Diffusion approximations for a risk process with the possibility of borrowing and investment

A diffusion approximation is constructed for an insurance risk model which was considered by Embrechts and Schmidli, where the company is allowed to borrow money if needed and to invest money for large surpluses. Besides the weak convergence of a sequence of such processes to a diffusion, the convergence of the finite- and infinite-time ruin probabilties is shown. The ruin probabilities of the diffusion are calculated and, for two examples, compared with the exact values. The convergence of the corresponding infinite-time ruin probabilities for a diffusion approximation for the classical process is also shown.