffusions constrained by Poisson process arrivals, providing explicit solutions and analyzing their properties as the Poisson signal intensity varies.

Contribution

It introduces a novel approach to solve two-sided control problems with Poisson constraints, deriving explicit solutions using minimal r-excessive mappings.

Findings

Derived a quasi-explicit unique solution for the control problem.

Provided verifiable conditions for the solution's existence and uniqueness.

Analyzed the limiting behavior of solutions as Poisson signal intensity changes.

Abstract

We study a class of two-sided optimal control problems of general linear diffusions under a so-called Poisson constraint: the controlling is only allowed at the arrival times of an independent Poisson signal processes. We give a weak and easily verifiable set of sufficient conditions under which we derive a quasi-explicit unique solution to the problem in terms of the minimal r-excessive mappings of the diffusion. We also investigate limiting properties of the solutions with respect to the signal intensity of the Poisson process. Lastly, we illustrate our results with an explicit example.