Stable windings at the origin

Entrance and exit at infinity for stable jump diffusions

“…) ĥ(X τ (−1,1) − x)/ ĥ(y − x)]; see similar calculations in [34]. It follows from the explicit formula (5.3) that x is absolutely continuous with respect to the Lebesgue measure, for each x ∈ (−1, 1), as well as that ( x , x ∈ (−1, 1)) forming a weakly continuous family of measures.…”

Windings of Planar Processes and Applications to the Pricing of Asian Options

“…For discussions concerning the small time asymptotic behavior (t → 0) of the windings of a Stable process, we address the interested reader to [7] for the case where our process is issued from a point different from the origin and to [14] when it is issued from the origin. For the latter, one can either apply classical scaling arguments, or use a novel method appealing to tools from the theory of self-similar Markov processes involving path transformations and time changes.…”

“…Concerning now the Stable process U (see also [3,7,14]), contrary to planar Brownian motion, we cannot define its winding number directly. However, we can consider a path on a finite time interval [0, t] and "fill in" the gaps with line segments in order to obtain the curve of a continuous function f : [0, 1] → C with f (0) = 1.…”

Windings of planar processes, exponential functionals and Asian options

“…to [25] for more details about planar Brownian motion. Contrary to planar Brownian motion, one cannot define the winding number directly for the isotropic planar Stable process U (see [3,13,20]). However, we can consider a path on a finite time interval [0, t] and "fill in" the gaps with line segments in order to obtain the curve of a continuous function f : [0, 1] → C such that f (0) = 1.…”