The compound binomial model with randomized decisions on paying dividends

Consider a discrete time risk process based on the compound binomial model. The insurer pays a dividend of 1 with a probability q0 when the surplus is greater than or equal to a non-negative integer x. We will derive recursion formulas and an asymptotic estimate for the ruin probability, the probability function of the surplus prior to the ruin time, and the severity of ruin, etc.