Strulovici, Bruno and Szydlowski, Martin
(2012):
*On the Smoothness of Value Functions.*

Preview |
PDF
MPRA_paper_36326.pdf Download (415kB) | Preview |

## Abstract

We prove that under standard Lipschitz and growth conditions, the value function of all optimal control problems for one-dimensional diffusions is twice continuously differentiable, as long as the control space is compact and the volatility is uniformly bounded below, away from zero. Under similar conditions, the value function of any optimal stopping problem is continuously differentiable. For the first problem, we provide sufficient conditions for the existence of an optimal control, which is also shown to be Markov. These conditions are based on the theory of monotone comparative statics.

Item Type: | MPRA Paper |
---|---|

Original Title: | On the Smoothness of Value Functions |

Language: | English |

Keywords: | Super Contact; Smooth Pasting; HJB Equation; Optimal Control; Markov Control; Comparative Statics; Supermodularity; Single-Crossing; Interval Dominance Order |

Subjects: | C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis |

Item ID: | 36326 |

Depositing User: | Bruno Strulovici |

Date Deposited: | 01 Feb 2012 07:26 |

Last Modified: | 28 Sep 2019 09:22 |

References: | Aliprantis, C., Border, K. (2006) Innite Dimensional Analysis, Second Edition, Springer-Verlag. Bailey, P. (1968) Nonlinear Two Point Boundary Value Problems, Academic Press. Barlow, M (1982) \One Dimensional Stochastic Dierential Equations with No Strong Solution," Journal of the London Mathematical Society, Vol s2-26, pp. 335-347. Hartman, P. (2002) Ordinary Dierential Equations, Second Edition, Classics in Applied Mathematics, SIAM. Josephy, M. (1981) \Composing Functions of Bounded Variation," Proceedings of the American Mathematical Society, Vol. 83, pp. 354{356. Karatzas, I., Shreve, S. (1998) Brownian Motion and Stochastic Calculus, Second Edition, Springer. Karlin, S., Rubin, H. (1956) \The Theory of Decision Procedures for Distributions With Monotone Likelihood Ratio," Annals of Mathematical Statistics, Vol. 27, pp. 272-299. Krylov, N. (1980) Controlled Diusion Processes, Springer Verlag, Springer. Lehmann, E. (1988) \Comparing Location Experiments," Annals of Statistics, Vol. 16, pp. 521533. Milgrom, P., Shannon, C. (1994) \Monotone Comparative Statics," Econometrica, Vol. 62, pp. 157{180. Nakao, S. (1972) \On the Pathwise Uniqueness of Solutions of One-Dimensional Stochastic Differential Equations," Osaka Journal of Mathematics, Vol. 9, pp. 513-518. Quah, J., Strulovici, B. (2009) \Comparative Statics, Informativeness, and the Interval Dominance Order," Econometrica, Vol. 77, pp. 1949-1992. Revuz, D., Yor, M. (2001) Continuous Martingales and Brownian motion, Third Edition (Corrected), Springer-Verlag. Schrader, K. (1969) \Existence theorems for second order boundary value problems," Journal of Dierential Equations, Vol. 5, pp. 572{584. Topkis, D. (1978) \ Minimizing a Submodular Function on a Lattice," Operations Research, Vol. 26, 305{321. Veinott, A. (1989) \Lattice Programming," Unpublished Lecture Notes, Stanford University. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/36326 |