Advertisement

The High Contact Principle in Optimal Stopping and Stochastic Waves

  • Bernt Øksendal
Chapter
Part of the Progress in Probability book series (PRPR, volume 18)

Abstract

The high contact principle in optimal stopping states that at the boundary ∂D of the continuation region D the reward function g has a smooth fit with the optimal expected reward function g*, in the sense that
$$\begin{gathered} g = {g^*}on{\text{ }}\partial D \hfill \\ \nabla g = \nabla g*on{\text{ }}\partial D \hfill \\ \end{gathered} $$
Thus this principle gives the crucial link between optimal stopping and free boundary problems.

Keywords

Variational Inequality Stochastic Differential Equation Free Boundary Problem Reward Function Geometric Brownian Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. A. Bather: Optimal stopping problems for brownian motion. Advances in Appl. Prob. 2 (1970), 259–286.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Bensoussan & J. L. Lions: Applications of Variational Inequalities in Stochastic Control. North-Holland 1982.Google Scholar
  3. [3]
    E. B. Dynkin: Markov Processes, Vol. I. Springer-Verlag 1965.Google Scholar
  4. [4]
    E. B. Dynkin: Markov Processes, Vol. II. Springer-Verlag 1965.Google Scholar
  5. [5]
    E. B. Dynkin & R. J. Vanderbei: Stochastic waves. Transactions Amer. Math. Soc. 275 (1983), 771–779.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    A. Friedman: Stochastic Differential Equations and Applications, Vol. II. Academic Press 1976.Google Scholar
  7. [7]
    H. P. McKean: A free boundary problem for the heat equation arising from a problem of mathematical economics. Industrial managem. review 6 (1965), 32–39.MathSciNetGoogle Scholar
  8. [8]
    R. C. Merton: The theory of rational option pricing. Bell J. of Economic & Management Science 4 (Spring) (1973), 141–183.MathSciNetCrossRefGoogle Scholar
  9. [9]
    C. Miranda: Partial Differential Equations of Elliptic Type. (2. ed.) Springer-Verlag 1970.Google Scholar
  10. [10]
    B. Øksendal: Stochastic Differential Equations (2. ed.) Springer-Verlag 1989.Google Scholar
  11. [11]
    P. A. Samuelson: Rational theory of warrant pricing. Industrial managem. review 6 (1965), 13–32.Google Scholar
  12. [12]
    A. N. Shiryaev: Optimal Stopping Rules. Springer-Verlag 1978.Google Scholar
  13. [13]
    P. Van Moerbeke: An optimal stopping problem with linear reward. Acta Mathematica 132 (1974), 111–151.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • Bernt Øksendal
    • 1
  1. 1.Dept. of MathematicsUniversity of California, San DiegoLa JollaUSA

Personalised recommendations