Simultaneous Ruin Probability for Two-Dimensional Fractional Brownian Motion Risk Process over Discrete Grid

Abstract

We derive the asymptotic behavior of the following ruin probability:P∃t∈Gδ:BHt−c1t>q1uBHt−c2t>q2u,H∈01,u→∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathrm{P}\\left\\{\\exists t\\in G\\left(\\delta \\right):{B}_H(t)-{c}_1t>{q}_1u,{B}_H(t)-{c}_2t>{q}_2u\\right\\},\\kern0.72em H\\in \\left(0,1\\right),u\ o \\infty, $$\\end{document}where BH is a standard fractional Brownian motion, c1, q1, c2, q2> 0, and G(δ) denotes the regular grid {0, δ, 2δ, . } for some δ > 0. The approximation depends on H, δ (only when H ≤ 1/2) and the relations between parameters c1, q1, c2, q2.