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},
year = {2013},
publisher = {EDP Sciences},
volume = {19},
number = {1},
doi = {10.1051/cocv/2012001},
mrnumber = {3023063},
zbl = {1259.49043},
language = {en},
url = {https://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 - https://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 https://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, Tome 19 (2013) no. 1, pp. 112-128. doi: 10.1051/cocv/2012001
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