Lectures on Risk Theory by Prof. Dr. sc. math. Klaus D. Schmidt (auth.)

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

"... in particular now, the place from the facet of mathematical finance curiosity is usually proven for insurance-related items, a booklet like this one will certainly be instrumental in speaking the elemental mathematical types to non-experts in coverage. I for this reason welcome this e-book for its meant audience." P. Embrechts. Mathematical reports, Ann Arbor "... [The publication] comes in handy as an in depth theoretical supplement to 1 of the classical introductory texts on probability thought ...". M. Schweizer. Zentralblatt für Mathematik, Berlin "... The author's pursuits are sincerely proclaimed on the outset, and they're pursued with patience and integrity. the result's a e-book that is an quintessential entire, unique in a few respects, with attention-grabbing contributions. And no error - no longer even a unmarried misprint. i like to recommend it to each train of chance concept as a resource of mathematically good proofs and whole explorations of definite elements of the subject." R. Norberg. Metrika, Heidelberg

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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.

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