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Renewal point processes show up in many different fields of science and engineering. In some cases the renewal points become the only observable parts of an anticipated hidden random variation of some physical quantity. The hypothesis might be that a hidden random process originating from zero or some other low value only becomes visible at the time of first crossing of some given value level, and that the process is restarted from scratch immediately after the level crossing. It might then be of interest to reveal the defining properties of this hidden process from a sample of observed first-passage times. In this paper the hidden process is first anticipated as a non-stationary Ornstein–Uhlenbeck (OU) process with unknown parameters that have to be estimated only by use of the information contained in a sample of first-passage times. The estimation method is a direct application of the Fortet integral equation of …