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Applied Mathematics & Optimization

, Volume 71, Issue 1, pp 125–163 | Cite as

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

  • Olivier Bokanowski
  • Athena Picarelli
  • Hasnaa Zidani
Article

Abstract

This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton–Jacobi–Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.

Keywords

Hamilton–Jacobi equations Oblique Neuman boundary condition Error estimate Viscosity notion Reachable sets under state constraints Lookback options Maximum cost 

Mathematics Subject Classification

49J20 49L25 65M15 35K55 

Notes

Acknowledgments

This work was partially supported by the EU under the 7th Framework Programme Marie Curie Initial Training Network “FP7-PEOPLE-2010-ITN”, SADCO project, GA number 264735-SADCO.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Olivier Bokanowski
    • 1
  • Athena Picarelli
    • 2
  • Hasnaa Zidani
    • 3
  1. 1.Laboratoire Jacques-Louis Lions, Université Paris-Diderot (Paris 7) UFR de Mathématiques - Bât. Sophie GermainParis Cedex 13France
  2. 2.Projet Commands, INRIA Saclay & ENSTA ParisTechPalaiseau CedexFrance
  3. 3.Unité de Mathématiques appliquées (UMA), ENSTA ParisTechPalaiseau CedexFrance

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