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.