The Kramers–Moyal Coefficients of Non-stationary Time Series and in the Presence of Microstructure (Measurement) Noise

Abstract

Most real world time series have transient behaviours and are non-stationary. They exhibit different type of non-stationarities, such as trends, cycles, random-walking, and generally exhibit strong intermittency. Therefore local stochastic characteristics of time series, such as the drift and diffusion coefficients, as well as the jump rate and jump amplitude, will provide very important information for understanding and quantifying “real time” variability of time series. For diffusive processes the systems have a longer memory and a higher correlation time scale and, therefore, one expects the stochastic features of dynamics to change slowly. In contrast, a rapid change of dynamics with jumps will cause strong ramp events (abrupt changes) in small time scales. Beside nonstationarity, in real world data, such as stock market index, pattern of cosmic background radiation, genomic data, etc., there is unique trajectory. Therefore, we require analysing techniques to estimate local stochastic behaviour of time series, which will be applicable to stationary and nonstationary time series and those with unique trajectory. This chapter contains technical aspects of the approach for real-time estimating of the Kramers–Moyal (KM) coefficients, and the drift and diffusion coefficients, as well as jump contributions for time series. We present the Kernel method (Nadaraya-Watson estimator) to estimate the time-dependent KM coefficients, which can be used in analysing stationary and nonstationary time series. We will also provide the details of estimating the KM coefficients in the presence of microstructure (measurement) noise, and show how the statistical properties of the noise can be determined from vanishing \(\tau \) limit behaviour of the KM conditional moments.