Optimal stopping and impulse control in the presence of an anticipated regime switch

We consider a class of stochastic optimal stopping and impulse control problems where the agent solving the problem anticipates that a regime switch will happen at a random time in the future. We assume that there are only two regimes, the regime switching time is exponentially distributed, the unde...

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Bibliographic Details
Published in:Mathematical methods of operations research (Heidelberg, Germany) Germany), 2023-10, Vol.98 (2), p.205-230
Main Authors: Alvarez E, Luis H. R, Sillanpää, Wiljami
Format: Article
Language:English
Subjects:
Brownian motion
Business and Management
Calculus of Variations and Optimal Control
Optimization
Diffusion
Markov analysis
Markov chains
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations research
Operations Research/Decision Theory
Original Article
Policies
Random variables
Stochastic models
Stochastic processes
Switching
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Summary:We consider a class of stochastic optimal stopping and impulse control problems where the agent solving the problem anticipates that a regime switch will happen at a random time in the future. We assume that there are only two regimes, the regime switching time is exponentially distributed, the underlying stochastic process is a linear, regular, time-homogeneous diffusion in both regimes and the payoff may be regime-dependent. This is in contrast with most existing literature on the topic, where regime switching is modulated by a continuous-time Markov chain and the underlying process and payoff belong to the same parametric family in all regimes. We state a set of easily verifiable sufficient conditions under which the solutions to these problems are given by one-sided threshold strategies. We prove uniqueness of the thresholds and characterize them as solutions to certain algebraic equations. We also study how anticipation affects optimal policies i.e. we present various comparison results for problems with and without regime switching. It may happen that the anticipative value functions and optimal policies coincide with the usual ones even if the regime switching structure is non-trivial. We illustrate our results with practical examples.
ISSN:1432-2994
1432-5217
DOI:10.1007/s00186-023-00838-9