Exponential drawdown and the supremum transform

The drawdown at an independent exponential clock has exponential law. Independence and the process transform determine the supremum transform below its apparent pole; real analyticity extends the denominator-free identity and gives its diagonal value.

Main results, with Φ = q^{1/θ}: law_supReflected_expTime (the drawdown is exponential with parameter Φ), integral_exp_neg_eval_expTime, integral_exp_neg_supRunning_expTime (E e^{-λ S_{e_q}} = q (Φ - λ) / (Φ (q - λ^θ))), its diagonal value 1/θ, the Brownian case law_supRunning_expTime_two, and law_supRunning_expTime_ne_exp for θ < 2.

References

• J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapters IV and VI.
• A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed., Springer, 2014, Chapter 6 (Wiener–Hopf factorization).