Equivalent of Panjer formula

[Nobody1111]

If S (aggregate claim) is a random variable with compound Poisson distribution and X is a random variable describing amount of claim (we assume that realisations of X are positive integers) then using Pajner formula we can calculate P(S = k), k is positive integer. Panjer formula is a recursive formula and in order to find the value P(S=k) we need values P(S=0),...,P(S=k-1).
But this theorem is correct only for compound Poisson distribution. My question is: Is there any similar formula for other distributions (I mean a kind of any recursive formula)? I'm particularly interested in case if S is compound geometric distribution, but generalization for other distribution is also welcome. About random variable X we also assume that its realisations are positive integers.

Thank you for any comments.
Nobody1111

[ctperng]

If S (aggregate claim) is a random variable with compound Poisson distribution and X is a random variable describing amount of claim (we assume that realisations of X are positive integers) then using Pajner formula we can calculate P(S = k), k is positive integer. Panjer formula is a recursive formula and in order to find the value P(S=k) we need values P(S=0),...,P(S=k-1).
But this theorem is correct only for compound Poisson distribution. My question is: Is there any similar formula for other distributions (I mean a kind of any recursive formula)? I'm particularly interested in case if S is compound geometric distribution, but generalization for other distribution is also welcome. About random variable X we also assume that its realisations are positive integers.

Thank you for any comments.
Nobody1111

If you check out the book "Loss Models" by three authors (Klugman, Panjer, and Willmot), you will see a recursive formula (4.20 on p.91). Specializing to the (a,b,0) class, you get Theorem 4.48, which gives a recursive relation for the cases of Poisson, Negative Binomial, Binomial and Geometric.

ctperng

PS. X is (a,b,0) class if and only if X is either Poisson, Geometric, Binomial or Negative Binomial.

[Nobody1111]

Thank you very much. I didn't know about the fact that Panjer formula may be applied to other types of distributions. However the proof of the theorem for general case is much more complicated than for example for compund Poisson.

Nobody1111