Iterative scheme for an elliptic non-local free boundary problem

Abstract

In this paper we treat a non-local free boundary problem arising in financial bubbles, where the model is set in the framework of viscosity solutions. We suggest an iterative scheme which consists of a sequence of obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration scheme. The suggested approach gives a solution to the theoretical problem and paves the way for the numerical implementation, as done in the text below.

Keywords:

• Free boundary problem
• non-local obstacle-type problem
• financial bubbles
• iterative scheme
• viscosity solution

AMS Subject Classifications:

• 35R35
• 35D40
• 35J66

Acknowledgements

The authors would like to thank Prof. Henrik Shahgholian for suggesting the problem and for several helpful discussions during the work.

Notes

No potential conflict of interest was reported by the authors.

1 It should be remarked that even though we call this an obstacle, it is indeed not the only obstacle function. The obstacle is actually given by .

Additional information

Funding

This work was supported by Qatar National Research Fund (a member of Qatar Foundation) [grant number NPRP 5-088-1-021]. The statements made herein are solely the responsibility of the authors.