The Compound Binomial Risk Model With Random Income

The seminal paper by Gerber [Mathematical fun with the compound binomial process, ASTIN Bulletin,18:161-168 (1988)] investigates a risk model known as the (classical) compound binomial risk model, in which time, claim amounts, premiums, and initial surplus are assumed to be integer valued. In his model, the premium rate is assumed to be one per period. However, the premium income may depend on time and not always be constant. Thus, to describe a more realistic situation, we assume that the premium income is also a binomial process.Three different models are given in Section 1, namely the (classical) compound binomial risk model, the compound binomial model with random income and the time-correlated binomial model with random income, respectively. We could see that the (classical) compound binomial risk model is a special case of the compound binomial model with random income, and the compound binomial model with random income is a special case of the time-correlated binomial model with random income. The notations used in this paper are also given in Section 1.In section 2, we focus on the compound binomial model with random income.First, we derive a defective renewal equation for the Gerber-Shiu discounted penalty function mν+(u), then we investigate the solution of the defective renewal equation and the asymptotic behavior of mν+(u). It is notable that the roots of Lundberg fundamental equation play an important role in deriving the above results. Second, we obtain an explicit expression for the finite time survival probabilityφ+(u,k). An explicit expression for the generating function of ultimate survival probabilityφ+(u) is also obtained.In section 3, we focus on the time-correlated compound binomial model with random income. By introducing an auxiliary process, we obtain recursive equations for the finite time survival probabilityφ(?)(u,k) and the joint distribution of the surplus one period prior to ruin and the deficit at ruinφ+(u,x,y). The (ultimate) ruin probability is also discussed in some details.Finally, we show that the premium income could be generalized to a more general case in Section 4.We can see remarks in the paper to get a better understanding for the relations among the above three models. The technique of generating function is an important tool in studying ruin-related quantities in ruin theory and plays a central role in deriving our results.