An approach for solving perpetual optimal stopping problems driven by Lévy processes

Abstract

In this paper, we expose an approach for solving perpetual optimal stopping problems for a general class of payoff functions under Lévy processes. This approach was inspired by the work of Boyarchenko and Levendorski. In contrast to Boyarchenko and Levendorsikii, our approach does not appeal to a free boundary problem associated to the optimal stopping problem nor to the theory of pseudodifferential operators to solve the problem. Instead, we introduce an averaging problem from which we obtain using the Wiener–Hopf factorization a fluctuation identity for overshoots of a Lévy process. This identity constitutes the main principle in solving the optimal stopping problem. If a solution to the averaging problem can be found and has certain monotonicity properties, we show using the fluctuation identity that an optimal solution to the optimal stopping problem can be written in terms of such monotone function. Using the optimal solution, we give sufficient and necessary conditions for the smooth pasting condition to occur in the considered problem. Our conclusion over the smooth pasting condition extends further the recent result of Alili and Kyprianou into a more general payoff function.

Keywords:

• Optimal stopping problem
• American put-type options
• Principles of smooth and continuous pasting
• Lévy processes and Wiener–Hopf factorization

AMS Subject Classification::

• 60G40
• 60J75
• 91B70
• 60G51

Acknowledgements

The author is grateful to Richard Gill and Andreas Kyprianou for a number of very useful comments. The author is also grateful to a number of anonymous referees for comments on an earlier draft of this paper.

Notes

If h is in the Schwartz class of rapidly decreasing functions, then using integration by parts it can be checked straightforwardly from equation (3.6) that the function admits the estimate , for C>0, as for any integer . This is the reason that the class is useful in studying Fourier transform since whenever . We refer to Hormander [11] for more details on general theory of Fourier integral operators.

See for instance Refs. [1,5,13] and the literature therein for details.

We refer to Chapter VII in Bertoin [4].

This is a process of pure jumps whose characteristic exponent is given for and by where as in the strip . This type of Lévy process was considered by Boyarchenko and Levendorskii [6]. They showed in Ref. [6] that under some regularity conditions imposed on the Wiener–Hopf factor is of the form (5.5).