Appears in collection : COSA : Interactions in applied mathematics for economics, finance and Insurance / Interactions en mathématiques appliquées à l'économie, la finance et l'assurance

Nowadays, insurance companies place their capital reserves in stock markets to obtain additional gain. The collective risk theory assuming absence of risky investment became obsolete. To replace it, a new theory emerged.

In 1993 J. Paulsen suggested to describe the reserve evolution by a generalized Ornstein–Uhlenbeck process and the development was concentrated around this model. The problem is how to explain the results to master students: the existing studies are based on advanced mathematical tools and cumbersome estimates.

In the recent notes [1], [2] it was developed a new approach how to get and solve IDE for the ruin probabilities and obtain their asymptotic behavior in the case where risky asset follows gBm and the business process has one-sided jumps.

In the current study it was extended a still special case of the OU process where the business process is a Lévy process of bounded variation with one-side jumps.