[Paper Review] Explicit identities for Lévy processes associated to symmetric stable processes

M. E. Caballero, Juan Carlos Pardo+1 more|ArXiv.org|Nov 3, 2009

Stochastic processes and financial applications12 references4 citations

TL;DR

This paper introduces hypergeometric-stable Lévy processes—Lévy processes derived from symmetric stable processes via Lamperti transformations and involving the Gauss hypergeometric function. It characterizes their Lévy measures, derives Wiener–Hopf factorizations, and establishes explicit joint distributions for first passage and last passage times, with key results involving multivariate densities and potential kernels expressed through hypergeometric functions.

ABSTRACT

In this paper we introduce a new class of Lévy processes which we call hypergeometric-stable Lévy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.

Motivation & Objective

• To define and characterize a new class of Lévy processes—hypergeometric-stable Lévy processes—derived from symmetric stable processes through Lamperti's representation.
• To determine the Lévy measure of these processes and establish their path properties, including regularity for intervals and scaling behavior.
• To derive explicit Wiener–Hopf factorizations and joint distributions for first hitting and last passage times of the radial process associated with symmetric stable Lévy processes.
• To compute potential kernels for the underlying Lévy process killed upon entering (−∞,0), expressing them via integrals involving the Gauss hypergeometric function.

Proposed method

• Lamperti's transformation is applied to the radial process $ R_t = \|Z_t\| $ of a symmetric stable Lévy process to construct a Lévy process $ \xi $, with $ R_t = x \exp(\xi_{\tau(t x^{-\alpha})}) $, where $ \tau $ is the inverse local time of the exponential functional of $ \xi $.
• The Lévy measure of the resulting hypergeometric-stable Lévy process is characterized using the scaling and self-similarity properties of the radial process and the Lamperti representation.
• The Wiener–Hopf factorization is derived by exploiting the Lamperti representation and the fluctuation theory of Lévy processes, particularly through the use of last passage times and first hitting times.
• Joint density formulas for quadruple laws at last passage times are obtained using results from Kyprianou et al. and Millar, involving the Gauss hypergeometric function $ {}_2\mathcal{F}_1 $ in the density expressions.
• Potential kernels for the Lévy process killed upon entering $ (-\infty,0) $ are computed via an integral representation involving $ (1 - e^{-2y})^{\alpha/2 - 1} $ and $ (e^{2z} - 1)^{\alpha/2 - 1} $, with the final expression involving the hypergeometric function.
• The radial process potential measure is derived by time-changing the Lévy process potential via $ \mathbb{E}_x[\sigma_1^{-}] = \mathbb{E}_{\log x}[\int_0^{T_0^-} e^{\alpha \xi_t} dt] $, leading to a beta-type integral in terms of $ u^{d/2 - 1}(1 - u)^{\alpha/2 - 1} $.

Experimental results

Research questions

• RQ1 What is the explicit form of the Lévy measure for the class of hypergeometric-stable Lévy processes derived from symmetric stable processes?
• RQ2 How can the Wiener–Hopf factorization be explicitly computed for the Lévy process associated with the radial part of a symmetric stable Lévy process?
• RQ3 What are the joint distributions of the last passage time, future infimum, and overshoots for the radial process and its associated Lévy process?
• RQ4 What is the potential kernel of the Lévy process killed upon entering $ (-\infty,0) $, and how can it be expressed in terms of special functions?
• RQ5 How does the potential measure of the radial process killed upon hitting 1 relate to the potential kernel of the underlying Lévy process?

Key findings

• The Lévy measure of the hypergeometric-stable Lévy process is characterized via a density involving the Gauss hypergeometric function $ {}_2\mathcal{F}_1\left(\frac{\alpha + d}{2}, \frac{\alpha}{2} + 1; \frac{d}{2}; \cdot \right) $, with a normalization constant involving gamma functions and $ \sin(\alpha\pi/2) $.
• A quadruple law at the last passage time for the Lévy process $ \xi $ is derived, with a joint density involving $ (e^{2v} - 1)^{\alpha/2 - 1} $, $ (e^{2(w - u)} - 1)^{\alpha/2 - 1} $, and $ (1 - e^{-2(x + v - y)})^{\alpha/2 - 1} $, all multiplied by the hypergeometric function.
• The potential kernel of the Lévy process killed upon entering $ (-\infty,0) $ is expressed as a double integral involving $ (1 - e^{-2y})^{\alpha/2 - 1} $, $ e^{(2 - d)z}(e^{2z} - 1)^{\alpha/2 - 1} $, and a shift in the argument of $ f $, with a constant $ k $ and normalization involving $ \Gamma(\alpha/2)^{-2} $.
• The first passage time potential for the radial process $ R $ killed upon hitting 1 is given by $ \mathbb{E}_x[\sigma_1^{-}] = k \frac{x^\alpha}{2\Gamma(\alpha)} \int_{x^{-2}}^1 u^{d/2 - 1}(1 - u)^{\alpha/2 - 1} du $, a beta-type integral.
• The joint density of $ (1/F_0, R_{L_b-}, R_{L_b}, F_{L_b}) $ is derived as a quadruple law with a density involving $ v^{1 - d} $, $ yw^{1 - d - \alpha} $, $ u^{d - \alpha - 1} $, and $ (y^2 - (bv)^{-2})^{\alpha/2 - 1} $, with the hypergeometric function $ {}_2\mathcal{F}_1\left(\frac{\alpha + d}{2}, \frac{\alpha}{2} + 1; \frac{d}{2}; (y/bw)^2\right) $.