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

摘要

In this paper, we study a deterministic infinite horizon, mixed continuous and impulse control problem in Rn, 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 hH”, including a natural one. We prove that the approximate value function uh 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 h goes to zero; the rate of convergence is proved to be in hσ, where 0<σ<1/2.