Computer treatment of the integro-differential equations of collective non-ruin; the finite time case

An important problem of collective non-ruin is the estimation of the probabilities R(z, t) and R(z) of the finite and ultimate non-ruin, respectively, where t is time and z the initial reserve. The governing equations are first-order Volterra integro-differential equations, partial (PVIDEs) in the finite time case and ordinary (VIDEs) in the ultimate non-ruin case, respectively.
In this paper, computational results for the finite time case are presented. These are obtained by solving numerically the VIDE satisfied by the Laplace transform of R(z, t) with respect to t using polynomial spline collocation methods and then inverting numerically.
Comparisons are made with results obtained by other approaches which invert the analytical solution of the corresponding VIDE satisfied by the Laplace transform of R(z, t).
The proposed fully numerical method can be applied for cases when the VIDE is difficult to solve analytically.