  Let $\left\{\sum_{i=1}^n \lambda_i X_i(t), t\in [0,T]\right\}$ be an
aggregate Gaussian risk process with $X_i, i\leq n$ 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 $$\mathbb{P}\left(\sup_{t\in[0,T]}\left(\sum_{i=1}^n
\lambda_i X_i(t)- g(t) \right)>u\right)$$ as $u\to\infty$ 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.
