We describe a general method for analyzing a nonstationary stochastic process $X\left(t\right)$ which, unlike many of the previous analysis methods, does not require $X\left(t\right)$ to have any scaling feature. The method is used to study the fluctuations in the daily price of oil. It is shown that the returns time series, $y\left(t\right)=\ln [X(t+1)∕X\left(t\right)]$, is a stationary and Markov process, characterized by a Markov time scale $t_M$. The coefficients of the Kramers-Moyal expansion for the probability density function $P(y,t\mid y_0,t_0)$ are computed. $P(y,t\mid ,y_0,t_0)$ satisfies a Fokker-Planck equation, which is equivalent to a Langevin equation for $y\left(t\right)$ that provides quantitative predictions for the oil price over times that are of the order of $t_M$. Also studied is the average frequency of positive-slope crossings, ${\nu }_{\alpha }^{+}=P(y_i>\alpha ,y_{i-1}<\alpha )$, for the returns, where $T\left(\alpha \right)=1∕{\nu }_{\alpha }^{+}$ is the average waiting time for observing $y\left(t\right)=\alpha$ again.

• Received 10 January 2007

DOI:https://doi.org/10.1103/PhysRevE.75.060102