Problem is taken from exams for Polish Actuaries from 10th October 2005, Exercice 1, General Insurance Mathematics. Here is the problem:

Damage which occurred in year t is liquidated:

- in the same year with probability 0.3

- in the next year with probabilitu 0.3

- in the year t+k (k>1) with probability 0.8 * (0.5)^k

We don't care in which period of the year the damage occurred - e.g. if one damage occurred on the 1st January and any other damage occurred on 30th December the probability that they are liquidated by the end of the k-th year is the same.

Let R(t) denotes the number of damages which were NOT liquidated at the end of the year t (R(t) consists both of the unliquidated damages which occurred in year t as well as of the unliquidated damages which occurred before the year t and weren't liquidated at the end of year t-1).

Let n(t) denotes the number of damages in year t.

Assuming that

R(t-1) = 1100

n(t) = 900

n(t-1) = 800

find the expected value of R(t) under the three conditions given above.

Answer is R(t) =1220.

The most sensible way of solving the problem seems to be finding the recursion formula for R(t) as the function of (R(t-1), n(t) and n(t-1).

As I was trying to solve the problem I met with the problem that n(t-2) which value is not given couldn't be reduced. I suppose I do it in the wrong way and don't know what to do with it.

If anyone could help me with this exercice I'll be extremely grateful.

Nobody1111