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

Abstract

We generalize the Ornstein–Uhlenbeck (OU) process using Doob’s theorem. We relax the Gaussian and stationary conditions, assuming a linear and time-homogeneous process. The proposed generalization retains much of the simplicity of the original stochastic process, while exhibiting a somewhat richer behavior. Analytical results are obtained using transition probability and the characteristic function formalism and compared with empirical stock market data, which are notorious for the non-Gaussian behavior. The analysis focus on the decay patterns and the convergence study of the first four cumulants considering the logarithmic returns of stock prices. It is shown that the proposed model offers a good improvement over the classical OU model.