On the time-homogeneous Ornstein–Uhlenbeck process in the foreign exchange rates

• Gaussianity and stationarity assumptions replaced by linearity and time-homogeneity.
• We revisit the time-homogeneous Ornstein–Uhlenbeck (THOU) process.
• We employ the THOU process to analyze foreign exchange rates against the US dollar.
• The first four cumulants patterns from data can be described by the THOU process.

Since Gaussianity and stationarity assumptions cannot be fulfilled by financial data, the time-homogeneous Ornstein–Uhlenbeck (THOU) process was introduced as a candidate model to describe time series of financial returns [1]. It is an Ornstein–Uhlenbeck (OU) process in which these assumptions are replaced by linearity and time-homogeneity. We employ the OU and THOU processes to analyze daily foreign exchange rates against the US dollar. We confirm that the OU process does not fit the data, while in most cases the first four cumulants patterns from data can be described by the THOU process. However, there are some exceptions in which the data do not follow linearity or time-homogeneity assumptions.