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We prove the dynamic programming principle (DPP) in a class of problems wherean agent controls a $d$-dimensional diffusive dynamics via both classical andsingular controls and, moreover, is able to terminate the optimisation at atime of her choosing, prior to a given maturity. The time-horizon of theproblem is random and it is the smallest between a fixed terminal time and thefirst exit time of the state dynamics from a Borel set. We consider both thecases in which the total available fuel for the singular control is eitherbounded or unbounded. We build upon existing proofs of DPP and extend resultsavailable in the traditional literature on singular control (e.g., Haussmannand Suo, SIAM J. Control Optim., 33, 1995) by relaxing some key assumptions andincluding the discretionary stopping feature. We also connect with more generalversions 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 abstractrequirements therein.

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