This work deals with the reliability and maintenance of structures that are damaged by shocks arriving randomly in time. The degradation is modeled as a cumulative stochastic point process. Previous studies mostly adopted expected cost rate criterion for optimizing the maintenance policies, which ignores practical implications of discounting of maintenance cost over the life cycle of the system. Therefore, detailed analysis of expected discounted cost criterion has been done, which provides a more realistic basis for optimizing the maintenance. Examples of maintenance policies combining preventive maintenance with age-based replacement are analyzed. Derivation for general cases involving preventive maintenance damage level have been discussed. Specific cases are also considered.