On Threshold Strategies and the Smooth-Fit Principle for Optimal Stopping Problems

In this paper we investigate sufficient conditions that ensure the optimality of threshold strategies for optimal stopping problems with finite or perpetual maturities. Our result is based on a local-time argument that enables us to give an alternative proof of the smooth-fit principle. Moreover, we present a class of optimal stopping problems for which the propagation of convexity fails.

[1]  Luis H. R. Alvarez,et al.  On the properties of $r$-excessive mappings for a class of diffusions , 2003 .

[2]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .

[3]  Daniel B. Kotlow.  A free boundary problem connected with the optimal stopping problem for diffusion processes , 1973 .

[4]  E. Prescott,et al.  Investment Under Uncertainty , 1971 .

[5]  N. Karoui.  Les Aspects Probabilistes Du Controle Stochastique , 1981 .

[6]  Paul Dupuis,et al.  On the convergence from discrete to continuous time in an optimal stopping problem , 2005, math/0505241.

[7]  Paavo Salminen,et al.  Optimal Stopping of One‐Dimensional Diffusions , 1985 .

[8]  Savas Dayanik,et al.  On the optimal stopping problem for one-dimensional diffusions , 2003 .

[9]  Erik Ekström.  Properties of American option prices , 2004 .

[10]  S. Shreve,et al.  Robustness of the Black and Scholes Formula , 1998 .

[11]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[12]  Thomas Mariotti,et al.  Irreversible investment in alternative projects , 2006 .

[13]  Stéphane Villeneuve,et al.  Exercise regions of American options on several assets , 1999, Finance Stochastics.

[14]  Svante Janson,et al.  Volatility time and properties of option prices , 2003 .

[15]  Propagation of Convexity by Markovian and Martingalian Semigroups , 1999 .

[16]  P. Mandl.  Analytical treatment of one-dimensional Markov processes , 1968 .

[17]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[18]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[19]  R. McDonald,et al.  The Value of Waiting to Invest , 1982 .

[20]  A. Kyprianou,et al.  Perpetual options and Canadization through fluctuation theory , 2003 .