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 q 0 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.