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(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.

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},
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     year = {2004},
     doi = {10.1051/m2an:2004034},
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     url = {https://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. https://www.numdam.org/articles/10.1051/m2an:2004034/

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