Sojourns of locally self-similar Gaussian processes

Given a Gaussian risk process $R(t)=u+c(t)-X(t),t\ge 0$, the cumulative Parisian ruin probability on a finite time interval $[0,T]$ with respect to $L \geq 0$ is defined as the probability that the sojourn time that the risk process $R$ spends under the level 0 on this time interval $[0,T]$ exceeds $L$. In this contribution we derive exact asymptotic approximations of the cumulative Parisian ruin probability for a general class of Gaussian processes introduced in [9] assuming that $X$ is locally self-similar. We illustrate our findings with several examples. As a byproduct we show that Berman's constants can be defined alternatively by a self-similar Gaussian process which could be quite different to the fractional Brownian motion.