Error Estimates of Penalty Schemes for Quasi-Variational Inequalities Arising from Impulse Control Problems

Abstract

This paper proposes penalty schemes for a class of weakly coupled systems of Hamilton--Jacobi--Bellman quasi-variational inequalities (HJBQVIs) arising from stochastic hybrid control problems of regime-switching models with both continuous and impulse controls. We show that the solutions of the penalized equations converge monotonically to those of the HJBQVIs. We further establish that the scheme is half-order accurate for HJBQVIs with Lipschitz coefficients and first-order accurate for equations with more regular coefficients. Moreover, we construct the action regions and optimal impulse controls based on the error estimates and the penalized solutions. The penalty schemes and convergence results are then extended to HJBQVIs with possibly negative impulse costs. We also demonstrate the convergence of monotone discretizations of the penalized equations and establish that policy iteration applied to the discrete equation is monotonically convergent with an arbitrary initial guess in an infinite-dimensional setting. Numerical examples for infinite-horizon optimal switching problems are presented to illustrate the effectiveness of the penalty schemes over the conventional direct control scheme.