The Pólya-Aeppli process and ruin problems
The investigations of this paper are motivated by a standard model of an insurance company described by a counting process \(N\left( t\right) ,t\geq 0,\) interpreted as the number of claims on the company during the interval \(\left[ 0,t\right] ,\) a sequence of mutually i.i.d. random variables \(\left\{ Z_{k}\right\} _{k=1}^{\infty }\) with common distribution function which is independent of \(N\left( t\right) ,\) a positive real constant \(c\) representing the gross premium rate and the risk process \( \left\{ X\left( t\right) ,t\geq 0\right\} \)\ defined on a complete probability space \(\left( \Omega ,\mathcal{F},P\right) \) by: \[ X\left( t\right) =ct-\sum_{k=1}^{N\left( t\right) }Z_{k} \] \(S\left( t\right) =\sum_{k=1}^{N\left( t\right) }Z_{k}\) is the aggregate claim up to time \(t.\) \ If the number of claims \(N\left( t\right) \) forms a renewal process the model is called a \textit{renewal risk model}. In the classical risk model the counting process is a stationary Poisson counting process. The author is interested in the generalization of the counting process to a family of probability distributions, called \textit{inflated parameter distributions}, which she has introduced in earlier papers, generalizing the classical Poisson, binomial, negative binomial and logarithmic distributions by adding a new parameter \(\rho \in [ 0,1).\) An interpretation of \(\rho \) would be probability of 'failure' where there are two types of claims possible on the insurance company, 'success' or 'failure' with the probability of the latter being \(1-\rho .\) \ If the r.v. \(N \) is the number of trials until the \(i^{th}\) successive claim appears, it is distributed according to the negative binomial with parameters \(1-\rho \) and \(i.\) The r.v. \(N\) can be represented as the sum of \(i\) independent identically \(Ge_{1}\left( 1-\rho \right) \) distributed r.v.'s \(\left\{ X_{j}\right\} _{j=1}^{i}.\) \ Supposing that \(i\) is an outcome of the r.v. \(Y\) ,\(\;\)independent of \(\left\{ X_{j}\right\} _{j=1}^{i},\) with a Poisson distribution with parameter \(\lambda ,\) then \(N\) has an inflated-parameter distribution which coincides with the Polya-Aeppli distribution. A counting process \(\left\{ N\left( t\right) ,t\geq 0\right\} \) is called a \textit{Polya-Aeppli process} if it starts at zero, has independent stationary increments and for each \(t>0,\) \(N\left( t\right) \) is \textit{Polya-Aeppli distributed,} i.e., the following probability formula holds: \[ \begin{matrix} P\left( N\left( t\right) =n\right) &=&e^{-\lambda t},n=0 \\ \text{and } &=&e^{-\lambda t}\sum_{i=1}^{n}\left( _{i-1}^{n-1}\right) \frac{ \left[ \lambda \left( 1-\rho \right) t\right] ^{i}}{i!}\rho ^{n-i},n=1,2,.. \end{matrix} \] The risk process \(\left\{ X\left( t\right) ,t\geq 0\right\} \) is accordingly called a \textit{Polya-Aeppli risk process}. If the insurance company has initial capital \(u\geq 0\), the \textit{survival probability} or \textit{non-ruin probability} \[ \Phi \left( u\right) =P\left\{ \inf_{t\geq 0}X\left( t\right) \geq -u\right\} \] and the \textit{ruin probability} is \(1-\Phi \left( u\right) .\) The author studies the ruin probability of a Polya Aeppli risk process under some different cases and compare their ruin probabilities.
scientific article; zbMATH DE number 2148076