We study the partial differential equation max{Lu - f, H(Du)} = 0 where u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Hölder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.

Keywords: HJB equation, gradient constraint, free boundary problem, singular control, penalty method, viscosity solutions

@article{COCV_2013__19_1_112_0, author = {Hynd, Ryan}, title = {Analysis of {Hamilton-Jacobi-Bellman} equations arising in stochastic singular control}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {112--128}, publisher = {EDP-Sciences}, volume = {19}, number = {1}, year = {2013}, doi = {10.1051/cocv/2012001}, mrnumber = {3023063}, zbl = {1259.49043}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012001/} }

TY - JOUR AU - Hynd, Ryan TI - Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 112 EP - 128 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012001/ DO - 10.1051/cocv/2012001 LA - en ID - COCV_2013__19_1_112_0 ER -

%0 Journal Article %A Hynd, Ryan %T Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 112-128 %V 19 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012001/ %R 10.1051/cocv/2012001 %G en %F COCV_2013__19_1_112_0

Hynd, Ryan. Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 112-128. doi : 10.1051/cocv/2012001. http://www.numdam.org/articles/10.1051/cocv/2012001/

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