On a solution of the optimal stopping problem for processes with independent increments

Abstract

We discuss a solution of the optimal stopping problem for the case when a reward function is a power function of a process with independent stationary increments (random walks or Levy processes) on an infinite time interval. It is shown that an optimal stopping time is the first crossing time through a level defined as the largest root of the Appell function associated with the maximum of the underlying process.

Keywords:

• Optimal stopping
• Lévy processes
• Nonlinear reward function

MSC::

• 62L15
• 62Cxx
• 62Mxx

Acknowledgements

The authors would like to thank two anonymous referees for their helpful comments and suggestions.

Notes

^¶Supported by ARC Discovery grant.

^§Supported by INTAS grant 03-51-50-18

Additional information

Notes on contributors

Alexander Novikov

¶ ¶Supported by ARC Discovery grant.

Albert Shiryaev

§ §Supported by INTAS grant 03-51-50-18