This paper deals with the Cramér-Lundberg model with stochastic premiums. The authors consider the Lundberg exponential inequality and prove it using standard martingale techniques. Two examples of finding adjustment (Lundberg) coefficients for exponential premiums and claims with gamma or a mixture of exponential distributions are discussed. The authors extend the de Vylder approach to the approximation of ruin probabilities to models with stochastic premiums. Diffusion approximation of ruin probabilities in finite/infinite intervals is considered. Using general results due to the strong invariance principle for random sums, the authors prove the strong invariance principle for risk processes with stochastic premiums. Special cases, i.e., claims with finite second moment and large claims attracted to \(\alpha\)-stable laws, are studied separately. Various modifications of the law of the iterated logarithm and the Erdös-Renyi-Révész-Csörgő-type strong law of large numbers for risk processes are proved. Cases of small claims, large claims with finite variance and large claims attracted to asymmetric stable laws are discussed.