The value function of a finite fuel problem for a new class of singular stochastic controls

We investigate, via the dynamic programming approach, a finite fuel nonlinear singular stochastic control problem of Bolza type. We prove that the associated value function is continuous and that its continuous extension to the closure of the domain coincides with the value function of a non singular control problem, for which we prove the existence of an optimal control. Moreover such a continuous extension is characterized as the unique viscosity solution of a quasi variational inequality with suitable boundary conditions of mixed type.