Supremum potential and reflected occupation resolvents

For a spectrally positive stable process of index θ ∈ (1, 2] reflected at its running supremum, this module identifies the law of the supremum at an independent exponential clock and derives the resulting potentials and resolvent bounds.

Main results

• stableScale, stableSeries: the scale function and the q-scale series.
• map_supRunning_expClockLaw: the density of S_{e_q}, the supremum at an exponential clock.
• supExpTail, supPotential: the tail and the potential, with Φ h_q(0) = q^{2/θ-1}.
• supOccupationResolvent_eq: the reflected resolvent formula, with its uniform convergence (tendstoUniformlyOn_supOccupationResolvent).
• supCorrection and abs_supCorrection_le: the correction term and its bound.
• condExp_reflectedResolvent_tail, resolventMarkov_supReflectedFrom: the conditional resolvent identity and the resolvent Markov property.
• integral_sq_discountedOccupation_supOccupation_le_two: second-moment bounds.

The proofs combine the scale series (SupremumPotentialBasic), the density and potential identities (SupremumPotentialDensity, SupremumPotentialIdentity), the occupation convolution (SupremumPotentialConvolution) and the generic resolvent moment theorem.

References

• J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapter VII.
• A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed., Springer, 2014, Chapter 8 (scale functions).
• A. Kuznetsov, A. E. Kyprianou and V. Rivero, The theory of scale functions for spectrally negative Lévy processes, in Lévy Matters II, Lecture Notes in Math. 2061, Springer, 2012.