A transport firm has effected an insurance contract for a fleet of vehicles. The premium payment is due at the beginning of each year. There are four possible premium classes with a premium payment of Pi in class i, where Pi+1 <>i for i = 1, 2, 3. If no damage is claimed in the year just ended and the last premium charged is Pi, the next premium payment is Pi+1 (with P5 = P4); otherwise, the highest premium P1 is due. 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. In the case that a claim is made, the insurance company compensates the accumulated damage minus an own risk which amounts to ri for premium classic. The sizes of the damages in successive years are independent random variables that are exponentially distributed with mean 1/η. The claim strategy of the firm is characterized by four given numbers α1,…,α4 with for all i. If the current premium class is i, then the firm claims at the end of the year only damages larger than αi; otherwise, nothing is claimed. How do you calculate the long-run fraction of time the firm is in premium class i? Also, give an expression for the long-run average yearly cost.
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