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Randomized dividends in the compound binomial model with a general premium rate. (English) Zbl 1164.91032
In the compound binomial model, the interclaim times \(\{W_{j}\}_{j=1}^{\infty}\) form a sequence of i.i.d. random variables with probability mass function \(f_{W}(j)=q(1-q)^{j-1}\). The individual claim amounts \(\{X_{j}\}_{j=1}^{\infty}\) are assumed to be a sequence of strictly positive, integer-valued and i.i.d. random variables. It is supposed that the random variables \(\{X_{j}\}_{j=1}^{\infty}\) are distributed like a generic random variable \(X\) with probability mass function \(f\). The random variables \(\{X_{j}\}_{j=1}^{\infty}\) and \(\{W_{j}\}_{j=1}^{\infty}\) are mutually independent. The total claim amount process \(\bar S=\{S_{k}, k\in N\}\) is defined as \(S_{k}=\sum_{j=1}^{N_{k}}X_{j}\). The dividend-modified surplus process \(\bar U=\{U_{k}, k\in N\}\) is defined as \(U_0=u\) and \(U_{k}=u+ck-S_{k}-Z_{k}\), where \(u\in N\) is the initial surplus level, \(c\in N^{+}\) is the level premium rate received per period and \(Z_{k}=Z_{k-1}+\sum_{i=1}^{n+1}D_{i,k}1_{\{U_{k-1}\in[b_{i-1},b_{i})\}}\) represents the total amount of dividends paid in the first \(k\) periods, where \(Z_0=0\), \(D_{i,k}\) represents the amount of dividends paid at time \(k\) if the surplus level at time \(k-1\) resides into the \(i\)-th surplus layer. Using the roots of generalization of Lundberg’s fundamental equation and the general theory on difference equations, the author derives an explicit expression for the Gerber-Shiu discounted penalty function with any initial surplus. An explicit expression is also obtained for the expected discounted dividend payment before ruin.

MSC:
91B30 Risk theory, insurance (MSC2010)
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