In this paper, the expected discounted penalty function is considered in the risk process with the time-correlated claims, that is, every main claim can cause a by-claim but the occurrence of the by-claim may be delayed. By the renewal argument, it is shown that the expected value satisfies a system of integro-differential equations. Moreover, the explicit expression for the Laplace transform of the expected value is derived by means of Rouche＇s theorem. A numerical example is also given for illustrating the result.
