A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments

Let ψ(y) be the probability of ultimate ruin in the classical risk process compounded by a linear Brownian motion. Here y is the initial capital. We give sufficient conditions for the survival probability function ϕ=1−ψ to be four times continuously differentiable, which in particular implies that ϕ is the solution of a second order integro-differential equation. Transforming this equation into an ordinary Volterra integral equation of the second kind, we analyze properties of its numerical solution when basically the block-by-block method in conjunction with Simpsons rule is used. Finally, several numerical examples show that the method works very well.