zbMATH — the first resource for mathematics

Inhomogeneous perturbations of a renewal equation and the Cramér-Lundberg theorem for a risk process with variable premium rates. (Ukrainian, English) Zbl 1224.91065
Teor. Jmovirn. Mat. Stat. 78, 54-65 (2008); translation in Theory Probab. Math. Stat. 78, 61-73 (2009).
The author deals with a time inhomogeneous perturbation of the classical renewal equation with continuous time generated by the probability measure \(G\) \[ x(t)=y(t)+\int_{\mathbb R}x(t-s)G(ds),\quad t\in\mathbb R, \] with an unknown bounded Borel function \(x\in B_0(\mathbb R)\) and a bounded function \(y\in L_1^0(\mathbb R)\). The inhomogeneous perturbation of the renewal equation is described by the generalized integral Volterra equation \[ x(t)=y(t)+\int_{\mathbb R}x(t-s)F(t,ds),\quad t\in\mathbb R, \] with a measurable nonnegative kernel \(F\) that asymptotically approaches the probability distribution \(G\) in variation. The author prove that, under some mild conditions, this equation has a unique solution \(x\) in the class \(B_0(\mathbb R)\) for every \(y\in L_1^0(\mathbb R)\), and this solution coincides with the sum of the series \(x(t)=\sum_{n\geq0}F^n[y](t)\) that uniformly converges on finite intervals. Here, \(F^n[y]\) is the \(n\)-fold product of the bounded integral operator \(F\) acting on \(B_0(\mathbb R)\) as follows \[ F[y](t)= \int_{\mathbb R}y(t-s)F(t,ds). \] The proofs in the first part of the paper use the ideas of the author’s dissertation and the paper by H. Schmidli [Ann. Appl. Probab. 7, No. 1, 121–133 (1997; Zbl 0876.60072)].
In the second part of the paper, the obtained results are used for the investigation of the asymptotic behaviour of the ruin function for the classical ruin process where the premium rate depends on the current capital of an insurance company. It is proved that the risk function is exponential with the Lundberg index evaluated from the limit premium rate. This generalization of the Cramér-Lundberg theorem is proved under a minimal assumption that the difference between the premium rate and the limit rate is integrable at infinity.

91B30 Risk theory, insurance (MSC2010)
60K05 Renewal theory
62P05 Applications of statistics to actuarial sciences and financial mathematics
60A05 Axioms; other general questions in probability
60J45 Probabilistic potential theory
Full Text: DOI