A Study on the Properties of a Class of Optimal Switching Problems

Abstract

The theory of optimal switching constitutes a fundamental domain in economics and finance. A book by Professor Pham has analyzed some properties of the solution of the classical optimal switching problem under a general case. The results are elegant, with significant financial implications and valuable contributions to the study of market switching problems. However, we found a small gap in the proof of the conclusion. Although easily overlooked, this gap is critical since it undermines the reliability of judging switching region properties solely based on a specific condition involving key market-related parameters. We present a counterexample to directly refute that the aforementioned condition alone suffices to ensure a non-empty switching region. Moreover, we reformulate a double-obstacle variational inequality to consider the properties, instead of directly using the systems of variational inequalities as in Pham’s book. We also provide an equivalent characterization of the switching region, and some equivalent and sufficient conditions for the switching region to be non-empty.

Keywords

Optimal switching problem / Double-obstacle problem / Switching region / 91B70 / 90C15 / 93E20 / 49N90

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Funding

National Natural Science Foundation of China(12071292)

RIGHTS & PERMISSIONS

Shanghai University