Generalized Ornstein–Uhlenbeck process by Doob’s theorem and the time evolution of financial prices

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

“…Thus, the THOU process allows arbitrary linear and Markovian noises instead of considering only Gaussian distributions. Previously, we found that the exponential decay patterns of the first four cumulants given by the THOU model can be found in some financial data, as the intraday logarithmic returns of stocks (Coca-Cola, Shell, Xerox and Petrobras) traded on the NYSE [1].…”

“…Similarly to the stationary OU process, u(t) is Markovian if, and only if, A(τ ) = e −γ τ , where γ ≥ 0 [1]. In this case, the characteristic function of u(t + τ )|u(t) is a limiting case (n → +∞) of a …”

“…. are positive constants, and {g n (·)} is a sequence of arbitrary characteristic functions [1]. Thus, Eq.…”

“…While originally proposed to describe stationary Gauss-Markov process [2][3][4][5], a number of generalizations of the OrnsteinUhlenbeck (OU) process have been suggested under less restrictive assumptions [6][7][8][9][10][11][12]. Indeed, non-Gaussian processes of the OU-type are becoming popular given their flexibility in modelling financial data [7].…”

“…Particularly, as Gaussianity and stationarity properties are not likely to be found in financial data [13], elsewhere [1] some of us have suggested the time-homogeneous Ornstein-Uhlenbeck (THOU) process, where these properties were relaxed to linearity and time-homogeneity. Thus, the THOU process allows arbitrary linear and Markovian noises instead of considering only Gaussian distributions.…”