Abstract
In this paper, we continue to study a diffusion-type, finite-fuel singular stochastic control problem and the related stochastic differential equations with discontinuous paths and reflecting boundary conditions as denned in the previous work of the author [15]. The measurable dependence of the solution with respect to the initial state and the underlying probability measures (with Prohorov metric) is derived. Also, the approximation of certain complete class of controls by those with continuous paths is proved to be possible in a weak sense. With the help of these results, we prove the Dynamic Programming Principle (Bellman principle) rigorously and show that the value function is the viscosity solution of certain Hamilton-Jacobi-Bellman equation
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