Journal of Applied Probability

On threshold strategies and the smooth-fit principle for optimal stopping problems

Stéphane Villeneuve

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Appl. Probab. Volume 44, Number 1 (2007), 181-198.

First available in Project Euclid: 30 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65]

Optimal stopping principle of smooth fit local time


Villeneuve, Stéphane. On threshold strategies and the smooth-fit principle for optimal stopping problems. J. Appl. Probab. 44 (2007), no. 1, 181--198. doi:10.1239/jap/1175267171.

Export citation


  • Alvarez, L. R. H. (2003). On the properties of $r$-excessive mappings for a class of diffusions. Ann. Appl. Prob. 13, 1517--1533.
  • Borodin, A. N. and Salminen, P. (1996). Handbook on Brownian Motion---Facts and Formulae. Birkhäuser, Basel.
  • Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173--212.
  • Décamps, J. P., Mariotti, T. and Villeneuve, S. (2006). Irreversible investment in alternative projects. Econometric Theory 28, 425--448.
  • Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press.
  • Dupuis, P. and Wang, H. (2005). On the convergence from discrete to continuous time in an optimal stopping problem. Ann. Appl. Prob. 15, 1339--1366.
  • Ekström, E. (2004). Properties of American option prices. Stoch. Process. Appl. 114, 265--278.
  • El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. In Ninth Saint Flour Prob. Summer School (Lecture Notes Math. 876), Springer, Berlin, pp. 74--239.
  • El Karoui, N., Jeanblanc, M. and Shreve, S. (1998). Robustness of the Black and Scholes formula. Math. Finance 8, 93--126.
  • Itô, K. and McKean, H. P., Jr. (1965). Diffusion Processes and Their Sample Paths. Springer, Berlin.
  • Janson, S. and Tysk, J. (2003). Volatility time and properties of option prices. Ann. Appl. Prob. 13, 890--913.
  • Jönsson, H., Kukush, A. G. and Sylvestrov, D. S. (2005). Threshold structure for optimal stopping strategy for American type option. I. Theory Prob. Math. Statist. 71, 93--103.
  • Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.
  • Karatzas, I. and Shreve, S. (1998). Methods of Mathematical Finance. Springer, Berlin.
  • Kotlow, D. B. (1973). A free boundary problem connected with the optimal stopping problem for diffusion processes. Trans. Amer. Math. Soc. 184, 457--478.
  • Kyprianou, A. E. and Pistorius, M. R. (2003). Perpetual options and Canadization through fluctuation theory. Ann. Appl. Prob. 13, 1077--1098.
  • Mandl, P. (1968). Analytical Treatment of One-Dimensional Markov Processes. Springer, New York.
  • Martini, C. (1992). Propagation of convexity by Markovian and martingalian semigroups. Potential Anal. 10, 133--175.
  • McDonald, R. and Siegel, D. (1986). The value of waiting to invest. Quart. J. Econom. 101, 707--727.
  • Øksendal, B. (1998). Stochastic Differential Equations: An Introduction with Applications, 5th edn. Springer, Berlin.
  • Salminen, P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85--101.
  • Villeneuve, S. (1999). Exercise regions of American options on several assets. Finance Stoch. 3, 295--322.