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.
Classification : 35J15, 49L25, 35R35, 49L20
Mots clés : 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},
zbl = {1259.49043},
mrnumber = {3023063},
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 DA - 2013/// SP - 112 EP - 128 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012001/ UR - https://zbmath.org/?q=an%3A1259.49043 UR - https://www.ams.org/mathscinet-getitem?mr=3023063 UR - https://doi.org/10.1051/cocv/2012001 DO - 10.1051/cocv/2012001 LA - en ID - COCV_2013__19_1_112_0 ER -
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. http://www.numdam.org/articles/10.1051/cocv/2012001/
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