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The -finite measure which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s -transform processes with respect to these functions are utilized for the construction of .
Penalisation involving the one-sided supremum for a stable Lévy process with index ∈(0, 2] is studied. We introduce the analogue of Azéma–Yor martingales for a stable Lévy process and give the law of the overall supremum under the penalised measure.
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