| Issue |
ESAIM: COCV
Volume 19, Number 1, January-March 2013
|
|
|---|---|---|
| Page(s) | 112 - 128 | |
| DOI | https://doi.org/10.1051/cocv/2012001 | |
| Published online | 01 March 2012 | |
Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control∗
Courant Institute of Mathematical Sciences, New York
University, 251 Mercer
Street, New York,
10012-1185
NY,
USA
rhynd@cims.nyu.edu
Received:
22
June
2011
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.
Mathematics Subject Classification: 35J15 / 49L25 / 35R35 / 49L20
Key words: HJB equation / gradient constraint / free boundary problem / singular control / penalty method / viscosity solutions
© EDP Sciences, SMAI, 2012
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