  In [Azimzadeh, P., and P. A. Forsyth. "Weakly chained matrices, policy
iteration, and impulse control." SIAM J. Num. Anal. 54.3 (2016): 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.
