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The probability of ruin in a kind of Cox risk model with variable premium rate. (English) Zbl 1142.62096
Extending the approach due to H. Jasiulewicz [Insur. Math. Econ. 29, No. 2, 291–296 (2001; Zbl 0999.91048)] the authors investigate the ruin probability under the assumptions that the premium rate varies according to the intensity of the claims and the occurrence of claims is described by a Cox process. In other words, they deal with a continuous time Cox risk model where the reserve process is defined by $U(t)=u+\int_0^t c(\lambda_s)\,ds+\sum_{k=1}^{N(t)}Z_k,\quad t\geq 0,$ where $$u\geq 0$$ is the insurer’s initial capital, the i.i.d. r.v.s $$\{Z_k, k\geq 1\}$$ represent the values of successive claims, $$N(t)$$ is a Cox process with intensity $$\lambda(t)$$, $$t\geq 0$$, and $$c(l)$$, $$l>0$$, is a positive real function. The integral equation satisfied by the probability of ruin is derived by “backward differential arguments” and the solution of the equation is obtained in the case when the intensity process is a homogeneous $$n$$-state Markov process. An example when the claims are exponentially distributed and $$\lambda(t)$$ is a two-state Markov process is discussed.

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B30 Risk theory, insurance (MSC2010) 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60J75 Jump processes (MSC2010)
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