Optimal stopping of a risk process when claims are covered immediately

Abstract

The optimal stopping problem for the risk process with interests rates and when claims are covered immediately is considered. An insurance company receives premiums and pays out claims which have occured according to a renewal process and which have been recognized by them. The capital of the company is invested at interest rate $α\in\Re^{+}$, the size of claims increase at rate $β\in\Re^{+}$ according to inflation process. The immediate payment of claims decreases the company investment by rate $α_1$. The aim is to find the stopping time which maximizes the capital of the company. The improvement to the known models by taking into account different scheme of claims payment and the possibility of rejection of the request by the insurance company is made. It leads to essentially new risk process and the solution of optimal stopping problem is different.