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On a perturbed by diffusion compound Poisson risk model with delayed claims and multi-layer dividend strategy. (English) Zbl 1294.91073

In a perturbed risk process, there are two types of claims. Claims of the first type \(\{X_k\}\) occur according to a Poisson process \(\{N(t)\}\) with occurrence times \(\{T_k\}\). Each of these claims causes a claim of the second type \(\{Y_k\}\). With probability \(\theta\) this claim occurs at the same time \(T_k\) and with probability \(1-\theta\) at the occurrence of the next claim \(T_{k+1}\). Let \(I_k = 1\) with probability \(\theta\) and \(I_k = 0\), otherwise. Further let \(\{B(t)\}\) be a standard Brownian motion. It is assumed that all stochastic quantities are independent, and that \(\{X_k\}\) and \(\{Y_k\}\), respectively, are i.i.d. Then the surplus process becomes \[ U(t) = u + c t - \sum_{k=1}^{N(t)} (X_k + I_k Y_k) - \sum_{k=1}^{N(t)-1} (1-I_k) Y_k + \sigma B(t). \] The time of ruin is defined as \(\tau = \inf\{t > 0: U(t) < 0\}\). The quantity of interest is the Gerber-Shiu discounted penalty function \[ \phi(u) = E\bigl[ e^{-\delta \tau} w(U(\tau-),|U(\tau)|) \bigm| U(0) = u\bigr]. \]
Defining \(A(t) = I_{N(t)}\) then \(\{(U(t), A(t))\}\) becomes a Markov process. Defining \[ \phi_k(u) = E\bigl[ e^{-\delta \tau} w(U(\tau-),|U(\tau)|) \bigm| U(0) = u, A(0) = k\bigr], \] we have \(\phi(u) = \phi_1(u)\). The functions \(\phi_k(u)\) solve then the usual system of integro-differential equations. Taking the Laplace transform, using \(\phi_k(u) = w(0,0)\), the Laplace transforms can be expressed using the unknown values \(\phi_k'(0)\). Because the denominator of the Laplace transforms have two positive roots, the numerator must have the same roots. This determines \(\phi_k'(0)\). One therefore has found an explicit expression for the Laplace transforms. If the Laplace transform of the claim size distributions is rational, the Laplace transform may be inversed. It is now possible to perform the same operations as for the classical model. For example, a defective renewal equation can be found.
A second model considers dividend payments that depend on the surplus level. More specifically, there are \(n\) layers \(0 = \beta_0 < \beta_1 < \cdots < \beta_{n+1} = \infty\). If \(U_\beta(t) \in [\beta_{k-1},\beta_k)\) then the premium rate is \(c_k\), where \(c_1 > c_2 > \cdots > c_{n+1} \geq 0\). One can interpret \(d_k = c_1 - c_k\) as a rate at which dividends are paid. Similar techniques as above yield then the Gerber-Shiu expected penalty function in this case. For both models, numerical results for exponentially distributed claim sizes are given.
In an appendix, it is proved that \(\phi_k(u)\) is twice continuously differentiable provided the densities of the claim size distributions as well as the functions \(w_k(x) = E[w(U(\tau-), |U(\tau)|) \mid U(\tau-) = x, A(\tau-) = k]\) are twice continuously differentiable.

MSC:

91B30 Risk theory, insurance (MSC2010)
60J25 Continuous-time Markov processes on general state spaces
60J65 Brownian motion
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