Cramer Lundberg

Overview


Compound Poisson

Cramer Lundberg Process


The Cramer Lundberg process models the reserve {% R(t) %} of an insurance company over time.
{% R(t) = R(0) + ct - \sum_{k=1}^{N(t)} V_k %}
where

  • {% c %} - is the premium rate (here taken to be a constant)
  • {% N(t) %} is the number of claims by time t, usually taken to be driven by a Poisson distribution with rate {% \lambda %}
  • {% V_k %} is the size of the kth claim, driven by some distribution


All random variables (each claim size and the number of claims) are assumed to be independent.

For convenience, the process is redefined in terms of excess reserve.
{% Excess \, Reserve = R(t) - R(0) %}
Ruin occurs when the excess reserve hits zero.

Calculation Techniques


  • Analytic
  • Monte Carlo - provides a reasonably simple method to calculate an approximation of the excess reserve distribution, even when the assumptions of the model are relaxed.