Optimal reflection of diffusions and barrier options pricing under constraints

We introduce a new class of control problems in which the gain depends on the solu- tion of a stochastic differential equation (SDE) reflected at the boundary of a bounded domain, along directions which are controlled by a bounded variation process. We provide a PDE characterization of the associated value function. This study is motivated by applications in mathematical finance, where such equations are related to the pricing of barrier options under portfolio constraints.