Joint asymptotic distribution of certain path functionals of the\n reflected process

2013/06/28 by Aleksandar Mijatović, Mijatovic, Aleksandar, Martijn Pistorius +1 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F05 #60G17 #60G51 #Advanced Statistical Process Monitoring #FOS: Mathematics #Probability (math.PR) #Statistical Methods in Clinical Trials #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1306.6746

openalex publication_date 2013/06/28 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Let \τ(x) be the first time the reflected process Y of a Levy processes\nX crosses x>0. The main aim of the paper is to investigate the asymptotic\ndependence of the path functionals: Y(t) = X(t) - \inf_{0\≤ s\≤ t}X(s),\nM(t,x)=\sup_{0\≤ s\≤ t}Y(s)-x and Z(x)=Y(\τ(x))-x. We prove that\nunder Cramer's condition on X(1), the functionals Y(t), M(t,y) and Z(x+y)\nare asymptotically independent as \min t,y,x \→\∞. We also\ncharacterise the law of the limiting overshoot Z(\∞) of the reflected\nprocess. If, as \min t,x \→\∞, the quantity t te-\γ x has a\npositive limit (\γ denotes the Cram 'er coefficient), our results\ntogether with the theorem of Doney & Maller (2005) imply the existence and the\nexplicit form of the joint weak limit (Y(\∞),M(\∞),Z(\∞)).\n

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