Page 113 - A First Course In Stochastic Models

THE EQUILIBRIUM PROBABILITIES 105

contract is for a whole fleet of vehicles, the transport firm has obtained the option
to decide only at the end of the year whether the accumulated damage during
that year should be claimed or not. If a claim is made, the insurance company
compensates the accumulated damage minus an own risk which amounts to r i
for premium class i. The total damages in the successive years are independent
random variables having a common probability distribution function G(s) with
density g(s). What is a reasonable claim strategy and what is the long-run average
cost per year?
An obvious claim strategy is the rule characterized by four parameters α 1 , . . . , α 4 .
If the current premium class is class i, then the transport firm claims at the end of the
year only damages larger than α i , otherwise nothing is claimed. Consider now a given
claim rule (α 1 , . . . , α 4 ) with α i > r i for i = 1, . . . , 4. For this rule the average
cost per year can be obtained by considering the stochastic process which describes
the evolution of the premium class for the transport firm. Let

X n = the premium class for the firm at the beginning of the nth year.

Then the stochastic process {X n } is a Markov chain with four possible states
i = 1, . . . , 4. The one-step transition probabilities p ij are easily found. A one-
step transition from state i to state 1 occurs only if at the end of the present year
a damage is claimed, otherwise a transition from state i to state i + 1 occurs (with
state 5 ≡ state 4). Since for premium class i only cumulative damages larger than
α i are claimed, it follows that

p i1 = 1 − G(α i ), i = 1, . . . , 4,
p i,i+1 = G(α i ), i = 1, 2, 3 and p 44 = G(α 4 ).

The other one-step transition probabilities p ij are equal to zero. The Markov chain
has no two disjoint closed sets. Hence the equilibrium probabilities π j , 1 ≤ j ≤ 4,
are the unique solution to the equilibrium equations

π 4 = G(α 3 )π 3 + G(α 4 )π 4 ,

π 3 = G(α 2 )π 2 ,
π 2 = G(α 1 )π 1 ,

π 1 = {1 − G(a 1 )}π 1 + {1 − G(α 2 )}π 2 + {1 − G(α 3 )}π 3 + {1 − G(α 4 )}π 4
together with the normalizing equation π 1 +π 2 +π 3 +π 4 = 1. These linear equations
can be solved recursively. Starting with π 4 := 1, we recursively compute π 3 , π 2
and π 1 from the first three equations. Next we obtain the true values of the π j

4
from π j := π j / k=1 π k . Denote by c(j) the expected costs incurred during a year
in which premium P j is paid. Then by Theorem 3.3.3 we have that the long-run