Singular consumption control ($dw=\cdots -dC,d\xi =-\lambda \xi dt+dC$)

In singular stochastic control, an unrestricted consumption rate can approximate instantaneous transfers from portfolio wealth into a consumption satisfaction stock. In the relaxed problem a nondecreasing control $C$ records these transfers. A transfer $\Delta C$ decreases portfolio wealth and increases consumption satisfaction by the same amount. Rate controls need not attain the supremum when the relaxed optimum has a jump.