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[Last passage times arise in a number of areas of applied probability, including risk theory and degradation models. Such times are obviously not stopping times since they depend on the whole path of the underlying process. We consider the problem of finding a stopping time that minimises the L1-distance to the last time a spectrally negative Lévy process X is below zero. Examples of related problems in a finite horizon setting for processes with continuous paths are by Du Toit et al. (Stochastics Int J Probab Stochastics Process 80(2–3):229–245, 2008) and Glover and Hulley (SIAM J Control Optim 52(6):3833–3853, 2014), where the last zero is predicted for a Brownian motion with drift, and for a transient diffusion, respectively.]
Published: Oct 17, 2020
Keywords: Lévy processes; Optimal prediction; Optimal stopping; 60G40; 62M20
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