For a class of Bellman equations in bounded domains we prove that sub- and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with ergodic control with state constraints. In this context, our condition on the behavior of the operator near the boundary ensures some invariance property of the domain for the associated controlled diffusion process.
DOI : 10.1051/cocv/2015033
Keywords: Hamilton-Jacobi-Bellman equations, degenerate elliptic PDEs, stochastic control, exit-time problems, ergodic control with state constraints, viscosity solutions
Bardi, Martino 1 ; Cesaroni, Annalisa 1, 2 ; Rossi, Luca 1
@article{COCV_2016__22_3_842_0,
author = {Bardi, Martino and Cesaroni, Annalisa and Rossi, Luca},
title = {Nonexistence of nonconstant solutions of some degenerate {Bellman} equations and applications to stochastic control},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {842--861},
year = {2016},
publisher = {EDP Sciences},
volume = {22},
number = {3},
doi = {10.1051/cocv/2015033},
mrnumber = {3527947},
zbl = {1346.35067},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2015033/}
}
TY - JOUR AU - Bardi, Martino AU - Cesaroni, Annalisa AU - Rossi, Luca TI - Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 842 EP - 861 VL - 22 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2015033/ DO - 10.1051/cocv/2015033 LA - en ID - COCV_2016__22_3_842_0 ER -
%0 Journal Article %A Bardi, Martino %A Cesaroni, Annalisa %A Rossi, Luca %T Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 842-861 %V 22 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2015033/ %R 10.1051/cocv/2015033 %G en %F COCV_2016__22_3_842_0
Bardi, Martino; Cesaroni, Annalisa; Rossi, Luca. Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 842-861. doi: 10.1051/cocv/2015033
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