Markov analysis and Kramers-Moyal expansion of nonstationary stochastic processes with application to the fluctuations in the oil price

We describe a general method for analyzing a nonstationary stochastic process X(t) which, unlike many of the previous analysis methods, does not require X(t) 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(t)=In[X(t+1)/X(t)], 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 vertical bar y(0),t(0)) are computed. P(y,t vertical bar,y(0),t(0)) satisfies a Fokker-Planck equation, which is equivalent to a Langevin equation for y(t) 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(alpha)=1/nu(+)(alpha) is the average waiting time for observing y(t)=alpha again.