A haulier operates a fleet of trucks. His contract with an insurance company covers his whole fleet and has the following structure ('bonus malus system' in car insurance): The haulier has to pay his...

A haulier operates a fleet of trucks. His contract with an insurance company covers his whole fleet and has the following structure ('bonus malus system' in car insurance): The haulier has to pay his premium at the beginning of each year. There are 3 premium levels: λ₁, λ₂
and λ₃
with λ_3
2
1. If no claim had been made in the previous year and the premium level was λ_i,  then the premium level in the current year is λ_i+1
or λ₃
if λ_i
= λ₃. If a claim had been made in the previous year, the premium level in the current year is λ₁. The haulier will claim only then if the total damage a year exceeds an amount of c_i
given the premium level λ_i
in that year; i = 1, 2, 3. In case of a claim, the insurance company will cover the full amount minus a profitincreasing amount of

_i, 0 ≤

_i
i. The total damages a year are independent random variables, identically distributed as M.

Given a vector of claim limits (c₁, c₂, c₃), determine the haulier's long-run mean loss cost a year.

Hint Introduce the Markov chain {X₁, X₂, ...}, where X_n
= I if the premium level at the beginning of year n is λ_i
and make use of theorem 4.10.

(Loss cost = premium plus total damage not refunded by the insurance company.)