On the optimal stopping problem for one-dimensional diffusions

By Savas Dayanik, Ioannis Karatzas. 2003

272 citations recorded by the source corpus.

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.

Subfields: Demography, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/438bf9fcab448df2fbddcc029218be1351f2b1eb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Demography

1. Stéphane Villeneuve
2. Erik Ekström

In Finance

1. Stéphane Villeneuve
2. Julien Hugonnier
3. Erik Ekström
4. Erwan Morellec
5. Semyon Malamud
6. Gustavo Manso

In Modeling and Simulation

1. Erik Ekström
2. Stéphane Villeneuve

Citations in the corpus

• Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit (2014, unknown): “Among other related studies, Ekstrom et al. (2011) analyze the optimal single liquidation timing under the OU model with zero long-run mean and no transaction cost. The current paper extends their model in a number of ways. First, we analyze the optimal entry timing as well as the optimal liquidatio”
• Capital Supply Uncertainty, Cash Holdings, and Investment (2014, background): “The third claim is formula (6.2) in Dayanik and Karatzas (2003).”
• On the value of optimal stopping games (2006, methodology): “1 in [6], it can therefore be shown that Z(t) := e−βt∧τ ∗ V (X(t ∧ τ∗)) is a sub-martingale provided Ex ( sup 0≤s≤t Z(s) ) <∞”
• On Threshold Strategies and the Smooth-Fit Principle for Optimal Stopping Problems (2007, background): “[13] Karatzas, I. and Shreve, S. (1988).”
• A harmonic function technique for the optimal stopping of diffusions (2011, unknown): “Let us just mention [2,4,10,13].”