On Singular Control Problems, the Time-Stretching Method, and the Weak-M1 Topology

We consider a general class of singular control problems with state constraints. Budhiraja and Ross [Ann. Appl. Probab., 16 (2006), pp. 2235--2255] established the existence optimal controls for relaxed version this by using so-called time-stretching method J1-topology. show that weak-M1 topology is better suited establishing existence, since it allows to bypass time-transformations, without any additional effort. Furthermore, we reveal how time-scaling feature in definition distance embeds method's scheme. This case study suggests one can benefit from working other frameworks, such as queueing under heavy traffic.