Probability of entering an orthant by correlated fractional Brownian motion with drift: exact asymptotics

Abstract

For {BH(t)=(BH,1(t),…,BH,d(t))⊤,t≥0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\varvec{B}_{H}(t)= (B_{H,1}(t) ,\\ldots ,B_{H,d}(t))^{{\ op }},t\\ge 0\\}$$\\end{document}, where {BH,i(t),t≥0},1≤i≤d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{B_{H,i}(t),t\\ge 0\\}, 1\\le i\\le d$$\\end{document} are mutually independent fractional Brownian motions, we obtain the exact asymptotics of P(∃t≥0:ABH(t)-μt>νu),u→∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb P (\\exists t\\ge 0: A \\varvec{B}_{H}(t) - \\varvec{\\mu }t >\\varvec{\ u }u), \\ \\ \\ \\ u\\rightarrow \\infty ,$$\\end{document}where A is a non-singular d×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\ imes d$$\\end{document} matrix and μ=(μ1,…,μd)⊤∈Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\mu }=(\\mu _1,\\ldots , \\mu _d)^{{\ op }}\\in \\mathbb {R}^d$$\\end{document}, ν=(ν1,…,νd)⊤∈Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ u }=(\ u _1, \\ldots , \ u _d)^{{\ op }} \\in \\mathbb {R}^d$$\\end{document} are such that there exists some 1≤i≤d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le i\\le d$$\\end{document} such that μi>0,νi>0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _i>0, \ u _i>0.$$\\end{document}