Optimal stopping of expected profit and cost yields in an investment under uncertainty

Abstract

We consider a finite horizon optimal stopping problem related to trade-off strategies between expected profit and cost cash flows of an investment under uncertainty. The optimal problem is first formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We then construct both a minimal solution and a maximal solution using an approximation scheme of the associated system of reflected backward stochastic differential equations (SDEs). We also address the question of uniqueness of solutions of this system of SDEs. When the dependence of the cash flows on the sources of uncertainty, such as fluctuation market prices, assumed to evolve according to a diffusion process, is made explicit, we obtain a connection between these solutions and viscosity solutions of a system of variational inequalities with interconnected obstacles.

Keywords:

• optimal stopping
• Snell envelop
• backward stochastic differential equations
• merger and acquisition

AMS Subject Classification::

• 60G40
• 93E20
• 62P20
• 91B99

Acknowledgements

The authors are grateful to Paavo Salminen and the two anonymous reviewers for their insightful remarks and suggestions that improved the content of the work. This work was completed while the first author was visiting the Department of Mathematics of Université du Maine. Financial support from MATPYL (RPL) is gratefully acknowledged.

Notes

1. [email protected]

2. [email protected]

Additional information

Notes on contributors

Said Hamadène

1 1. [email protected]

Marie-Amélie Morlais

2 2. [email protected]