 Panjer recursion

For positive integer severities with probability mass function $f_j$ and independent count in the Panjer claim-count class, the aggregate probability mass function starts at $g_0=p_0$ and satisfies the displayed recurrence. If $P,F,G$ are the count, severity and aggregate probability generating functions, respectively, then $G=P\circ F$ and $(1-aF)G^{\prime }=(a+b)F^{\prime }G$. Equating the coefficient of $z^{r-1}$ gives $r{g}_r={\sum }_{j=1}^r[a(r-j)+(a+b)j]f_j{g}_{r-j}$. The positivity assumption excludes zero severities; otherwise the initial value and recursion denominator change.

Panjer recursion is a recursive algorithm used in actuarial science and insurance mathematics to calculate the distribution of the sum of independent random variables, particularly in the context of risk management and insurance claims. Named after Hendrik Panjer, this method is particularly useful for computing the probabilities associated with different outcomes of aggregate claims. ### Key Elements of Panjer Recursion: 1. **Assumptions**: - The random variables (e.g., claims) are independent.