Weakly chained matrices, policy iteration, and impulse control

This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the HJBQVI problem. Scheme (i) takes the form of a Bellman problem involving an operator which is not necessarily contractive. We consider the well-posedness of the Bellman problem and give sufficient conditions for convergence of the corresponding policy iteration. To do so, we use weakly chained diagonally dominant matrices, which give a graph-theoretic characterization of nonsingular weakly diagonally dominant M-matrices. We compare schemes (i)–(iii) under the following examples: (a) optimal control of the exchange rate, (b) optimal consumption with fixed and proportional transaction costs, and © pricing guaranteed minimum withdrawal benefits in variable annuities. We find that one should abstain from using scheme (i).