Details

Autor(en) / Beteiligte

• Asmussen, Søren
• Aurzada, Frank
• Pihlsgård, Mats
• Andersen, Lars Nørvang
• Maejima, Makoto
• Simon, Thomas
• Glynn, Peter W

Titel

Persistence Probabilities and Exponents

Ist Teil von

• Lévy Matters V, 2015, Vol.2149, p.183-224

Ort / Verlag

Switzerland: Springer International Publishing AG

Erscheinungsjahr

2015

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• This article deals with the asymptotic behavior as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$t \rightarrow +\infty $$ \end{document} of the survival function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{P}[T > t]$$ \end{document}, where T is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behavior is of the type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{P}[T > t] = t^{-\theta +o(1)}$$ \end{document} for a known or unknown positive parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\theta$$ \end{document} which is called the persistence exponent. The problem is well understood for random walks or Lévy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.