On the optimal stopping problem for one-dimensional diffusions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:107

Article

Dayanik, S ; Karatzas, L

关键词: optimal stopping;    diffusions;    principle of smooth-fit;    convexity;   

DOI  :  10.1016/S0304-4149(03)00076-0

来源: Elsevier

【 摘 要 】

A new characterization of excessive functions for arbitrary one-dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as the smallest nonnegative concave majorant of the reward function and allows us to generalize results of Dynkin and Yushkevich for standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process to an undiscounted optimal stopping problem for standard Brownian motion. The concavity of the value functions also leads to conclusions about their smoothness, thanks to the properties of concave functions. One is thus led to a new perspective and new facts about the principle of smooth-fit in the context of optimal stopping. The results are illustrated in detail on a number of non-trivial, concrete optimal stopping problems, both old and new. (C) 2003 Elsevier B.V. All rights reserved.

【 授权许可】

Free