The survival probability in finite time period in fully discrete risk model (Q1288288)

Consider the surplus of an insurance company in discrete time and with discrete claims, i.e. \(U_n=u+n-S_n\), \(n\geq 0\), where \(u\) denotes the initial surplus, \(n\) is the premium income up to time \(n\), and \(S_n\) is a compound binomial aggregate claim sequence. Most emphasis in this paper is put on determining \[ s(u;k,x)= P(U_j>0;\;j=1,\dots, k-1,\;U_k=x \mid U_0=u), \] the probability of survival up to time \(k\) with a surplus \(x\). Two methods of proof are applied, an analytical deduction and a probabilistic argument. Other probability laws related to \(s(u;k,x)\) are also investigated.