Convergence of Implicit Schemes for Hamilton--Jacobi--Bellman Quasi-Variational Inequalities

In [P. Azimzadeh and P. A. Forsyth, SIAM J. Numer. Anal., 54 (2016), pp. 1341--1364], we outlined the theory and implementation of computational methods for implicit schemes for Hamilton--Jacobi--Bellman quasi-variational inequalities (HJBQVIs). No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles--Souganidis-type analysis. Our results rely only on a well-known comparison principle and are independent of the specific form of the intervention operator.