Sample path large and moderate deviation principles for Markov modulated risk models with delayed claims are proved by the exponential martingale method. As applications, asymptotic estimates and exponential bounds of the ruin probability are also studied.

We study asymptotic behavior of the empirical conditional value-at-risk (CVaR). In particular, the Berry–Essen bound, the law of iterated logarithm, the moderate deviation principle and the large deviation principle for the empirical CVaR are obtained. We also give some numerical examples.