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 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). (An erratum is attached.)


  1. Hamilton--Jacobi--Bellman equation
  2. combined stochastic and impulse control
  3. policy iteration
  4. weakly chained diagonally dominant matrix
  5. optimal exchange rate
  6. optimal consumption
  7. GMWB

MSC codes

  1. 65N06
  2. 93E20

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 1341 - 1364
ISSN (online): 1095-7170


Submitted: 12 October 2015
Accepted: 18 March 2016
Published online: 5 May 2016


  1. Hamilton--Jacobi--Bellman equation
  2. combined stochastic and impulse control
  3. policy iteration
  4. weakly chained diagonally dominant matrix
  5. optimal exchange rate
  6. optimal consumption
  7. GMWB

MSC codes

  1. 65N06
  2. 93E20



Funding Information

Natural Sciences and Engineering Research Council of Canadahttp://dx.doi.org/10.13039/501100000038: 36828

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