I will give an introduction to large deviations for stochastic processes 
with regularly varying tails. First we derive a functional large deviation 
result for a system that perturbed by a small but heavy-tailed 
(regularly varying) noise. We extend the result to small perturbations of
 a stochastic integral equation and a large deviation result for hitting
 probabilities is obtained. This is applied to an insurance problem to 
obtain the asymptotic decay of finite time ruin probabilities.

If there is time I will also talk about large deviations for a stationary
sequence of random variables with regularly varying tails. For this sequence 
it is possible that large values arrive in clusters. A limiting measure, 
on the space of point measures, describes the joint limiting behavior of 
all the large values of the sequence.