Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times
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.
scientific article; zbMATH DE number 5821392