[Paper Review] The time at which a Lévy process creeps

Philip S. Griffin, Ross Maller|arXiv (Cornell University)|Jun 29, 2011

Probability and Risk Models6 references4 citations

TL;DR

This paper establishes that the time at which a Lévy process creeps over a level $ u $ is characterized by the left derivative of the renewal function $ V(t,u) $ of the bivariate ascending ladder process, scaled by the reciprocal of the drift of the height component $ H $. The key contribution is identifying the missing creeping term in Doney and Kyprianou’s quintuple law and deriving a generalized Laplace transform identity that extends the second factorization identity of Percheskii and Rogozin.

ABSTRACT

We show that if a Lévy process creeps then, as a function of $u$, the renewal function $V(t,u)$ of the bivariate ascending ladder process $(L^{-1},H)$ is absolutely continuous on $[0,\infty)$ and left differentiable on $(0,\infty)$, and the left derivative at $u$ is proportional to the (improper) distribution function of the time at which the process creeps over level $u$, where the constant of proportionality is $ md_H^{-1}$, the reciprocal of the (positive) drift of $H$. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an application, we derive a Laplace transform identity which generalises the second factorization identity. We also relate Doney and Kyprianou's extension of Vigon's équation amicale inversée to creeping. Some results concerning the ladder process of $X$, including the second factorization identity, continue to hold for a general bivariate subordinator, and are given in this generality.

Motivation & Objective

• To characterize the distribution of the time at which a Lévy process creeps over a level $ u $, particularly in terms of the bivariate ascending ladder process.
• To identify the missing creeping term in the quintuple law of Doney and Kyprianou (2006), which had previously omitted the contribution from paths that creep.
• To generalize the second factorization identity of Percheskii and Rogozin (1969) using a Laplace transform identity derived from the creeping mechanism.
• To extend results on the bivariate ladder process to general bivariate subordinators, revealing that the second factorization identity is a special case of a broader transform result.
• To relate the creeping phenomenon to Doney and Kyprianou’s extension of Vigon’s équation amicale inversée, clarifying its role in fluctuation theory.

Proposed method

• Analyzes the renewal function $ V(t,u) $ of the bivariate ascending ladder process $ (L^{-1}, H) $, showing it is absolutely continuous and left differentiable on $ (0,ty) $ when creeping occurs.
• Derives that the left derivative of $ V(t,u) $ at $ u $ is proportional to the (improper) distribution function of the creeping time, with proportionality constant $ ext{d}_H^{-1} $, the reciprocal of the drift of $ H $.
• Uses the Wiener-Hopf factorization and duality to relate the joint distribution of ladder height and inverse local time to the creeping mechanism via the dual process $ {X} $.
• Applies the compensation formula to express the jump distribution of the ladder process in terms of the Lévy measure $ { {P}}_{L^{-1},H} $, linking it to the jump measure of the original process.
• Derives a Laplace transform identity for the creeping time distribution by integrating the bivariate renewal function and relating it to the dual process’s ladder structure.
• Establishes a general transform result for bivariate subordinators, showing that the second factorization identity is a special case of this broader framework.

Experimental results

Research questions

• RQ1 What is the precise relationship between the creeping time distribution and the renewal function of the bivariate ascending ladder process?
• RQ2 How can the missing creeping term in the quintuple law of Doney and Kyprianou (2006) be explicitly identified and expressed?
• RQ3 Can the second factorization identity of Percheskii and Rogozin (1969) be generalized to include creeping contributions via a Laplace transform identity?
• RQ4 To what extent do results on the bivariate ladder process of a Lévy process extend to general bivariate subordinators?
• RQ5 How does the creeping mechanism relate to Doney and Kyprianou’s extension of Vigon’s équation amicale inversée?

Key findings

• The renewal function $ V(t,u) $ of the bivariate ascending ladder process is absolutely continuous and left differentiable on $ (0,ty) $ when the Lévy process creeps.
• The left derivative of $ V(t,u) $ at $ u $ is proportional to the cumulative distribution function of the creeping time, with proportionality constant $ ext{d}_H^{-1} $, the reciprocal of the drift of the height component $ H $.
• The missing creeping term in the quintuple law is identified as $ ext{d}_H^{-1} imes ext{CDF of creeping time} $, completing the fluctuation theory framework.
• A generalized Laplace transform identity is derived that extends the second factorization identity of Percheskii and Rogozin (1969) to include creeping paths.
• The second factorization identity is shown to be a special case of a general transform result for bivariate subordinators, a fact previously unnoticed in the literature.
• The creeping mechanism is formally linked to Doney and Kyprianou’s extension of Vigon’s équation amicale inversée, clarifying its role in the fluctuation theory of Lévy processes.