Convergence of implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities

Abstract

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.

Made minor corrections to proofs of Proposition 3, Theorems 4, 9, and 14. Aligned Definition 16 with its cited formulation and updated the proof of Proposition 17 accordingly. Main results unchanged. Minor bibliography/typesetting updates