An estimate of ruin probabilities for long range dependence models. (Ukrainian, English) Zbl 1125.60068

Mishura, Yuliya

Teor. Jmovirn. Mat. Stat. 72, 93-100 (2005); translation in Theory Probab. Math. Stat. 72, 103-111 (2006).

The author deals with the model of risk of an insurance company \[ Y(t,x,\widehat{\varphi})=R_1(t)+R_2(t)=x+ct-\sum_{i=1}^{N_t}X_i+\int_0^t \widehat{\varphi}_s\,dS_s. \] Here \(R_1(t)=x+ct-\sum_{i=1}^{N_t}X_i\) is the classical component. The second component \(R_2(t)=\int_0^t\widehat{\varphi}_s\,dS_s\) describes the case where a company buys a risky asset (security) \(S_t\) that is a semimartingale \(dS_t=S_t(dM_t+dA_t)\), \(t\geq0\), where \(M_t\) is a square integrable continuous martingale with absolutely continuous characteristics with respect to the Lebesgue measure and \(A_t\) is a process of integrable variation. An estimate of the ruin probability for the considered model is obtained. The result is applied to some long range dependence models, in particular, to the fractional Brownian motion and mixed models.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60H30	Applications of stochastic analysis (to PDEs, etc.)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
91B30	Risk theory, insurance (MSC2010)
60G44	Martingales with continuous parameter
60G15	Gaussian processes

Keywords:

insurance company; ruin probability; estimate; semimartingale