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• W2005965612 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 $P(y,tensuremath{mid}{y}_{0},{t}_{0})$ are computed. $P(y,tensuremath{mid},{y}_{0},{t}_{0})$ 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, ${ensuremath{nu}}_{ensuremath{alpha}}^{+}=P({y}_{i}>ensuremath{alpha},{y}_{iensuremath{-}1}<ensuremath{alpha})$, for the returns, where $T(ensuremath{alpha})=1∕{ensuremath{nu}}_{ensuremath{alpha}}^{+}$ is the average waiting time for observing $y(t)=ensuremath{alpha}$ again." @default.
• W2005965612 created "2016-06-24" @default.
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• W2005965612 date "2007-06-18" @default.
• W2005965612 modified "2023-09-25" @default.
• W2005965612 title "Markov analysis and Kramers-Moyal expansion of nonstationary stochastic processes with application to the fluctuations in the oil price" @default.
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• W2005965612 doi "https://doi.org/10.1103/physreve.75.060102" @default.
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