2021/09/27 by Konrad Krystecki, Krystecki, Konrad
Decision Sciences · Economics, Econometrics and Finance · Social Sciences · #Probability and Risk Models #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management
paper · pdf · doi:10.48550/arxiv.2109.12906
Let (W1(s), W2(t)), s,t\≥ 0 be a bivariate Brownian motion with standard\nBrownian motion marginals and constant correlation \ρ \∈ (-1,1). In this\ncontribution we derive precise approximations for cumulative Parisian ruin\nconditioned on the occurrence of the ruin of the aforementioned two-dimensional\nBrownian motion, i.e.\n
mathbbP
left(∫[0,1]\n
mathbf1(W1^*(s) · gt;u)ds · gt;H1(u)
int[0,1]\n
mathbf1(W2^*(t) · gt;au)dt · gt;H2(u)
|
existsv,w
in\n[0,1]\W1(v)-c1v · gt;u
W2(w)-c2w · gt;au
).\n We study the asymptotics for specific functions boldsymbolH(u) for u\nbeing proportional to initial position of the Brownian motion, which determines\nhow long does the process need to spend over the barrier.\n