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Solving Impulse-Control Problems with Control Delays

  • Kumar MuthuramanEmail author
  • Qi Wu
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 19)

Abstract

Several classes of stochastic control problems, wherein the controller can bring about an instantaneous change in state, lead to free boundary problems. Moving boundary methods are a class of computational methods that have been developed recently to solve such free boundary problems. The goal of this paper is to provide a detailed description of the methodology. We specifically focus on stochastic impulse-control problems which arise when the cost of control includes a fixed cost. The inclusion of a fixed cost, very common in financial applications, and makes the control effect finite changes in state, bringing about discontinuities in the state evolution. These problems are, hence, more complicated than problems wherein controls have only proportional costs (singular control) or wherein controls simply terminate the process (optimal stopping). We show how the impulse-control problem is transformed to a Quasi Variational Inequality and then describe the moving boundary method. We demonstrate problems with no control delay, fixed delay and stochastic delay. We also review all the theoretical guarantees that have been established. This paper summarizes and presents an implementation focused description of the research presented in (Feng and Muthuraman, A computational method for stochastic impulse control problems. Mathematics of Operations Research 35(4), 830–850, 2010 and Muthuraman et al., Inventory management with stochastic lead times. Working Paper, 2011) for solving impulse-control problems.

Keywords

Lead Time Inventory Level Free Boundary Problem Inventory Position Stochastic Control Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bar-Ilan, A., Sulem, A.: Explicit solution of inventory problems with delivery lags. Math. Oper. Res. 20(3), 709–720 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bar-Ilana, A., Perryb, D., Stadjec, W.: A generalized impulse control model of cash management. J. Econ. Dynam. Contr. 28, 1113–1133 (2004)Google Scholar
  3. 3.
    Brennan, M.J., Schwartz, E.S.: Evaluating natural resource investments. J. Business 58(2), 135–157 (1985)CrossRefGoogle Scholar
  4. 4.
    Constantinides, G.M., Richard, S.F.: Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time. Oper. Res. 26(4), 620–636 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Feng, H., Muthuraman, K.: A computational method for stochastic impulse control problems. Math. Oper. Res. 35(4), 830–850 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Kumar, S., Muthuraman, K.: A numerical method for solving stochastic singular control problems. Oper. Res. 52(4), 563–582 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Muthuraman, K.: A moving boundary approach to American option pricing. J. Econ. Dynam. Contr. 32(11), 3520–3537 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Muthuraman, K., Kumar, S.: Multi-dimensional portfolio optimization with proportional transaction costs. Math. Finance 16(2), 301–335 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Muthuraman, K., Kumar, S.: Solving free-boundary problems with application in finance. Foundations and Trends in Stochastic Systems 1(4), 259–341 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Muthuraman, K., Seshadri, S., Wu, Q.: Inventory management with stochastic lead times. Working Paper (2011)Google Scholar
  11. 11.
    Muthuraman, K., Zha, H.: Simulation based portfolio optimization for large portfolios with transaction costs. Math. Finance 18(1), 115–134 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Richard, S.F.: Optimal impulse control of a diffusion process with both fixed and proportional costs of control. SIAM J. Contr. Optim. 15, 77–91 (1977)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sulem, A.: A solvable one-dimensional model of a diffusion inventory system. Math. Oper. Res. 11(1), 125–133 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Zipkin, P.H.: Stochastic lead times in continuous-time inventory models. Nav. Res. Logist. Q. 33(4), 763–774 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Zipkin, P.H.: Foundations of Inventory Management. McGraw-Hill/Irwin, NY (2000)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.McCombs School of BusinessUniversity of TexasAustinUSA

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