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gradient constraint

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Analysis of Hamilton-Jacobi-Bellman equations arising instochastic singular control ∗
Ryan Hynd
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 19 / Issue 1 / January 2013

Published online by Cambridge University Press:

01 March 2012, pp. 112-128

Print publication:

January 2013
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We study the partial differential equation
max{Lu − f, H(Du)} = 0
where u is the unknownfunction, L is a second-order elliptic operator, f is agiven smooth function and H is a convex function. This is a modelequation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. Weestablish the existence of a unique viscosity solution of the Dirichlet problem that has aHölder continuous gradient. We also show that if H is uniformlyconvex, the gradient of this solution is Lipschitz continuous.