no. 4
A fast algorithm for the two dimensional HJB equation of stochastic control
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 4, pp. 723-735.

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(p max ) operations, where p max 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.

DOI : 10.1051/m2an:2004034
Classification : 49L99, 93E20
Mots clés : stochastic control, finite differences, viscosity solutions, consistency, HJB equation, Stern-Brocot tree 
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     author = {Bonnans, J. Fr\'ed\'eric and Ottenwaelter, \'Elisabeth and Zidani, Housnaa},
     title = {A fast algorithm for the two dimensional {HJB} equation of stochastic control},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {723--735},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {4},
     year = {2004},
     doi = {10.1051/m2an:2004034},
     mrnumber = {2087732},
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     url = {http://www.numdam.org/articles/10.1051/m2an:2004034/}
}
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Bonnans, J. Frédéric; Ottenwaelter, Élisabeth; Zidani, Housnaa. A fast algorithm for the two dimensional HJB equation of stochastic control. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 4, pp. 723-735. doi : 10.1051/m2an:2004034. http://www.numdam.org/articles/10.1051/m2an:2004034/

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