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.
Mots clés : HJB equation, gradient constraint, free boundary problem, singular control, penalty method, viscosity solutions
@article{COCV_2013__19_1_112_0,
author = {Hynd, Ryan},
title = {Analysis of {Hamilton-Jacobi-Bellman} equations arising in stochastic singular control},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {112--128},
publisher = {EDP-Sciences},
volume = {19},
number = {1},
year = {2013},
doi = {10.1051/cocv/2012001},
mrnumber = {3023063},
zbl = {1259.49043},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2012001/}
}
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 SP - 112 EP - 128 VL - 19 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012001/ DO - 10.1051/cocv/2012001 LA - en ID - COCV_2013__19_1_112_0 ER -
%0 Journal Article %A Hynd, Ryan %T Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 112-128 %V 19 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012001/ %R 10.1051/cocv/2012001 %G en %F COCV_2013__19_1_112_0
Hynd, Ryan. Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 112-128. doi : 10.1051/cocv/2012001. http://archive.numdam.org/articles/10.1051/cocv/2012001/
[1] , Viscosity solutions : a primer. Viscosity solutions and applications, Lecture Notes in Math. 1660. Springer, Berlin (1997) 1-43. | MR | Zbl
[2] , and , User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. | MR | Zbl
[3] , A second-order elliptic equation with gradient constraint. Comm. Partial Differential Equations 4 (1979) 555-572. | MR | Zbl
[4] , Partial differential equations. Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998). | MR | Zbl
[5] and , Controlled Markov processes and viscosity solutions, Stochastic Modeling and Applied Probability 25, 2nd edition. Springer, New York (2006). | MR | Zbl
[6] and , Elliptic Partial Differential Equations of Second Order. Springer (1998). | Zbl
[7] and , Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Comm. Partial Differential Equations 8 (1983) 317-346. | MR | Zbl
[8] and , Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin (1998). | MR | Zbl
[9] and , 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. | MR | Zbl
[10] and , Regularity of the value function for a two-dimensional singular stochastic control problem. SIAM J. Control Optim. 27 (1989) 876-907. | MR | Zbl
[11] , The C1,1-character of solutions of second order elliptic equations with gradient constraint. Comm. Partial Differential Equations 6 (1981) 361-371. | MR | Zbl
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