Singular optimal control for stochastic differential equations

Studies the control problem where the state is governed by an Ito stochastic differential equation allowing both classical control 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. The dynamic programming principle for the problem is established. When assuming Lipschitz continuity on the data, it is shown that the value function is continuous and is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation