Analytical solution for the equal-rate deterministic premium-and-claim model

Determine an analytical solution for the probability of non-ruin in the stochastic-premium risk process with Poisson claim and premium arrival intensities \(\lambda_c=\lambda_p=1\), deterministic claim size \(z=1\), and deterministic premium size \(p=\pi-1\).

Background

The paper studies a risk process in which claims and premiums arrive according to independent Poisson streams and have deterministic sizes. For the special case z=p=1z=p=1, an analytical formula for the non-ruin probability is available. The authors then consider the parameters λc=λp=1\lambda_c=\lambda_p=1, z=1z=1, and p=π−1p=\pi-1, and use successive approximations to compute the solution numerically. They explicitly state that an analytical solution for this parameter choice is not known, leaving its derivation unresolved.

References

Fig. 6.2 shows the form of the solution obtained by the method of successive approximations after 50 iterations for \lambda_c = \lambda_p = 1, z = 1, p = \pi - 1. In this case, an analytical solution is not known.

— Method of successive approximations for solving integral equations of actuarial mathematics  (2609.03279 - Norkin, 3 Sep 2026) in Chapter 6, Numerical Experiments, discussion of Fig. 6.2