Mathematical investigation of the Gerber–Shiu function in the case of dependent inter-claim time and claim size

In this paper we investigate the well-known Gerber–Shiu expected discounted penalty function in the case of dependence between the inter-claim times and the claim amounts. We set up an integral equation for it and we prove the existence and uniqueness of its solution in the set of bounded functions. We show that if δ > 0 , the limit property of the solution is not a regularity condition, but the characteristic of the solution even in the case when the net profit condition is not fulfilled. It is the consequence of the choice of the penalty function for a given density function. We present an example when the Gerber–Shiu function is not bounded, consequently, it does not tend to zero. Using an operator technique we also prove exponential boundedness.