Weakly chained matrices, policy iteration, and impulse control

Abstract

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 (c) pricing guaranteed minimum withdrawal benefits in variable annuities. We find that one should abstain from using scheme (i).

Changed Sec. 6.1 to study a two-sided intervention problem. Made minor corrections to Sec. 5.3 boundary term and proofs of Lemmas 4.1/4.6. Required X to be nonempty in Lemma A.2. Main theoretical results unchanged. Minor bib/LaTeX updates

References in corpus (1)

• The existence of optimal bang-bang controls for GMxB contracts

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