Approximation of passage times of \(\gamma\)-reflected processes with FBM input (Q2923431)

Let \((X_t)_{t\geq 0}\) be a centered Gaussian process with almost surely continuous sample paths and covariance function \(\text{cov}(X_t,X_s)=(t^{2H}+s^{2H}-|t-s|^{2H})/2\), \(t,s\geq 0\), i.e., \((X_t)_{t\geq 0}\) is a standard fractional Brownian motion with Hurst index \(H\in(0,1)\).NEWLINENEWLINEThe \(\gamma\)-reflected process with input process \(Y_t:=X_t-ct\) is defined by \(W_t:=Y_t-\gamma \inf_{s\in[0,t]}Y_s\), \(t\geq 0\), where \(\gamma\in[0,1]\) and \(c>0\) are two fixed constants.NEWLINENEWLINEThe first and last passage times of the process \((W_t)_{t\geq 0}\) to a constant threshold \(u>0\) are \(\tau_1(u):=\inf\{t\geq 0:\,W_t>u\}\), \(\tau_2(u):=\sup\{t\geq 0:\,W_t>u\}\). Put \((\tau_1^*(u),\tau_2^*(u)):=(\tau_1(u),\tau_2(u))\) under the condition that \(\tau_1(u)<\infty\). The main result of this article is the convergence in distribution \(((\tau_1^*(u)-\tilde t_0 u)/A(u),(\tau_2^*(u)-\tilde t_0 u)/A(u))\to_D (\xi,\xi)\) as \(u\to\infty\), where \(\xi\) is a standard normal distributed random variable, \(\tilde t_0=H/(c(1-H))\) and \(A(\cdot)\) is a suitable norming function.NEWLINENEWLINEThis joint convergence implies in particular \((\tau_2^*(u)-\tau_1^*(u))/A(u)\to 0\) in probability as \(u\to\infty\).NEWLINENEWLINEThe paper also links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by \((Y_t)_{t\geq 0}\).