Optimal consumption for general diffusions with absorbing and reflecting barriers.

*(English)*Zbl 0535.93071Singular control refers to control of a stochastic process by adding to it an adapted, monotone or bounded variation (typically not absolutely continuous) process. This kind of model, studied recently by a number of authors, has been used for storage, inventory and cash flow processes. The underlying process is usually a Brownian motion with possibly nonzero drift, this paper being a notable exception in that it studies singular control of a general one-dimensional diffusion. Two versions of the infinite horizon model are considered: one with absorption and the other with reflection at the origin. The value function is computed, and a necessary and sufficient condition for the existence of an optimal policy is given. When they exist, optimal policies are found to exercise the minimal amount of action necessary to keep the controlled process inside an interval.

##### MSC:

93E20 | Optimal stochastic control |

60J60 | Diffusion processes |

49K45 | Optimality conditions for problems involving randomness |

90B05 | Inventory, storage, reservoirs |

60K30 | Applications of queueing theory (congestion, allocation, storage, traffic, etc.) |