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.

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.

DOI : 10.1051/cocv/2012001
Classification : 35J15, 49L25, 35R35, 49L20
Mots clés : HJB equation, gradient constraint, free boundary problem, singular control, penalty method, viscosity solutions
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     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},
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     url = {http://www.numdam.org/articles/10.1051/cocv/2012001/}
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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://www.numdam.org/articles/10.1051/cocv/2012001/

[1] M. Crandall, Viscosity solutions : a primer. Viscosity solutions and applications, Lecture Notes in Math. 1660. Springer, Berlin (1997) 1-43. | MR | Zbl

[2] M.G. Crandall, H. Ishii and P.-L. Lions, 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] L.C. Evans, A second-order elliptic equation with gradient constraint. Comm. Partial Differential Equations 4 (1979) 555-572. | MR | Zbl

[4] L.C. Evans, Partial differential equations. Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998). | MR | Zbl

[5] W. Fleming and H. Soner, Controlled Markov processes and viscosity solutions, Stochastic Modeling and Applied Probability 25, 2nd edition. Springer, New York (2006). | MR | Zbl

[6] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer (1998). | Zbl

[7] H. Ishii and S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Comm. Partial Differential Equations 8 (1983) 317-346. | MR | Zbl

[8] R.T. Rockafellar and R. Wets, Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin (1998). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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