Small and Large Time Stability of the Time taken for a Lévy Process to Cross Curved Boundaries

2011/10/13 by Philip S. Griffin, Ross Maller, Griffin, Philip S. +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F15 #60F25 #60G51 #60K05 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1110.3064

openalex publication_date 2011/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper is concerned with the small time behaviour of a Lévy process X. In particular, we investigate the \it stabilities of the times, \Tstarb(r) and \Tbarb(r), at which X, started with X₀=0, first leaves the space-time regions \(t,y)∈\R²: y≤ rt^b, t≥ 0\ (one-sided exit), or \(t,y)∈\R²: |y|≤ rt^b, t≥ 0\ (two-sided exit), 0≤ b<1, as r\dto 0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L^p. In many instances these are seen to be equivalent to relative stability of the process X itself. The analogous large time problem is also discussed.

Citations

• Foundations of Modern Probability

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