The law of the supremum of a stable Lévy process with no negative jumps

By Violetta Bernyk, Robert C. Dalang, Goran Peskir. 2007

75 citations recorded by the source corpus.

Let X = (Xt)t‚0 be a stable Levy process of index fi 2 (1;2) with no negative jumps, and let St = sup0•st Xs denote its running supremum for t>0 . We show that the density function ft of St can be characterized as the unique solution to a weakly singular Volterra integral equation of the flrst kind, or equivalently, as the unique solution to a flrst-order Riemann-Liouville fractional difierential equation satisfying a boundary condition at zero. This yields an explicit series representation for ft . Recalling the familiar relation between St and the flrst entry time ?x of X into (x;1) , this further translates into an explicit series representation for the density function of ?x .

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/7052de93202ecbefb986f727ec0ba3e22aac1578

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Citations in the corpus

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes (2010, methodology): “In the spectrally positive case a convergent series representation for the density of the supremum was first obtained by Bernyk, Dalang and Peskir [4], and a complete asymptotic expantion was derived by Patie [39].”
Bernstein-gamma functions and exponential functionals of Levy Processes (2016, methodology): “The latter for example immediately recovers the asymptotic behaviour of the density of the supremum of a stable Lévy process as investigated in [8, 25, 40, 56].”
Some explicit identities associated with positive self-similar Markov processes. (2007, background): “Recently, Bernyk, Dalang and Peskir [ 1 ] computed the density of the supremum of a stable L´evy process with no negative jumps of index 2 (1,2).”
Suprema of Lévy processes (2011, background): “There are numerous other representations for the distribution of Mt, at least in the stable case, see [4, 7, 11, 12, 15, 16, 19, 20, 27, 28, 30, 36].”
Law of the absorption time of some positive self-similar Markov processes (2009, background): “[2] derive a representation of the distribution of T0 by inverting, in a non trivial way, the known expression of the Wiener-Hopf factorization of regular stable one-sided Lévy processes.”