We consider explosive Poisson shot noise processes as natural extensions of the classical compound Poisson process and investigate their asymptotic properties. Our main result is a functional central limit theorem with a self-similar Gaussian limit process which, in the classical case, is Brownian motion. The theorems are derived under regularity conditions on the moment and covariance functions of the shot noise process. The crucial condition is regular variation of the covariance function which implies the self-similarity of the limit process. The model is applied to delay in claim settlement in insurance portfolios. In this context we discuss some specific models and their properties. We also use the asymptotic theory for studying the ruin time and ruin probability for a risk process which is based on the Poisson shot noise process.