Asymptotically optimal dividend policy for regime-switching compound Poisson models
A Markov modulated risk model with dividends is considered, where the dividends are restricted to absolutely continuous payments with a maximal rate. The maximal rate is chosen, such that it is smaller than the average premium. Letting the environmental Markov chain \(\{\alpha(t)\}\) run faster (\(\alpha^\epsilon(t) = \alpha(t/\epsilon)\)), the risk process converges to the averaged process with intensity \(\lambda = \sum \nu_i \lambda_i\), claim size distribution \(F(x) = \sum \nu_i F_i(x)\) and premium intensity \(c = \sum \nu_i c_i\). Here \(\{\nu_i\}\) is the stationary distribution of \(\{\alpha(t)\}\), and \(\lambda_i\), \(F_i\), respectively, are the claim intensity and the claim size distribution when \(\alpha(t)\) is in state \(i\). The value of a dividend strategy \(\{\pi_i^\epsilon(t)\}\) is \[ J^\epsilon(x,i,\pi^\epsilon) = \mathbb{E}\Bigl[\int_0^\tau e^{-r t} \sum_i I_{\alpha^\epsilon(t)=i} \pi_i(t)\;d t\Bigr]\;, \] where \(\tau\) is the time of ruin. The value function is \(v^\epsilon(x,i) = \sup_{\pi^\epsilon} J^\epsilon(x,i,\pi^\epsilon)\). A strategy \(\pi^{\epsilon *}\) is called optimal if \(J^\epsilon(x,i,\pi^{\epsilon *}) = v^\epsilon(x,i)\). Now the limits of \(J^\epsilon(x,i,\pi^\epsilon)\), \(v^\epsilon(x,i)\) and \(\pi^{\epsilon *}\) as \(\epsilon \to 0\) are considered. Since the problem for the compound Poisson model is well studied, the results can be used as an approximation to the Markov modulated model. Here it had been interesting to make a numerical example to see how well the approximation works. A first problem is the differentiability of the value function. This is no problem if the minimal premium rate is larger than the maximal dividend rate. However, if there are states where more dividends can be paid than premia are earned it is possible that the derivative of the value function has jumps (see also the example in [\textit{P. Azcue} and \textit{N. Muler}, Math. Finance 15, 261--308 (2005; Zbl 1136.91016)]). A second problem is the definition of the time of ruin as the first time where a non-positive surplus is reached. The correct condition \(v(0,i) = 0\) suggests that \(v(0+,i) = 0\), which does not hold. It is a pity that the authors do not use the normal definition of ruin as entrance to a negative surplus, or at least warn the readers that \(v(0+,i) > 0\). A third problem is that several assumptions are made in the proofs. So it is assumed that the dividend policy is of feedback form and continuous (which does not hold for the optimal strategy) and that the claim sizes have a finite second moment. In the example at the end, it even is assumed that the dividend policy is Lipschitz-continuous as a function of the surplus. In this example, it should have been mentioned, that there is no optimal dividend policy, since the optimal policy jumps at points where the derivative of \(v^\epsilon(x,i)\) becomes one.
scientific article; zbMATH DE number 5808631