An estimate of the optimal \(Z^*\) value for threshold payments in the case when the insurance requirements assume two values (Q1975000)

All payments are made at discrete times \(t= 1,2,\dots\) . The essence of the threshold payments consists of the following: if at time \(t=n\) the capital of the company \(S_n\) exceeds a certain level \(Z\), the excess is removed. If \(S_n\) is positive but does not exceed \(Z\), the company continues to operate. If the company's capital is not sufficient to pay claims, the company goes bankrupt. Payments of claims are random variables. In general it is difficult to obtain a formula for \(Z^*\). In the present paper she obtains upper and lower estimates for \(Z^*\) in the case when the claims at each time \(i\) are given by random variables \(\xi_i\) assuming the two values 0 and \(M\) with probabilities \(q\) and \(p= 1-q\) respectively. It is assumed that all \(\xi_i\) are identically distributed.