Dynamic programming principle for classical and singular stochastic control with discretionary stopping

TL;DR

This paper establishes the dynamic programming principle for a class of stochastic control problems involving both classical and singular controls, with optional stopping and a random time horizon, extending existing theories.

Contribution

It extends the DPP framework to include discretionary stopping and unbounded control fuel, relaxing key assumptions in classical singular control literature.

Findings

Proves DPP for combined classical and singular controls with stopping.

Connects specific control problems to general DPP frameworks.

Extends classical results to more general, realistic settings.

Abstract

We prove the dynamic programming principle (DPP) in a class of problems where an agent controls a $d$-dimensional diffusive dynamics via both classical and singular controls and, moreover, is able to terminate the optimisation at a time of her choosing, prior to a given maturity. The time-horizon of the problem is random and it is the smallest between a fixed terminal time and the first exit time of the state dynamics from a Borel set. We consider both the cases in which the total available fuel for the singular control is either bounded or unbounded. We build upon existing proofs of DPP and extend results available in the traditional literature on singular control (e.g., Haussmann and Suo, SIAM J. Control Optim., 33, 1995) by relaxing some key assumptions and including the discretionary stopping feature. We also connect with more general versions of the DPP (e.g., Bouchard and Touzi, SIAM J. Control Optim., 49, 2011) by showing in detail how our class of problems meets the abstract requirements therein.