Spectral analysis of stable processes on the positive half-line

Abstract

We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectral expansion of the semigroup of a non-symmetric Lévy process killed on the first exit form the positive half-line.

Risk‐neutral pricing techniques and examples

“…Writing for the semigroup associated with , this property reads, for any and , as where is the dilation operator, that is, . Some well known instances of positive self‐similar Markov processes are squared Bessel processes (), ‐stable subordinators, reflected or killed ‐stable process, stable Lévy processes conditioned to stay positive, see for example, Bertoin and Yor (2002, 2002), Caballero and Chaumont (2006), Kuznetsov and Kwaśnicki (2018), Patie and Zhao (2017).…”

The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

“…Although the constant √ α 2 S 2 (αρ) Γ(1+αρ * ) was not specified in [13], using (1.10) and (1.19) from [13], we obtain that Even though the functions F and F * do not simultaneously belong to L 2 (0, ∞) (for ρ = 1/2), they can be understood as the generalized eigenfunctions of the semigroups Q * t and Q t , respectively (see Theorem 1.3 in [13]). Moreover, using Theorem 1.1 in [13] the transition probability density of the process X killed when exiting the positive half-line is given by…”

“…However, since the ladder time process (L * t ) −1 is ρ * -stable subordinator and n * (t < ζ) = π * (t, ∞), where π * is the Lévy measure of (L * t ) −1 we have n(t < ζ) = 1 Γ(1 − ρ * )t ρ * , t > 0, which gives the representations for f t (x). To find the constant in the other formula we use the relations S 2 (z)S 2 (1 + α − z) = 1 and 2 sin(πz/2)S 2 (z + 1) = S 2 (z) (see (A.7) and (1.9) in [13]).…”

“…The presented formulae for symmetric processes are given in terms of the corresponding Lévy-Kchintchin exponent Ψ(ξ) and the related generalized eigenfunctions introduced by M. Kwaśnicki in [14]. In the stable case, we based the calculations on the generalized eigenfunctions studied recently by A. Kuznetsov and M. Kwaśnicki in [13]. Then we used the formulae obtained for the entrance law densities to derive corresponding integral representations for supremum densities.…”

Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin

“…In our case, however, the functions F + and F − are no longer bounded; in fact, they grow exponentially fast, and thus the methods of [23] need to be substantially modified. Our approach is similar to that of [21], where similar problems for hitting a half-line are studied. However, we avoid the use of special functions.…”