Risk process with stochastic premiums

The Cramér-Lundberg model with stochastic premiums which is natural generalization of classical dynamic risk model is considered. Using martingale technique the Lundberg inequality for ruin probability is proved and characteristic equations for Lundberg coefficients are presented for certain classes of stochastic premiums and claims. The simple diffusion and de Vylder approximations for the ruin probability are introduced and investigated similarly to classical Cramér-Lundberg set-up. The weak and strong invariance principles for risk processes with stochastic premiums are discussed. Certain variants of the strong invariance principle for risk process are proved under various assumptions on claim size distributions. Obtained results are used for investigation the rate of growth of the risk process and its increments. Various modifications of the LIL and Erdös-Renyi-type SSLN are proved both for the cases of small and large claims.