Ruin probability for an insurer investing in several risky assets. (Ukrainian, English) Zbl 1199.62034

Bratyk, M. V.

Teor. Jmovirn. Mat. Stat. 77, 1-12 (2007); translation in Theory Probab. Math. Stat. 77, 1-13 (2008).

The author deals with the classical Cramér-Lundberg model of an insurance company with finite exponential moments. He obtains an estimate \(\Psi(x)\leq e^{-rx}\) for the ruin probability in the case when the insurance company invests in several risky assets modelled by either geometric Brownian motions or semi-martingales. The coefficient \(r\) in this bound is improved as compared to the cases where the insurer does not invest at all or when there is only a single risky asset. In particular, if the price processes are modelled by geometric Brownian motions, the above bound corresponds to the investment strategy that suggests that the insurer will invest constant amounts of money in every risky asset, independently of the current capital.
An optimal strategy and the ruin probability for an insurer whose activity is described by the Cramér-Lundberg model for which the insurance company is allowed to invest in a single asset are obtained by J. Gaier, P. Grandits and W. Schachermayer [Ann. Appl. Probab. 13, No. 3, 1054–1076 (2003; Zbl 1046.62113)]. An estimate of the ruin probability for the case when an insurance company invests in a single risky asset modelled by a semi-martingale with absolutely continuous characteristics with respect to the Lebesgue measure has been obtained by Y. Mishura [Teor. Jmovirn. Mat. Stat. 72, 93–100 (2005); translation in Theory Probab. Math. Stat. 72, 103–111 (2006; Zbl 1125.60068)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

62P05	Applications of statistics to actuarial sciences and financial mathematics
91B30	Risk theory, insurance (MSC2010)
60G48	Generalizations of martingales
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Keywords:

geometric Brownian motion; Cramér-Lundberg model

Citations:

Zbl 1046.62113; Zbl 1125.60068