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

Abstract

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, $y(t)=\mathrm{ln}[X(t+1)∕X(t)]$, 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 $P(y,t\ensuremath{\mid}{y}_{0},{t}_{0})$ are computed. $P(y,t\ensuremath{\mid},{y}_{0},{t}_{0})$ 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, ${\ensuremath{\nu}}_{\ensuremath{\alpha}}^{+}=P({y}_{i}>\ensuremath{\alpha},{y}_{i\ensuremath{-}1}<\ensuremath{\alpha})$, for the returns, where $T(\ensuremath{\alpha})=1∕{\ensuremath{\nu}}_{\ensuremath{\alpha}}^{+}$ is the average waiting time for observing $y(t)=\ensuremath{\alpha}$ again.

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