On optimal stopping of risk reserve process

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a ≥ 0, receives premiums and pays out claims that occur according to a renewal process {N(t), t ≥ 0}. The times between consecutive claims are independent and identically distributed (i.i.d.). The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate α ∈ [0, 1], claims increase at rate β ∈ [0, 1]. The aim is to find the stopping time that maximizes the capital of the company. The cases of immediate claim payout as well as payout at the end of periods are considered. Plan of presentatio