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

Ryan Hynd

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 1, page 112-128
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Hynd, Ryan. "Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control." ESAIM: Control, Optimisation and Calculus of Variations 19.1 (2013): 112-128. <http://eudml.org/doc/272884>.

@article{Hynd2013,
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.},
author = {Hynd, Ryan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {HJB equation; gradient constraint; free boundary problem; singular control; penalty method; viscosity solutions; Hamilton-Jacobi-Bellman equation},
language = {eng},
number = {1},
pages = {112-128},
publisher = {EDP-Sciences},
title = {Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control},
url = {http://eudml.org/doc/272884},
volume = {19},
year = {2013},
}

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
PB - EDP-Sciences
VL - 19
IS - 1
SP - 112
EP - 128
AB - 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.
LA - eng
KW - HJB equation; gradient constraint; free boundary problem; singular control; penalty method; viscosity solutions; Hamilton-Jacobi-Bellman equation
UR - http://eudml.org/doc/272884
ER -

References

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  8. [8] R.T. Rockafellar and R. Wets, Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin (1998). Zbl0888.49001MR1491362
  9. [9] S.E. Shreve and H.M. Soner, A free boundary problem related to singular stochastic control, Applied stochastic analysis (London, 1989), Stochastics Monogr. 5. Gordon and Breach, New York (1991) 265–301. Zbl0733.93083MR1108426
  10. [10] H.M. Soner and S. Shreve, Regularity of the value function for a two-dimensional singular stochastic control problem. SIAM J. Control Optim.27 (1989) 876–907. Zbl0685.93076MR1001925
  11. [11] M. Wiegner, The C1,1-character of solutions of second order elliptic equations with gradient constraint. Comm. Partial Differential Equations 6 (1981) 361–371. Zbl0458.35035MR607553

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