Finite-time ruin probability of aggregate Gaussian processes

Let $\{{\sum }_{i=1}^n{\lambda }_i{X}_i(t),t\in [0,T]\}$ be an aggregate Gaussian risk process with $X_i,i\le n$ independent Gaussian processes satisfying Piterbarg conditions and ${\lambda }_i$’s given positive weights. In this paper we derive exact asymptotics of the finite-time ruin probability given by $$\mathbb{P}(\underset{t\in [0,T]}{\sup }(\sum _{i=1}^n{\lambda }_i{X}_i(t)-g(t))>u)$$ as $u\to \mathrm{\infty }$ for some general trend function $g$. Further, we derive asymptotic results for the finite-time ruin probabilities of risk processes perturbed by an aggregate Gaussian process.