Singular stochastic control and optimal stopping with partial information of jump diffusions
We study partial information, possibly non-Markovian, singular stochastic control of jump diffusions and obtain general maximum principles. The results are used to find connections between singular stochastic control, reflected BSDEs and optimal stopping in the partial information case. Mathematics Subject Classification 2010: 93E20, 60H07, 60H10, 60HXX, 60J75
A General Stochastic Maximum Principle for Singular Control Problems
We consider the stochastic control problem in which the control domain need not be convex, the control variable has two components, the first being absolutely continuous and the second singular. The coefficients of the state equation are non linear and depend explicitly on the absolutely continuous component of the control. We establish a maximum principle, by using a spike variation on the absolutely continuous part of the control and a convex perturbation on the singular one. This result is a generalization of Peng's maximum principle to singular control problems.
Singular Optimal Stochastic Controls I: Existence
We apply the compactification method to study the control problem where the state is governed by an Ito stochastic differential equation allowing both classical and singular control. The problem is reformulated as a martingale problem on an appropriate canonical space after the relaxed form of the classical control is introduced. Under some mild continuity hypotheses on the data, it is shown by purely probabilistic arguments that an optimal control for the problem exists. The value function is shown to be Borel measurable.