Degenerate First-Order Quasi-variational Inequalities: An Approach to Approximate the Value Function

The originality of this paper is to deal with the particular case of null infimum jump costs in the infinite horizon impulse control problem. The value function of such problems is a viscosity solution of the classic quasi-variational inequality (QVI) associated, but not the unique one. This is a drawback to characterize it. In this paper, a new QVI for which the value function is the unique viscosity solution is given. This allows us to approximate the value function. So, we give some discrete approximations of the new QVI and prove that the approximate value function converges locally uniformly, toward the value function of the impulse control problem with zero lower bound of impulse cost. We choose the classic example of continuous time inventory control in $\mathbb{R}^n$ to illustrate the results of this paper.