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Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control∗

Published online by Cambridge University Press:  01 March 2012

Ryan Hynd
Ryan Hynd*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, 10012-1185 NY, USA. rhynd@cims.nyu.edu
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Abstract

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 equationgradient constraintfree boundary problemsingular controlpenalty methodviscosity solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 112 - 128
Copyright
© EDP Sciences, SMAI, 2012

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