On the role of Gittins index in singular stochastic control: semi-explicit solutions via the Wiener-Hopf factorisation

Abstract

This paper examines a class of singular stochastic control problems with convex objective functions. In Section 2, we use tools from convex analysis to derive necessary and sufficient first order conditions for this class of optimisation problems. The main result of this paper is Theorem 9 which uses results from optimal stopping to establish the link between singular stochastic control and Gittin's index without the need to appeal to the representation result in [5]. In Sections 3-5 we assume the singular control problem is driven by a L\'{e}vy process. Expressions for the Gittin's index are derived in terms of the Wiener-Hopf factorisation. This allows us to broaden the class of parameterised optimal stopping problems with explicit solutions examined in [6] and derive explicit solutions to the singular control problems studied in Section 2. In Section 4, we apply our results to the `monotone follower' problem which originates in [7] and [28]. In Section 5, we apply our results to an irreversible investment problem which has been studied in [9], [35] and [40].