Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin, Ross A. Maller

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

This paper is concerned with the small time behaviour of a Lévy process X. In particular, we investigate the stabilities of the times, T̄ b(r) and Tb* (r), at which X, started with X 0 = 0, first leaves the space-time regions {(t, y) ∈ ℝ2: y ≤ rtb, t ≥ 0} (one-sided exit), or {(t, y) ∈ ℝ2: |y| ≤ rtb, t ≥ 0} (two-sided exit), 0 ≤ b < 1, as r ↓ 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 Lp. 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.

Original languageEnglish (US)
Pages (from-to)208-235
Number of pages28
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume49
Issue number1
DOIs
StatePublished - Feb 2013

Keywords

  • Lévy process
  • Overshoot
  • Passage times across power law boundaries
  • Random walks
  • Relative stability

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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