Modelling persistent stationary processes in continuous time

This paper presents a novel approach to model continuous time processes that capture two countervailing features of many financial time series: persistency and long term stationarity. The processes introduced by our models exhibit persistent behaviors observed typically in non-stationary time series, but they remain stationary instead of tending towards explosive paths. We provide a relevant statistical theory and its implications on inference and forecasting, presenting both simulation evidence and empirical backing for the existence of, as well as the characteristic behavior for, such a series in real financial time series data.