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**Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times.**
*(English)*
Zbl 1202.91131

The authors analyse a compound Poisson risk model perturbed by a Brownian motion, that is
\[
U(t)=u+ct-S(t)+\sigma B(t),
\]
where \(u\geq0\) is the initial surplus, \(c>0\) the premium rate, \(S(t)\) the aggregate claim process, \(B(t)\) a standard Brownian motion starting from \(0\) and \(\sigma\) is the diffusion volatility.

In particular \[ S(t)=\sum_{i=1}^{N(t)}X_i, \] \(\{N(t), t\geq0\}\) being a Poisson process which depicts the number of claims up to \(t\) and \(\{X_i, i\geq 1\}\) being a sequence of strictly positive random variables which represent the individual claim sizes.

A central role is played by the sequence of random variables \(\{V_i, i\geq 1\}\), representing the interclaim times. In fact the paper focuses on the case of a perturbed risk model with dependence, i.e. under the assumption that the claim size and the interclaim time have a certain bivariate c.d.f. Within this context, modelling the c.d.f. by the Farlie-Gumbel-Morgenstern copula, the integro-differential equations involving the Gerber-Shiu functions are obtained.

Some examples are given in the case of exponential claims.

In particular \[ S(t)=\sum_{i=1}^{N(t)}X_i, \] \(\{N(t), t\geq0\}\) being a Poisson process which depicts the number of claims up to \(t\) and \(\{X_i, i\geq 1\}\) being a sequence of strictly positive random variables which represent the individual claim sizes.

A central role is played by the sequence of random variables \(\{V_i, i\geq 1\}\), representing the interclaim times. In fact the paper focuses on the case of a perturbed risk model with dependence, i.e. under the assumption that the claim size and the interclaim time have a certain bivariate c.d.f. Within this context, modelling the c.d.f. by the Farlie-Gumbel-Morgenstern copula, the integro-differential equations involving the Gerber-Shiu functions are obtained.

Some examples are given in the case of exponential claims.

Reviewer: Emilia Di Lorenzo (Napoli)

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91B70 | Stochastic models in economics |

60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |

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\textit{Z. Zhang} and \textit{H. Yang}, J. Comput. Appl. Math. 235, No. 5, 1189--1204 (2011; Zbl 1202.91131)

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### References:

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