System of variational inequalities with interconnected obstacles

Abstract

Our objective with this paper is to discuss multi-switching problems, arising as variational inequalities, that models decision under uncertainty. We prove general existence theory through monotone scheme and discuss iterative methods for numerical results. We also connect the recently developed models for asset bubbles (which is a non-local problem) to switching problems with two possible switching cases.

Keywords:

• Optimal switching
• free boundaries
• viscosity solution
• interconnected obstacles

2010 Mathematics Subject Classifications:

• 60G40
• 49L25
• 35R35
• 93E20

Acknowledgments

This work was partially 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.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Apply the maximum principle to $u_i^{\epsilon }\pm Cv$, for large values of C, and v solving $\mathrm{\Delta }v=1$, with zero boundary data.

Additional information

Funding

This work was partially supported by Qatar National Research Fund (a member of Qatar Foundation) [grant number NPRP 5-088-1-021].