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A fast algorithm for the two dimensionalHJB equation of stochastic control

Published online by Cambridge University Press:  15 August 2004

J. Frédéric Bonnans ,
Élisabeth Ottenwaelter  and
Housnaa Zidani
J. Frédéric Bonnans
Affiliation:
Projet Sydoco, Inria-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay, France. Frederic.Bonnans@inria.fr.
Élisabeth Ottenwaelter
Affiliation:
IUT de Paris and Projet Sydoco, Inria-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay, France. Elisabeth.Ottenwaelter@inria.fr.
Housnaa Zidani
Affiliation:
Projet Sydoco, Inria-Rocquencourt and Unité de Mathématiques Appliquées, ENSTA, 32 Boulevard Victor, 75739 Paris Cedex 15, France. zidani@ensta.fr.
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Abstract

This paper analyses the implementation of the generalized finite differences method for the HJB equation of stochastic control, introduced by two of the authors in [Bonnans and Zidani, SIAM J. Numer. Anal.41 (2003) 1008–1021]. The computation of coefficients needs to solve at each point of the grid (and for each control) a linear programming problem. We show here that, for two dimensional problems, this linear programming problem can be solved in O(pmax) operations, where pmax is the size of the stencil. The method is based on a walk on the Stern-Brocot tree, and on the related filling of the set of positive semidefinite matrices of size two.

Keywords

Stochastic controlfinite differencesviscosity solutionsconsistencyHJB equationStern-Brocot tree.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 4 , July 2004 , pp. 723 - 735
Copyright
© EDP Sciences, SMAI, 2004

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