Ruin probabilities for discrete time risk models with stochastic rates of interest

Consider a discrete time risk model Un=(Un-1+Xn)(1+In)-Yn,n=1,2,…, where U0≔M>0 is the initial reserve of an insurance company, Xn the total amount of premiums, Yn the total amount of claims, In the interest rate and Un the reserve at time n. The time of ruin is denoted by τM≔inf{n⩾1;Un<0}. In this paper, the recursive equations for finite time ruin probabilities and bounds for ultimate ruin probabilities are provided. When {Yn} are heavy tailed, we also give reasons for the asymptotic estimate P(τM<∞)≈M-λ, where λ is a specific positive parameter. A more general risk model from Nyrhinen [1999. On the ruin probabilities in a general economic environment, Stachastic Process. Appl. 83, 319-330] is also discussed, and similar asymptotic estimate for ultimate ruin probabilities is given.