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Analysis of Hamilton-Jacobi-Bellman equations arising instochastic 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. [email protected]
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Abstract

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.

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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References

M. Crandall, Viscosity solutions : a primer. Viscosity solutions and applications, Lecture Notes in Math. 1660. Springer, Berlin (1997) 1–43.
Crandall, M.G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. Google Scholar
Evans, L.C., A second-order elliptic equation with gradient constraint. Comm. Partial Differential Equations 4 (1979) 555–572. Google Scholar
L.C. Evans, Partial differential equations. Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998).
W. Fleming and H. Soner, Controlled Markov processes and viscosity solutions, Stochastic Modeling and Applied Probability 25, 2nd edition. Springer, New York (2006).
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer (1998).
Ishii, H. and Koike, S., Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Comm. Partial Differential Equations 8 (1983) 317–346. Google Scholar
R.T. Rockafellar and R. Wets, Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin (1998).
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.
Soner, H.M. and Shreve, S., Regularity of the value function for a two-dimensional singular stochastic control problem. SIAM J. Control Optim. 27 (1989) 876–907. Google Scholar
Wiegner, M., The C 1,1-character of solutions of second order elliptic equations with gradient constraint. Comm. Partial Differential Equations 6 (1981) 361–371. Google Scholar