A surplus process involving a compound Poisson counting process and applications

Applications of the classical compound Poisson model with claim sizes following a compound distribution

“…which recovers the relevant expression in Subsection 5.2.2 in Li and Sendova [6] and 𝜓(𝑢) may be again presented as a known linear combination of exponentials. Namely,…”

“…which is the result of Subsection 5.1.2 in Li and Sendova [6]. We may then write ψ(𝑠) = 𝑅(𝑠)/𝑃(𝑠), where 𝑅 and 𝑃 are known polynomials.…”

“…where {𝑀 (𝑡), 𝑡 ≥ 0} is a homogeneous Poisson process with intensity parameter 𝜆 > 0 and probability mass function (p.m.f.) given by It is noteworthy that Remark 2.1 in Li and Sendova [6] states that the process 𝑁 (𝑡) is over-dispersed regardless of the distribution of 𝑋. Definition 4.…”

“…Sendova and Minkova [9] propose the Poisson-Logarithmic process, where the compounded distribution is logarithmic and the compounding distribution is Poisson. Li and Sendova [6] introduce a surplus processes involving a Poisson-Negative-Binomial or a Poisson-Binomial counting process. To deduce specific expressions, exponential or Erlang claim amounts are assumed.…”

“…To deduce specific expressions, exponential or Erlang claim amounts are assumed. As noted in Remark 2.1 in Li and Sendova [6], any compound Poisson counting process is an over-dispersed process. Finally, there are studies of more general claim-counting processes than the compound Poisson one.…”