Homogeneous risk models with equalized claim amounts

We consider an homogeneous risk model on a fixed bounded time interval [0, t] and we denote by Nt the number of claims in that interval. The claim amounts are X1,X2,…,XNt. The homogeneous model is an extension of the classical actuarial risk model with Nt not necessarily Poisson distributed. In the model with equalized claim amounts, each amount Xk is replaced with Xk∼=(X1+⋯+XNt)/Nt. Let Ψ(t, u) be the ruin probability before t in the homogenous model, corresponding to the initial risk reserve u≥0 and let Ψ∼(t, u) be the corresponding ruin probability evaluated in the associated model with equalized claim amounts. The essence of the classical Prabhu formula is that Ψ(t, 0)=Ψ∼(t, 0). By rather systematic numerical investigations in the classical risk model, we verify that Ψ∼(t, u)≤Ψ(t, u) for any value of u≥0 and that Ψ∼(t, u) is an excellent approximation of Ψ(t, u). Then these conclusions must be valid in any homogeneous model and this is an interesting observation because Ψ∼(t, u) can be calculated numerically, whereas no algorithms are yet available for the numerical evaluation of Ψ(t, u) in general homogeneous risk models.