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.

[Type][CrossLinking]The original version of this chapter was revised: Belated correction has been incorporated. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-18472-8_24

Problems

17.1

Nadaraya-Watson Estimator, Ornstein–Uhlenbeck Process with Jumps

Use Nadaraya-Watson estimator (use a Gaussian kernel with bandwidth $h\simeq 0.1$) to estimate time dependent Kramers–Moyal coefficients and reproduce the results presented in Fig. 17.1. The drift and diffusion coefficients as well as jump amplitude and jump rate are given in the Sects. 17.1.1 and 21.2.1. Use the Euler-Maruyama scheme to integrate related jump-diffusion dynamical equation.

17.2

Local Drift and Diffusion of the Minimal Market Model

The minimal market model is given by a (Itô) Langevin dynamics with time dependent drift a(x, t) and multiplicative b(x, t) as,

$$ a(x,t) = \alpha \exp \{ \eta t \}, b(x,t) = \sqrt{\alpha \exp \{ \eta t \} x} $$

with scaling parameter $\alpha > 0$ and growth rate $\eta >0$.

(a) Use the Euler-Maruyama scheme to approximate this Langevin equation and generate $10^5$ data points of $x(t_n)$ with $x(t_0)= x_0=1$, $dt=0.001$, $\alpha =0.2$ and $\eta =0.01$.

(b) Use the Nadaraya-Watson estimator with Gaussian kernel (with different bandwidth $h \in \{0.01,0.1,0.6\}$) to estimate time dependence of the functions a and b. The first and second order Kramers–Moyal coefficients are given by $D^{(1)} (x,t) = a(x,t)$ and $b(x,t) = \sqrt{2D^{(2)} (x,t)}$.

17.3

Microstructure Noise

Derive the third- and fourth-order ($j=3,4$) Kramers–Moyal conditional moments $K^{(j)}(y, \tau )$ in terms of $K^{(j)}(x, \tau )$ and statistical moments of noise $\eta (t)$ in the weak noise limit. Assume that the noise to be uncorrelated with x(t), that it has finite statistical moments $\langle \eta ^{n} \rangle$ for $n=2,\cdots$.

17.4

Microstructure Noise

Consider a Ornstein–Uhlenbeck process x(t) with drift and diffusion coefficients as $D^{(1)} (x,t) = -10x$ and $D^{(2)} (x,t) = 1$, respectively, and synthesis a time series with $10^6$ data points (with $x(t=0)=0$ and $dt=0.01$).

Now generate a new time series with microstructure noise as,

$$ y(t) = x(t) + \sigma \zeta (t) $$

where $\sigma =0.01$ and $\zeta (t)$ is uncorrelated zero mean, unit variance Gaussian noise $\zeta \sim \mathcal{N} (0,\sigma _{\zeta } ^2=1)$.

(a) Estimate second order Kramers–Moyal conditional moment for process y(t), and using the method presented in Sect. 17.2, determine $\sigma$.

(b) Estimate drift and diffusion coefficients of original process x(t).

(c) Find the explicit expressions for $\gamma _{1,2}(y)$ in terms of y, $\sigma$ and variance of x [14].