Panjer recursion

The Panjer recursion is an algorithm to compute the probability distribution of a compound random variable

S = sum_{i=1}^N X_i.,

where both

N,

and

X_i,

are stochastic and of a special type. It was introduced in a paper of Harry Panjer . It is heavily used in actuarial science.

Preliminaries

We are interested in the compound random variable

S = sum_{i=1}^N X_i,

where

N,

and

X_i,

fulfill the following preconditions.

Claim size distribution

We assume the

X_i,

to be i.i.d. and independent of

N,

. Furthermore the

X_i,

have to be distributed on a lattice

h mathbb{N}_0,

with latticewidth

h>0,

f_k = P[X_i = hk].,

Claim number distribution

N,

is the "claim number distribution", i.e.

N in mathbb{N}_0,

Furthermore, $N,$ has to be a member of the Panjer class. The Panjer class consists of all counting random variables which fulfill the following relation: $P[N=k] = p_k= (a + frac{b}{k}) cdot p_{k-1},~~k ge 1.,$ for some $a,$ and $b,$ which fulfill $a+b ge 0,$ . the value $p_0,$ is determined such that $sum_{k=0}^infty p_k = 1.,$

Sundt proved in the paper that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to the Panjer class, depending on the sign of $a,$ . They have the parameters and values as described in the following table. $W_N(x),$ denotes the probability generating function.

Distribution	$P[N=k],$	$a,$	$b ,$	$p_0,$	$W_N(x),$	$E[N],$	$Var(N),$
Binomial	$binom{n}{k} p^k (1-p)^{n-k} ,$	$frac{-p}{1-p}$	$frac{p(n+1)}{1-p}$	$(1-p)^n,$	$(px+(1-p))^{n} ,$	$np,$	$np(1-p) ,$
Poisson	$e^{-lambda}frac{ lambda^k}{k!},$	$0,$	$lambda ,$	$e^{- lambda},$	$e^{lambda(s-1)} ,$	$lambda,$	$lambda ,$
negative binomial	$frac{Gamma(r+k)}{k!,Gamma(r)},p^r,(1-p)^k ,$	$1-p,$	$(1-p)(r-1),$	$p^r ,$	$left(frac{p}{1 - x(1-p)}right) ^r ,$	$frac{r(1-p)}{p} ,$	$frac{r(1-p)}{p^2} ,$