The time of ultimate recovery in Gaussian risk model

We analyze the distance $\mathcal{R}_T(u)$ between the first and the last passage time of $\{X(t)-ct:t\in [0,T]\}$ at level $u$ in time horizon $T\in(0,\infty]$, where $X$ is a centered Gaussian process with stationary increments and $c\in\mathbb{R}$, given that the first passage time occurred before $T$. Under some tractable assumptions on $X$, we find $\Delta(u)$ and $G(x)$ such that $$\lim_{u\to\infty}\mathbb{P}\left(\mathcal{R}_T(u)>\Delta(u)x\right)=G(x),$$ for $x\geq 0$. We distinguish two scenarios: $T<\infty$ and $T=\infty$, that lead to qualitatively different asymptotics. The obtained results provide exact asymptotics of the ultimate recovery time after the ruin in Gaussian risk model.