Discounted optimal stopping zero-sum games in diffusion type models with maxima and minima

Gapeev, Pavel V. ORCID: 0000-0002-1346-2074 (2024) Discounted optimal stopping zero-sum games in diffusion type models with maxima and minima. Advances in Applied Probability. ISSN 0001-8678

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Abstract

We present a closed-form solution to a discounted optimal stopping zero-sum game in a model based on a generalised geometric Brownian motion with coefficients depending on its running maximum and minimum processes. The optimal stopping times forming a Nash equilibrium are shown to be the first times at which the original process hits certain boundaries depending on the running values of the associated maximum and minimum processes. The proof is based on the reduction of the original game to the equivalent coupled free-boundary problem and the solution of the latter problem by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are partially determined as either unique solutions to the appropriate system of arithmetic equations or unique solutions to the appropriate first-order nonlinear ordinary differential equations. The results obtained are related to the valuation of the perpetual lookback game options with floating strikes in the appropriate diffusion-type extension of the Black-Merton-Scholes model.

Item Type:	Article
Official URL:	https://www.cambridge.org/core/journals/advances-i...
Additional Information:	© 2025 The Author(s)
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	18 Oct 2021 10:54
Last Modified:	22 Feb 2025 17:25
URI:	http://eprints.lse.ac.uk/id/eprint/112457

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