Deterministic impulse control problems: Two discrete approximations of the quasi-variational inequality

In this paper, we study a deterministic infinite horizon, mixed continuous and impulse control problem in RnRn, with general impulses, and cost of impulses. We assume that the cost of impulses is a positive function. We prove that the value function of the control problem is the unique viscosity solution of the related first order Hamilton–Jacobi quasi-variational inequality.1 We then propose time discretization schemes of this QVI, where we consider two approximations of the “Hamiltonian hHhH”, including a natural one. We prove that the approximate value function uhuh exists, that it is the unique solution of the approximate QVI and that it forms a uniformly bounded and uniformly equicontinuous family. We also prove that the approximate value function converges locally uniformly, towards the value function of the control problem, when the discretization step hh goes to zero; the rate of convergence is proved to be in hσhσ, where 0<σ<1/20<σ<1/2.