Numerical algorithms for Panjer recursion by applying Bernstein approximation (Q384623)

The authors consider the compound risk model in actuarial science. They compute the distribution of the total loss \(S=\sum_{i=1}^NX_i\) using Panjer recursion when the number of claims \(N\) belongs to \((a,b,l)\) family i.e., for the probability function \(p^n\) of \(N\) \(\frac{p_k}{p_{k-1}}=a+\frac{b}{k}\), \(k=l+1,l+2,\dots\), holds for any \(l\geq 0\) and some constants \(a\), \(b\). For total claims with continuous distribution, the numerical calculation of \(S\) by applying Panjer recursion involves an approximation of the integration. In order to simplify the numerical algorithms, the authors apply Bernstein approximation for the continuous distribution function and obtain approximated recursive equations which are used for computing the approximated values of the compound distribution. A theoretical error bound for the approximation is also obtained.