Optimal stopping zero-sum games in continuous hidden Markov models

Gapeev, Pavel V. ORCID: 0000-0002-1346-2074 (2025) Optimal stopping zero-sum games in continuous hidden Markov models. Advances in Applied Probability. ISSN 0001-8678 (In Press)

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Abstract

We study a two-dimensional discounted optimal stopping zero-sum (or Dynkin) game related to the perpetual redeemable convertible bonds expressed as game (or Israeli) options in a model of financial markets in which the behaviour of the ex-dividend price of a dividend paying asset follows a generalised geometric Brownian motion. It is assumed that the dynamics of the random dividend rate of the asset paid to shareholders are described by the mean-reverting filtering estimate of an unobservable continuous-time Markov chain with two states. It is shown that the optimal exercise (conversion) and withdrawal (redemption) times forming a Nash equilibrium are the first times at which the asset price hits either lower or upper stochastic boundaries being monotone functions of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundaries are regular for the stopping region relative to the resulting two-dimensional diffusion process and the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the optimal stopping boundaries are determined as a unique solution to the associated coupled system of nonlinear Fredholm integral equations among the couples of continuous functions of bounded variation satisfying certain conditions. We also give a closed-form solution to the appropriate optimal stopping zero-sum game in the corresponding model with an observable continuous-time Markov chain.

Item Type:	Article
Additional Information:	© 2025 The Author
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	28 Apr 2025 09:57
Last Modified:	09 May 2025 20:09
URI:	http://eprints.lse.ac.uk/id/eprint/127998

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