By Prof. Dr. sc. math. Klaus D. Schmidt (auth.)

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6 Theorem. Proof. 1, the claim number process is a Markov process. To prove homogeneity, consider kENo and h E R+ and an admissible pair (n, t) satisfying P[{Nt = n}] > O. Then we have Pn,n+k(t,t+h) = = P[{Nt+h = n+k}I{Nt = n}] P[{Nt+h-Nt = k}I{Nt-No = n}] P[{Nt+h-Nt = k}] P[{Nh-NO = k}] P[{Nh = k}]. Therefore, {NdtER+ is homogeneous. 1. 7 Corollary. If the claim number process is a Poisson process, then it is a homogeneous Markov process. \n}nEN of continuous functions R+ ~ (0,00) such that, for each admissible pair (n, t), (i) P[ {Nt (ii) the function R+ (iii) ~ [0, 1J : h t-> = n} J > 0, Pn,n(t, t+h) is continuous, An+!

T m , the family of increments {Nt,+h -Nt,_I+hLE{I, ... ,m}, and it is - a (homogeneous) Poisson process with parameter 0: E (0,00) if it has stationary independent increments such that PN, = P(o:t) holds for all t E (0,00). It is immediate from the definitions that a claim number process having independent increments has stationary increments if and only if the identity PN'+h -N, = PNh holds for all t, h E R+. 1 Lemma (Multinomial Criterion). Let are equivalent: (a) The claim number process {NdtER+ satisfies PN, = 0: E (0,00).

Proof. The result is obtained by straightforward calculation: • Assume first that (a) holds. Then we have P [n {Nt, -Nt,_l = kj }] J=I P [D{Nt, -Nt,_l = mn ! I . e-atm (o:t~)n n. 24 Chapter 2 The Claim Number Process Therefore, (a) implies (b) . • Assume now that (b) holds. Then we have PN , = P(at) as well as P LQ {Nt} - Nt}_l = k 1 j } P[{Ntm = n}] m j=! Therefore, (b) implies (a). 2 raises the question whether the Poisson process can also be characterized in terms of the claim arri val process or in terms of the claim interarrival process.