[1]Pyei Phyo Lin

[1]Joachim Peinke

[1]ForWind, Institute of Physics, University of Oldenburg, Oldenburg, Germany

2]Theoretical Physics/Complex Systems, ICBM, University of Oldenburg,
Oldenburg, Germany

3]Department of Physics, Sharif University of Technology, Tehran, 11155-9161

# Assessing the Suitability of the Langevin Equation for Analyzing Measured
Data Through Downsampling

<EMAIL_ADDRESS>Matthias Wächter<EMAIL_ADDRESS><EMAIL_ADDRESS>M. Reza Rahimi Tabar<EMAIL_ADDRESS>*  [  [

###### Abstract

The measured time series from complex systems are renowned for their intricate
stochastic behavior, characterized by random fluctuations stemming from
external influences and nonlinear interactions. These fluctuations take
diverse forms, ranging from continuous trajectories reminiscent of Brownian
motion to noncontinuous trajectories featuring jump events. The Langevin
equation serves as a powerful tool for generating stochasticity and capturing
the complex behavior of measured data with continuous stochastic
characteristics. However, the traditional modeling framework of the Langevin
equation falls short when it comes to capturing the presence of abrupt
changes, particularly jumps, in trajectories that exhibit non-continuity. Such
non-continuous changes pose a significant challenge for general processes and
have profound implications for risk management. Moreover, the discrete nature
of observed physical phenomena, measured with a finite sample rate, adds
another layer of complexity. In such cases, data points often appear as a
series of discontinuous jumps, even when the underlying trajectory is
continuous. In this study, we present an analytical framework that goes beyond
the limitations of the Langevin equation. Our approach effectively
distinguishes between diffusive or Brownian-type trajectories and trajectories
with jumps. By employing downsampling techniques, where we artificially lower
the sample rate, we derive a set of measures and criteria to analyze the data
and differentiate between diffusive and non-diffusive behaviors. To further
demonstrate its versatility and practical applicability, we have applied our
proposed method to real-world data in various scientific fields, turbulence,
optical tweezers for trapped particles, neuroscience, renewable energy, and
market price analysis.

###### keywords:

Stochastic processes, Langevin equation, Jump-diffusion process

##  1 Introduction

In a complex world, empirical data ranging from natural science (e.g.
physical, climatological, biological and ecological systems) to social
sciences (e.g. infectious diseases spread, finance and economic) can be
characterized by the stochastic dynamics. The classical method to approach
such systems is to approximate the data with the Langevin equation or
diffusion process [1, 2, 3]. The Langevin equation is a stochastic
differential equation which is driven by Gaussian white noise generating the
continuous (Brownian type) stochastic process [4, 5]. Recent studies have
highlighted the presence of jumps, which are discontinuous/non-Brownian
events, in complex systems. These jumps can have significant implications for
risk management and have received considerable attention in recent years [6].
Jumps introduce higher uncertainties in the stochastic features of the
underlying system and contribute to the non-Gaussian characteristics of the
increment statistics in short-time scales [7].

In the presence of discontinuous events, the traditional modeling framework of
the Langevin equation, which generates only continuous trajectories, is
inadequate for capturing these phenomena [5]. To address this limitation, an
effective approximation is to introduce an additional noise term known as
jump-noise in the Langevin equation with distributed jumps in time and
amplitude. This gives rise to jump-diffusion processes [7, 6, 5, 8, 9]. In
this framework, the jump noise represents the discontinuous paths within the
diffusion process.

The nature of experimental data is such that it is often measured with limited
sampling time or spatial resolution. As a result, the recorded trajectory may
appear discontinuous, despite the underlying process being continuous.
Furthermore, certain real-world datasets, such as those related to stock
market indices, glaciers, and sea levels, exhibit distinct trajectories. Here,
we present a powerful method to differentiate between diffusive and non-
diffusive behavior in empirical data by using downsampling. This method is
applicable to datasets with finite sampling intervals and those that display
unique trajectories.

##  2 Diffusive and non-diffusive behaviors

A given time series typically consists of three independent time scales: the
sampling interval $\tau_{s}$, the correlation timescale $T_{\rm C}$, and the
average timescale between jumps $T_{\rm J}$ (if present). The timescales
$T_{\rm C}$ and $T_{\rm J}$ can be estimated using Kramers–Moyal (KM)
coefficients of order one, four and six, see below [10]. For time series
$x(t)$, KM coefficients of order $l$, $K^{(n)}({x},t)$ are given by

$K^{(n)}(x,t)=\lim_{\tau_{s}\rightarrow
0}\frac{M^{(n)}(x,t,\tau_{s})}{\tau_{s}}$ (1)

where $M^{(n)}(x,t,\tau_{s})$ are KM conditional moments, defined as

$M^{(n)}(x,t,\tau_{s})=\int
dx^{\prime}(x^{\prime}-x)^{n}p(x^{\prime},t+\tau_{s}|x,t)~{}.$ (2)

In definition (1), we omitted the $n!$ term in the denominator to simplify the
expressions that allowing us to distinguish between diffusive and non-
diffusive processes.

For a diffusive process that generates a continuous trajectory like Brownian
motion [5], the dynamics of a state variable $x(t)$ can be described by a
Langevin equation that includes both deterministic and stochastic
contributions, given by [11]:

$\mathrm{d}x=D^{(1)}({x},t)\,\mathrm{d}t+\sqrt{D^{(2)}({x},t)}\,\mathrm{d}W_{t}~{},$
(3)

where $D^{(1)}(x,t)$ (drift) and $D^{(2)}(x,t)$ (diffusion coefficient) are
given functions of the state variable $x$ and “time” $t$. They are expressed
in terms of the first and second KM coefficients as
$D^{(1)}(x,t)=K^{(1)}(x,t)$ and $D^{(2)}(x,t)=K^{(2)}(x,t)$. All other KM
coefficients are zero, i.e. $K^{(n)}(x,t)=0$ for $n\geq 3$, for a diffusive
process [12]. In Eq. (3), the increment of Wiener process is defined by
$dW_{t}=\eta(t)dt$ where $\eta(t)$ represents a zero-mean Gaussian white noise
with unit intensity, satisfying
$\langle\eta(t)\eta(t^{\prime})\rangle=\delta(t-t^{\prime})$.

In practice, verifying the condition $K^{(n)}(x,t)=0$ for $n\geq 3$ is not
straightforward, as it requires knowledge of the expansions of KM conditional
moments $M^{(n)}(x,t,\tau_{s})$ in terms of the sampling time $\tau_{s}$. By
employing the expansion of KM conditional moments, it has been demonstrated
that the necessary condition for a process to belong to the diffusive category
is $M^{(4)}(x,t,\tau_{s})\simeq 3(M^{(2)}(x,t,\tau_{s}))^{2}$ [13]. In an
alternative representation, we define $\Theta(x,t,\tau_{s})$ as:

$\Theta(x,t,\tau_{s})=3\frac{(M^{(2)}(x,t,\tau_{s}))^{2}}{M^{(4)}(x,t,\tau_{s})}~{}.$
(4)

If $\Theta(x,t,\tau_{s})\approx 1$, it indicates that the data fulfills the
criterion for belonging to the diffusive processes. This criterion has been
examined for various linear and nonlinear drift and diffusion coefficients,
and we have observed $\Theta(x,t,\tau_{s})\approx 1$; for more details, refer
to Appendix A.4.

For a given time series that $\Theta(x,t,\tau_{s})$ has not satisfying
$\Theta(x,t,\tau_{s})\approx 1$, one can approximate its dynamics with a jump-
diffusion process. This can simply be done by adding to Langevin equations an
additional Poisson distributed jump term (jump-diffusion process), therefore
one can write,

$\mathrm{d}x=D^{(1)}({x},t)\,\mathrm{d}t+\sqrt{D^{(2)}({x},t)}\,\mathrm{d}W_{t}+\xi\,\mathrm{d}J_{t}~{}.$
(5)

In this model, we consider the noise $\xi\sim\mathcal{N}(0,\sigma_{\xi}^{2})$,
which represents the size of jumps and follows a normal distribution with
variance $\sigma_{\xi}^{2}$. The jump amplitude is also defined as,
$\sigma_{\xi}^{2}$ and can be state and time-dependent. Additionally, the
Poisson distributed jump process $J_{t}\sim P(\lambda t)$ is present, which is
a zero-one jump process with a jump rate of $\lambda(x,t)$. Indeed, the final
term in Eq. (5) simulates abrupt jumps in the trajectories of $x(t)$ with
varying amplitudes. In the jump-diffusion dynamical equation (5), we assume
that jump events are rare and can be modeled via a Poisson process.

Two unknown functions, jump rate $\lambda(x,t)$ and jump amplitude,
$\sigma_{\xi}^{2}$ can be found using the relations [7, 5],

$\displaystyle\sigma_{\xi}^{2}(x,t)=\frac{K^{(6)}(x,t)}{5K^{(4)}(x,t)},\hskip
14.22636pt\lambda(x,t)=\frac{K^{(4)}(x,t)}{3\sigma_{\xi}^{4}(x,t)}~{}.$ (6)

For jump-diffusion processes described by Eq. (5), expansion of KM conditional
moments can be done and one can define a function $Q(x,t,\tau_{s})$ (so-called
$Q$-criterion, [13]) as (see Appendix A.1),

$Q(x,t,\tau_{s})=\frac{M^{(6)}(x,t,\tau_{s})}{5M^{(4)}(x,t,\tau_{s})}~{}.$ (7)

The function $Q(x,t,\tau_{s})$ serves as a criterion for determining if a jump
is present in a given time series.

For small values of $\tau_{s}$ in the functions $\Theta(x,t,\tau_{s})$ and
$Q(x,t,\tau_{s})$, which are obtained from second, fourth, and sixth-order KM
conditional moments, we observe distinct behavior for three types of time
series (see Appendix A), as

$\Theta(x,t,\tau_{s})\approx\begin{cases}1,&\text{diffusive}\\\
\frac{(D^{(2)}(x,t)+\lambda(x,t)\sigma_{\xi}^{2}(x,t))^{2}}{\lambda(x,t)\sigma_{\xi}^{4}(x,t)}\tau_{s},&\text{jumpy}\\\
1,&\text{white noise at }x=0,{}\end{cases}$ (8)

and

$Q(x,t,\tau_{s})\approx\begin{cases}D^{(2)}(x,t)\,\tau_{s},&\text{diffusive},\\\
\sigma_{\xi}^{2}(x,t),&\text{jumpy},\\\ \sigma_{\eta}^{2},&\text{white noise
at},~{}x=0.\end{cases}$ (9)

Here, the behaviors of $\Theta(x,t,\tau_{s})$ and $Q(x,t,\tau_{s})$ for
various processes, including diffusive (Langevin dynamics), jump-diffusion
processes and pure white noise are given. It is worth noting that the
requirement for $x=0$ is needed only in the case of Gaussian white noise. For
all other situations, the aforementioned relations hold for any value of $x$.
Also $\sigma_{\eta}^{2}$ is the variance of white noise.

The function $\Theta(x,t,\tau_{s})$ exhibits a linear behavior with respect to
$\tau_{s}$ for jump-diffusion and pure jump processes, while it remains unity
for diffusive and white noise trajectories (at $x=0$). Simultaneously, the
function $Q(x,t,\tau_{s})$ takes on a constant value of
$\sigma_{\xi}^{2}(x,t)$ for small $\tau_{s}$ in the case of jump-diffusion
processes. However, for diffusive processes, it displays a linear relationship
with $\tau_{s}$, specifically given by $D^{(2)}(x,t)\tau_{s}$.

In the Appendix A.3, we have conducted numerical integration of corresponding
Langevin and jump-diffusion dynamics with various different values of
$\tau_{s}$. This allowed us to verify the behaviors of $\Theta(x,t,\tau_{s})$
and $Q(x,t,\tau_{s})$ in relation to $\tau_{s}$ for diffusive processes, jump-
diffusion processes and pure uncorrelated (white) noise. Therefore, by
utilizing the $\Theta$\- and $Q$-criterion, we can effectively distinguish
whether a stochastic time series “generated” from certain dynamical equations
originates from a diffusive (continuous) process or a jumpy (discontinuous)
process [13].

The results presented above for $\Theta(x,t,\tau_{s})$ and $Q(x,t,\tau_{s})$
are derived from the expansion of Kramers–Moyal coefficients for Langevin and
jump-diffusion dynamical equations with varying time steps, $\tau_{s}$. To
make them applicable to real-world time series, especially those displaying
unique trajectories, their expressions require modification.

In the following, we demonstrate that the obtained behaviors of $Q(x,t,\tau)$
hold true for time scales $\tau_{s}$ that are shorter than both the
correlation timescale $T_{\rm C}$ of the time series and the average timescale
between jumps $T_{\rm J}$. However, when dealing with a real-world time series
sampled with $\tau_{s}$, it is important to note that we typically lack direct
access to the ground truth values of $T_{\rm C}$ and especially $T_{\rm J}$.
Consequently, assessing the behaviors of $\Theta(x,t,\tau_{s})$ and
$Q(x,t,\tau)$ can potentially lead to misleading conclusions. Therefore, in
this study, we aim to investigate this matter more deeply to gain a thorough
understanding of the implications and limitations associated with using these
criteria in real-world scenarios.

Based on a given dynamical equation, it is possible to generate a set of
stochastic processes with different time scales $T_{\rm C}$ and $T_{\rm J}$ in
order to check the $\Theta$\- and $Q$-criteria for the corresponding time
series for different $\tau_{s}$. In order to verify the these criteria using
synthetic data, we conducted numerical integration of a Langevin equation with
specific parameters. The drift coefficient is given by $D^{(1)}(x,t)=-\gamma
x$, where $\gamma=1~{}{\rm s^{-1}}$. The diffusion coefficient is denoted as
$D^{(2)}(x,t)=D$, with $D=1~{}{\rm s^{-1}}$. Additionally, we considered an
additional jump term (jump-diffusion process), characterized by a jump rate of
$\lambda=100~{}{\rm s^{-1}}$ and a variance of $\sigma_{\xi}^{2}=1$. In the
cases involving pure noisy data, we employed normal Gaussian white noise with
a variance of $\sigma_{\eta}^{2}=1$. The chosen parameters in the Langevin and
jump-diffusion dynamics yield two timescales. The correlation timescale is
given by $T_{\rm C}=\frac{1}{\gamma}=1~{}\mathrm{s}$, and the average
timescale between jumps is denoted as $T_{\rm
J}=\frac{1}{\lambda}=0.01~{}\mathrm{s}$.

From the simulated data with different time step $\tau_{s}$-values, we
estimate $\Theta(x,\tau=\tau_{s})$ and $Q(x,\tau=\tau_{s})$ as shown in Fig.
1. Theoretical predictions as described in Eqs. (9) and (8) are also plotted
with the solid lines. As shown in Fig. 1, for jump-diffusion process, Eq. (9)
is valid only for $\tau_{s}<\frac{1}{\lambda}=T_{\rm J}$. For small $\tau_{s}$
as shown in Fig. 1, all those estimates are rather close to the preset values
and emphasize the accuracy of the predictions of $\Theta$ and $Q$-criteria for
four types of the synthetic data. The subplot (ii) in Fig. 1 exhibit a notable
departure from the expected theoretical prediction of the Kramers-Moyal
conditional moments’ expansion for the jump-diffusion process, specifically
observed in the case of $\tau_{\rm s}>T_{\rm J}$. To conclude this part, we
find that the behaviors reported in Eqs. (8) and (9) are valid only for
$\tau_{\rm s}<T_{\rm J}$ and $T_{\rm C}$. It is essential to consider these
constraints when applying the criteria to real-world time series data.

Figure 1:  $\Theta=\Theta(x=0,\tau=\tau_{\rm s})$ and $Q=Q(x=0,\tau=\tau_{\rm
s})$ are calculated for diffusion process, (a) and (d), jump-diffusion
process, (b) and (e), and standard normal Gaussian white noise, (c) and (f),
are evaluated for each $\tau_{\rm s}$. For the evaluation, OU processes are
generated for different $\tau_{\rm s}$ with $\gamma=1\,{\rm s^{-1}}$,
$D=1\,{\rm s^{-1}}$ and also with the additional jump terms with
$\lambda=100\,{\rm s^{-1}}$ and $\sigma_{\xi}^{2}=1$. The theoretical
predictions according to Eqs. (8) and (9) are presented with black solid
lines. The blue vertical line is the timescale $T_{\rm J}=\frac{1}{\lambda}$.
Standard error of the mean are shown with shadows, which is smaller than the
symbol size in these cases.

##  3 Downsampling of the Data

To analyze data that is originally sampled with a finite time step $\tau_{s}$
and data with unique trajectories, the only practical approach to study
diffusive and non-diffusive behaviors is downsampling. When downsampling data,
we encounter situations where the sampling interval $\tau_{\rm ds}$ (which
belongs to the set $\in{[\tau_{s},2\tau_{s},\ldots]}$) can be larger than both
the timescales $T_{\rm J}$ and $T_{\rm C}$, or it may lie between these two
timescales.

Initially, we focus on diffusive processes that are governed by Langevin
equations. In the Appendix B, we provide a proof that downsampling time series
generated from a Langevin equation does not alter the behavior of
$\Theta(x,t,\tau_{\rm ds})$. Consequently, diffusive processes retain their
diffusive nature even after downsampling. The key concept is that the Wiener
process for the downsampled time series remain Wiener process, with the only
difference being the change of $\tau_{s}$ to $\tau_{\rm ds}$.

However, downsampling behaves differently for processes that exhibit jump
discontinuities. Let us consider the case where time scales have the property
$T_{\rm J}<T_{\rm C}$. To this end, we note that on average, there are
$\lambda\tau_{\rm ds}$ jump events within each downsampled step of $\tau_{\rm
ds}$. Consequently, within each step, we observe multiple jumps with different
amplitudes. The average jump amplitude for the downsampled time interval
$\tau_{\rm ds}$ is given by:

$\sigma_{\xi,{\rm eff}}^{2}=\sum_{i=1}^{\lambda\tau_{\rm
ds}}\sigma_{\xi}^{2}=\sigma_{\xi}^{2}\,\lambda\tau_{\rm ds},$ (10)

where $\lambda$ represents the jump rate of the original time series (before
downsampling). From Eq. (10), we can observe that $\sigma_{\xi,{\rm eff}}^{2}$
now exhibits a linear dependence on $\tau_{\rm ds}$. As a result, it becomes
challenging to distinguish the jump discontinuity process from the diffusive
process using the $Q$-criterion presented in Eq. (9), which we have shown that
was applicable for small $\tau_{\rm s}$.

Similar to jump events, the Gaussian noise $\eta$ in the diffusion part of the
jump-diffusion process also accumulates over each time step within the
downsampled interval $\tau_{\rm ds}$. This results in an effective variance
given by $\sigma_{\eta,{\rm
eff}}^{2}=\sum_{i=1}^{N_{\tau}}\sigma_{\eta}^{2}=N_{\tau}=\frac{\tau_{\rm
ds}}{\tau_{\rm s}}$, where $\sigma_{\eta}^{2}=1$ represents the unit variance
of $\eta$, and $\tau_{\rm s}$ is the sampling time step of the original data.
As a result, the effective diffusion coefficient $D^{(2)}_{\rm eff}(x,t)$
satisfies the relation $D^{(2)}_{\rm eff}(x,t)\tau_{\rm ds}=D^{(2)}\tau_{\rm
s}N_{\tau}$, which implies that the effective diffusion coefficient remains
the same as the original diffusion coefficient ($D^{(2)}_{\rm
eff}(x,t)=D^{(2)}(x,t)$). A detailed proof can be found in the Appendix B.
Thus, the effective function $Q$ for the jump-diffusion in the range of
$T_{\rm J}<\tau_{ds}<T_{\rm C}$ can be written as

$Q_{\rm eff}=D^{(2)}~{}\tau_{\rm ds}+\lambda\sigma_{\xi}^{2}~{}\tau_{\rm
ds}~{}.$ (11)

Therefore, in Eq. (9), the condition for jumpy time series changes from a
constant $\sigma_{\xi}^{2}$ to a linear relation with $\tau_{\rm ds}$, given
by $(D^{(2)}+\lambda\sigma_{\xi}^{2})\tau_{\rm ds}$. If $\tau_{\rm ds}>T_{\rm
C}$, the downsampled data will resemble white noise. In this scenario, the
$Q$-criterion will have a value equal to the variance of the resulting time
series. We note also that in the case of $T_{\rm C}<T_{\rm J}$, Eqs. (8) and
(9) are valid for $\tau_{\rm ds}<T_{\rm C}$.

In order to check the aforementioned expressions, i.e. Eqs. (8), (9) and (11)
numerically, we generate one sample each of OU process and OU jump-diffusion
process with $\gamma=1~{}{\rm s^{-1}}$, $D=1~{}{\rm s^{-1}}$,
$\lambda=100~{}{\rm s^{-1}}$ and $\sigma_{\xi}^{2}=1$. For the case of $T_{\rm
J}>T_{\rm C}$, we generate additional sample with $\gamma=1000~{}{\rm
s^{-1}}$, $D=1000~{}{\rm s^{-1}}$, $\lambda=100~{}{\rm s^{-1}}$ and
$\sigma_{\xi}^{2}=10$. All the samples are simulated with the integrating time
step $\tau_{\rm s}=10^{-4}~{}{\rm s}$.

The processes are realized with different time-lags $\tau_{\rm ds}$ using
downsampling, and then the $\Theta$\- and $Q$-criterion are calculated. The
results are plotted in Fig. 2. The blue vertical line is the timescale $T_{\rm
J}=\frac{1}{\lambda}$ and the red vertical line $T_{\rm C}=\frac{1}{\gamma}$.

Figure 2:  $\Theta=\Theta(x=0,\tau=\tau_{\rm s})$ and $Q=Q(x=0,\tau=\tau_{\rm
s})$ are calculated for the downsampled OU (diffusion) process, (a) and (d),
downsampled OU jump-diffusion process for $T_{\rm J}<T_{\rm C}$, (b) and (e),
and $T_{\rm J}>T_{\rm C}$, (c) and (f), are evaluated for each downsampled
period $\tau_{\rm s}$. The theoretical predictions according to Eqs. (8), (9)
and (11) are presented with black solid lines. The blue vertical line is the
timescale $T_{\rm J}=\frac{1}{\lambda}$ and the red vertical line $T_{\rm
C}=\frac{1}{\gamma}$. Standard error of the mean are shown with shadows, which
is smaller than the symbol size in these cases. The green solid lines in (b)
and (e) the approximation up to $\mathcal{O}(\tau^{4})$ of conditional moments
while calculating $\Theta$ and $Q$ in Eqs. (8) and (9) as described in
Appendix A.1.

For diffusion process, we can observe the same diffusive behavior for
$\tau_{\rm ds}<T_{\rm C}=\frac{1}{\gamma}$ as described in Eq. (8) and (9)
shown by black solid lines in Fig. 2 (a) and (d). For downsampled time steps
larger than correlation length, $\tau_{\rm ds}>T_{\rm C}$, the results of the
criteria shows the behavior of uncorrelated Gaussian white noise in which
$\Theta=1$ and $Q=\sigma_{x}^{2}$ where $\sigma_{x}^{2}=0.5$ of OU process
$x$, shown by blue horizontal dashed in Fig. 2 (d) for this case.

In the first case of jump-diffusion process where $T_{\rm J}<T_{\rm C}$ shown
in Fig. 2 (b) and (e), we can see that the approximations in Eqs. (8) and (9)
are still valid for $\tau_{\rm ds}\ll T_{\rm J}$. (If we consider calculating
$\Theta$ and $Q$ in Eqs. (8) and (9) up to $\mathcal{O}(\tau^{4})$ of
conditional moments as described in Appendix A.1, we can have a better
approximation shown in solid green line.) In the range of $T_{\rm J}<\tau_{\rm
ds}<T_{\rm C}$, the effective value of $Q$ becomes the sum of effective
diffusion and jump terms described in Eq. (11), and $\Theta$ also becomes
unity after a few time steps in this range which apparently show the diffusive
behavior. These are shown with the blue dashed lines. After $\tau_{\rm
ds}>T_{\rm C}$, the downsampled process becomes uncorrelated with Gaussian
distribution, where $\Theta=1$ and $Q$ becomes the variance of the process
$x$, $Q=\sigma_{x}^{2}=1$. Theoretically, the variance of the OU process with
jump can be calculated with Eq. (76).

In the second case of jump-diffusion process where $T_{\rm J}>T_{\rm C}$ shown
in Fig. 2 (c) and (f), we can see that the approximations in Eqs. (8) and (9)
are still valid for $\tau_{\rm ds}\ll T_{\rm C}$ (shown by black solid lines).
For $\tau_{\rm ds}>T_{\rm J}$, $Q$, $\Theta$ and $\Lambda$ become constant. In
this range, the process becomes uncorrelated but not Gaussian distributed. It
can be seen from the probability density function (PDF) and autocorrelation
function (ACF) of process $x$ downsampled at $\tau_{\rm ds}>T_{\rm C}$ for
first case and $\tau_{\rm ds}>T_{\rm J}$ for second of the aforementioned jump
diffusion processes as shown in Fig. 3

Figure 3:  The PDF and ACF of downsampled OU jump-diffusion process $x$ at
$\tau_{\rm ds}>T_{\rm C}$ in the case of $T_{\rm J}<T_{\rm C}$, (a) and (c),
and at $\tau_{\rm ds}>T_{\rm J}$ in the case of $T_{\rm J}>T_{\rm C}$, (b) and
(d). The red solid line shows the PDF of Gaussian distribution with zero mean
and the same variance $\sigma_{x}^{2}$. The PDF is plotted in semi-logarithmic
scale.

##  4 Real-world examples

All the measured data $x(t)$ with the sampling time $\tau_{\rm s}$ are
subtracted with their means and divided with their standard deviations, so
that the normalized data have zero-means and standard deviations of one. The
$Q$-, $\Theta$\- and $\Lambda$-criterion are evaluated at $x=0$ by
downsampling of the given data with downsampling time step $\tau_{\rm ds}$. We
take five different data sets (including the dynamics of trapped particles in
optical tweezers, solar irradiance, turbulence, log return of DAX data and
intracranial electroencephalographic (iEEG) time series of the brain) and
separated into three categories.

For each time series sampled with timescale $\tau_{s}$, we apply the methods
described in the Appendix A to estimate the drift, diffusion, jump rate, and
jump amplitude. This involves analyzing the KM coefficients of order $1$, $4$,
and $6$ to derive the values of $T_{\rm C}$ and $T_{\rm J}$ for the respective
data sets.

In the first category, we analyze the data of measurements of the spatial
positions of a dielectric bead (polystyrene, diameter-1${\rm\mu m}$, Bangs
Laboratories Inc. USA) trapped in and optical tweezers [14]. The results of
$\Theta=\Theta(x=0,\tau_{\rm ds})$ and $Q=Q(x=0,\tau_{\rm ds})$ versus
$\tau_{\rm ds}$, for a single measured data with duration $3$ s, and with
sample rate $22$ kHz are plotted in Fig. 4 (a) and (b).

Figure 4:  $\Theta$\- and $Q$-criterion of downsampled displacement data of
dielectric bead in the optical tweezers, (a) and (b), log return of DAX data,
(c) and (d), iEEG brain data, (e) and (f), turbulence data, (g) and (h), and
solar clear sky index data, (i) and (j), are plotted in double logarithmic
scale. The vertical blue lines show the jump time scale $T_{\rm
J}=\frac{1}{\lambda}$ and the vertical red lines the correlation time scale
$T_{\rm C}=\frac{1}{\gamma}$. Standard errors of the mean are shown as gray-
shaded background and some of them are smaller than the symbol size.

The results from optical tweezers show the behavior of diffusion process. The
correlation time $T_{\rm C}=\frac{1}{\gamma}$ is first estimated from the
drift function, which is approximately $D^{(1)}(x)=-\gamma x$. Below the
correlation timescale $\tau_{\rm ds}<T_{\rm C}$, $Q$ has the linear behavior
and $\Theta=1$. For $\tau_{\rm ds}>T_{\rm C}$, $Q=\sigma_{x}^{2}=1$ and
$\Theta=1$ also, which show the behavior of uncorrelated Gaussian white noise.

In the second category, we analyze DAX stock market index data [15] and
epileptic brain data [7]. For the DAX stock market index, the log return is
used for the analysis. The results of $\Theta=\Theta(x=0,\tau_{\rm ds})$ and
$Q=Q(x=0,\tau_{\rm ds})$ verses $\tau_{\rm ds}$ are plotted in Fig. 4. (c) and
(d). The results of $\Theta=\Theta(x=0,\tau_{\rm ds})$ and $Q=Q(x=0,\tau_{\rm
ds})$ verses $\tau_{\rm ds}$ for epileptic brain data are plotted in Fig. 4
(e) and (f).

Since $\Theta$ are deviated from one, they do not fulfil the necessary
conditions for the diffusion process. Here we evaluated again the correlation
time $T_{\rm C}=\frac{1}{\gamma}$ obtained from the drift function, which is
approximately $D^{(1)}(x)=-\gamma x$. We also estimated $T_{\rm
J}=\frac{1}{\lambda}$ by Eq. (6). For small $\tau_{\rm ds}$, $Q$ shows the
constant behavior while $\Theta$ is not unity and also show the non-constant
behavior on $\tau$. Thus, one could consider the jump-diffusion process to
model these dynamics [7].

In the third category, we look into the data of free jet low temperature
helium turbulence [16] with Reynolds number $Re=757000$ and solar clear sky
index [17, 18]. The results of $\Theta=\Theta(x=0,\tau_{\rm ds})$ and
$Q=Q(x=0,\tau_{\rm ds})$ verses $\tau_{\rm ds}$ for the turbulence data are
plotted in Fig. 4 (g) and (h). The results of $\Theta=\Theta(x=0,\tau_{\rm
ds})$ and $Q=Q(x=0,\tau_{\rm ds})$ verses $\tau_{\rm ds}$ for the solar clear
sky index are plotted in Fig. 4 (i) and (j). As $\Theta$ is deviated from
unity, they do not fulfil the necessary conditions for the diffusion process.
Similarly to previous data analysis, we evaluated again the correlation time
$T_{\rm C}=\frac{1}{\gamma}$ where the drift function is defined by
$D^{(1)}(x)=-\gamma x$. We also estimated $T_{\rm J}=\frac{1}{\lambda}$ by Eq.
(6). In these two cases, $T_{\rm J}<T_{\rm C}$. For small $\tau_{\rm ds}$,
both $Q$ and $\Theta$ show the nonlinear behavior on $\tau$. Thus, one could
only conclude that they are not diffusive from the current study. Turbulence
is known to have much more complex nature, and it is also not Markovian in
time but is in scale [19, 2]. For the third category, we can only conclude
that these time series do not belong to the class of diffusion processes.

##  5 Conclusion

In many models describing the dynamics of natural measurements, it is common
to assume a standard Gaussian white noise-driven Langevin equation to account
for observed variability. However, it is not immediately clear whether this
assumption holds true. Evaluating the validity of a Langevin equation to
describe the data presents a significant challenge, especially when dealing
with finite data points or processes that exhibit unique trajectories [16, 20,
18].

In this study, we have developed a set of criteria to assess the suitability
of modeling the data using a Langevin equation. By recognizing the limitations
of Langevin dynamics, alternative modeling approaches become relevant. These
may include jump-diffusion stochastic dynamics [7], generalized Langevin
equations [21, 22], fractional Klein-Kramers equations [23], fractional
diffusion processes [24, 25] and Lévy-driven Langevin dynamics [5].

Revealing novel aspects of the data that cannot be captured by white-noise
driven Langevin equations can illuminate the presence of abrupt changes in
time series. These changes often arise from complex physical phenomena that
extend beyond the scope of Langevin equations. Consequently, this exploration
uncovers new phenomena, enhances our understanding of measured variability in
diverse datasets, and opens up avenues for further research.

##  Appendix A Expansion of Conditional Moments at Finite Time Step

The conditional probability distribution of the process $x=x(t)$ which
satisfies the Kramers-Moyal differential equation can be written as [13, 5],

$\frac{\partial p(x,t|x^{\prime},t^{\prime})}{\partial t}=\mathcal{L}_{\rm
KM}~{}p(x,t|x^{\prime},t^{\prime})$ (12)

with initial condition $p(x,t|x^{\prime},t)=\delta(x-x^{\prime})$ and the
Kramers-Moyal (KM) operator $\mathcal{L}_{\rm KM}$ is given by,

$\mathcal{L}_{\rm
KM}=\sum_{n=1}^{\infty}\frac{1}{n!}\left(-\frac{\partial}{\partial
x}\right)^{n}~{}K^{(n)}(x,t)\quad.$ (13)

The formal solution of (12) reads

$p(x,t+\tau|x^{\prime},t)=\exp\\{\tau~{}\mathcal{L}_{\rm
KM}\\}~{}\delta(x-x^{\prime})\quad.$ (14)

The $n^{th}$-order conditional moments $M^{(n)}(x,t,\tau)$ with finite $\tau$
can be written as,

$\displaystyle M^{(n)}(x_{i},t,\tau)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}(x-x_{i})^{n}\exp\\{\tau~{}\mathcal{L}_{\rm
KM}\\}~{}\delta(x-x_{i})dx$ (15) $\displaystyle=$
$\displaystyle\exp\\{\tau~{}\mathcal{L}_{\rm
KM}^{\dagger}\\}(x-x_{i})^{n}|_{x=x_{i}}$ (16)

where $\mathcal{L}_{\rm KM}^{\dagger}$ is the adjoint operator of
$\mathcal{L}_{\rm KM}$ and is given by,

$\mathcal{L}_{\rm
KM}^{\dagger}=\sum_{n=1}^{\infty}\frac{1}{n!}~{}K^{(n)}(x,t)\left(\frac{\partial}{\partial
x}\right)^{n}.$ (17)

where $K^{(n)}(x,t)=\lim_{\tau\to 0}\frac{1}{\tau}M^{(n)}(x,t,\tau)$ are the
$n^{th}$-order Kramers-Moyal coefficients and $M^{(n)}(x,t,\tau)$ are given by
Eq. (16). Next, the explicit cases of the diffusion and jump-diffusion
processes are discussed.

###  A.1 Conditional Moments of Diffusion and Jump-Diffusion Processes

Diffusion process, in general, can be described by Langevin equation in Eq.
(3). We can evaluate its conditional moments $M_{\rm d}^{(n)}(x,\tau)$ using
Eq. (16). For diffusion process or Langevin equation, the adjoint KM operator
$\mathcal{L}_{\rm KM}^{\dagger}$ becomes the adjoint Fokker-Planck (FP)
operator $\mathcal{L}_{\rm FP}^{\dagger}$. For better readability, let the
drift be $D^{(1)}(x,t)=K^{(1)}(x,t)=a$ and the diffusion
$D^{(2)}(x,t)=K^{(2)}(x,t)=b^{2}$ and the adjoint FP-operator becomes

$\mathcal{L}_{\rm FP}^{\dagger}=a~{}\frac{\partial}{\partial
x}+\frac{1}{2}~{}b^{2}~{}\frac{\partial^{2}}{\partial x^{2}}~{}.$ (18)

In this formulation, $a$ and $b$ can still be the function of $x$ and $t$. The
second-, fourth- and sixth-order conditional moments up to first non-vanishing
term can be derived from Eqs. (16) and (18) as follow [13]:

$\displaystyle M_{\rm d}^{(2)}(x,\tau)$ $\displaystyle=$ $\displaystyle
b^{2}\tau+\mathcal{O}(\tau^{2}),$ (19) $\displaystyle M_{\rm d}^{(4)}(x,\tau)$
$\displaystyle=$ $\displaystyle 3b^{4}\tau^{2}+\mathcal{O}(\tau^{3}),$ (20)
$\displaystyle M_{\rm d}^{(6)}(x,\tau)$ $\displaystyle=$ $\displaystyle
15b^{6}\tau^{3}+\mathcal{O}(\tau^{4}).$ (21)

In order to distinguish from the jump-diffusion process which will be
discussed next, the subscript “d” is used here for the diffusion process.

Ornstein-Uhlenbeck (OU) process is considered here to be more concrete such
that

$\mathrm{d}x=-\gamma x~{}\mathrm{d}t+\sqrt{D}~{}\mathrm{d}W_{t}~{}.$ (22)

where $\gamma$ and $D$ are positive real constants. Expansion of $M_{\rm
d}^{(n)}(x,t,\tau)$ of OU process for small $\tau$ up to
$\mathcal{O}(\tau^{3})$ (up to $\mathcal{O}(\tau^{4})$ for sixth-order
conditional moment) reads

$\displaystyle M_{\rm d}^{(2)}(x,\tau)$ $\displaystyle=$ $\displaystyle
D\tau+\left(\gamma^{2}x^{2}-\gamma D\right)\tau^{2}+\mathcal{O}(\tau^{3}),$
(23) $\displaystyle M_{\rm d}^{(4)}(x,\tau)$ $\displaystyle=$ $\displaystyle
3D^{2}\tau^{2}+\mathcal{O}(\tau^{3}),$ (24) $\displaystyle M_{\rm
d}^{(6)}(x,\tau)$ $\displaystyle=$ $\displaystyle
15D^{3}\tau^{3}+\mathcal{O}(\tau^{4}).$ (25)

Jump-diffusion process, in general, is described in Eq. (5). The KM-
coefficients of such jump-diffusion are

$\displaystyle K^{(1)}(x,t)$ $\displaystyle=$ $\displaystyle D^{(1)}(x,t)$
(26) $\displaystyle K^{(2)}(x,t)$ $\displaystyle=$
$\displaystyle\left[D^{(2)}(x,t)+\langle\xi^{2}\rangle\lambda(x)\right]$ (27)
$\displaystyle K^{(2n)}(x,t)$ $\displaystyle=$
$\displaystyle\langle\xi^{2n}\rangle\lambda(x),~{}~{}\text{for}~{}2n>2$ (28)

Therefore, the adjoint KM-operator $\mathcal{L}_{\rm KM}^{\dagger}$ for jump-
diffusion process becomes

$\displaystyle\mathcal{L}_{\rm KM}^{\dagger}$ $\displaystyle=$
$\displaystyle\underbrace{D^{(1)}(x,t)}_{\textbf{A}}\frac{\partial}{\partial
x}+\underbrace{\frac{[D^{(2)}(x,t)+\langle\xi^{2}\rangle\lambda(x)]}{2!}}_{\textbf{B}}\frac{\partial^{2}}{\partial
x^{2}}$ $\displaystyle+$
$\displaystyle\underbrace{\frac{\langle\xi^{4}\rangle\lambda(x)}{4!}}_{\textbf{C}}\frac{\partial^{4}}{\partial
x^{4}}+\underbrace{\frac{\langle\xi^{6}\rangle\lambda(x)}{6!}}_{\textbf{D}}\frac{\partial^{6}}{\partial
x^{6}}$ $\displaystyle+$
$\displaystyle\underbrace{\frac{\langle\xi^{8}\rangle\lambda(x)}{8!}}_{\textbf{E}}\frac{\partial^{8}}{\partial
x^{8}}+\cdots\quad.$ (30)

Again for better readability, we use the substitution of the coefficients with
$\textbf{A},\textbf{B},\textbf{C},\textbf{D},\textbf{E},\cdots$, which can
still be the function of $x$ and $t$ in this formulation. With it, we can
derive the conditional moments of the jump-diffusion equation Eq. (5) for
second-, fourth and sixth- orders of the time interval $\tau$ using Eq. (16)
and (30) such that

$\displaystyle M_{\rm j}^{(2)}(x,\tau)$ $\displaystyle=$ $\displaystyle
2\textbf{B}\tau+\mathcal{O}(\tau^{2}),$ (31) $\displaystyle M_{\rm
j}^{(4)}(x,\tau)$ $\displaystyle=$ $\displaystyle
4!\textbf{C}\tau+\mathcal{O}(\tau^{2}),$ (32) $\displaystyle M_{\rm
j}^{(6)}(x,\tau)$ $\displaystyle=$ $\displaystyle
6!\textbf{D}\tau+\mathcal{O}(\tau^{2}),$ (33)

where the subscript “j” denotes the jump-diffusion process [13]. Here, they
are derived up to first non-vanishing term.

OU process with an additional jump term with constant jump rate and jump
amplitude is now considered. It is a linear jump-diffusion process, and one
finds,

$\mathrm{d}x=-\gamma
x~{}\mathrm{d}t+\sqrt{D}~{}\mathrm{d}W_{t}+\xi~{}\mathrm{d}J_{t}~{}.$ (34)

Adding the jump term in OU process creates the discontinuities in the
trajectory. For finite time interval $\tau$, the conditional moments $M_{\rm
j}^{(n)}(x,\tau)$ of an OU jump-diffusion process can be determined up to
$\mathcal{O}(\tau^{4})$ using Eqs. (16) and (30) as follows:

$\displaystyle M_{\rm j}^{(2)}(x,\tau)$ $\displaystyle=$
$\displaystyle(D+\lambda\sigma_{\xi}^{2})\tau+\left(\gamma^{2}x^{2}-\gamma(D+\lambda\sigma_{\xi}^{2})\right)\tau^{2}$
(35) $\displaystyle+$
$\displaystyle\left(\frac{1}{3}\gamma^{2}(D+\lambda\sigma_{\xi}^{2})-\frac{1}{2}\gamma^{3}x^{2}\right)\tau^{3}+\mathcal{O}(\tau^{4}),$
(36) $\displaystyle M_{\rm j}^{(4)}(x,\tau)$ $\displaystyle=$ $\displaystyle
3\lambda\sigma_{\xi}^{4}\tau+3\left((D+\lambda\sigma_{\xi}^{2})^{2}-2\gamma\lambda\sigma_{\xi}^{4}\right)\tau^{2}$
(37) $\displaystyle+$
$\displaystyle(6x^{2}\gamma^{2}(D+\lambda\sigma_{\xi}^{2})+8\gamma^{2}\lambda\sigma_{\xi}^{4}$
(38) $\displaystyle-$ $\displaystyle
6\gamma(D+\lambda\sigma_{\xi}^{2})^{2})\tau^{3}+\mathcal{O}(\tau^{4}),$ (39)
$\displaystyle M_{\rm j}^{(6)}(x,\tau)$ $\displaystyle=$ $\displaystyle
15\lambda\sigma_{\xi}^{6}\tau+45\left(\lambda\sigma_{\xi}^{4}(D+\lambda\sigma_{\xi}^{2})-\gamma\lambda\sigma_{\xi}^{6}\right)\tau^{2}$
(40) $\displaystyle+$
$\displaystyle(45\gamma^{2}\lambda\sigma_{\xi}^{4}x^{2}+15(D+\lambda\sigma_{\xi}^{2})^{3}$
(41) $\displaystyle+$ $\displaystyle
540\gamma^{2}\lambda\sigma_{\xi}^{6}-810\gamma\lambda\sigma_{\xi}^{4}(D+\lambda\sigma_{\xi}^{2}))\tau^{3}+\mathcal{O}(\tau^{4}),$
(42)

###  A.2 Conditional Moments of Gaussian White Noise

For a Gaussian distributed white noise with zero mean and variance
$\sigma_{\eta}^{2}$, $x=\eta\sim N(0,\sigma_{\eta}^{2})$, the $n^{\rm th}$
order conditional moments can be calculated as

$\displaystyle M_{\rm g}^{(n)}(x,\tau)$ $\displaystyle=$
$\displaystyle\left\langle\left(x(t+\tau)-x(t)\right)^{n}|_{x(t)=x}\right\rangle,$
(44) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left(x^{\prime}-x\right)^{n}\cdot
p(x^{\prime},t^{\prime}|x,t)\cdot{\rm d}x^{\prime},$ (45) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left(x^{\prime}-x\right)^{n}\cdot
p(x^{\prime},t^{\prime})\cdot{\rm d}x^{\prime}.$ (46)

For $x=0$ and
$p(x^{\prime},t^{\prime})=p(x^{\prime})=\frac{1}{\sqrt{2\pi\sigma_{\eta}^{2}}}\exp\left(-\frac{{x^{\prime}}^{2}}{2\sigma_{\eta}^{2}}\right)$,
we can use Wick’s theorem [26, 27] to solve integral in Eq. (46). The
$(2n+1)^{\textrm{th}}$ and the $2n^{\textrm{th}}$ conditional moments of
Gaussian white noise become

$\displaystyle M_{\rm g}^{(2n+1)}(x=0,\tau)$ $\displaystyle=$ $\displaystyle
0,$ (47) $\displaystyle M_{\rm g}^{(2n)}(x=0,\tau)$ $\displaystyle=$
$\displaystyle\frac{(2n)!}{2^{n}n!}\sigma_{\eta}^{2n}~{},$ (48)

where $\sigma_{\eta}^{2}$ is its variance.

###  A.3 $Q$-criterion of Jump-Diffusion Process at Time Scale $\tau>T_{\rm
J}$

As a first criterion, the $Q$-criterion is introduced in [13] to distinguish
whether given synthetic data simulated with different integration time step
$\tau_{\rm s}$ are diffusive or jumpy. Here, this criterion is also applied to
the uncorrelated Gaussian white noise.

For a small $\tau$, we obtain the function $Q(x,\tau)$ as described in Eq. (9)
considering up to the first non-vanishing order term [13] which is derived
from the fourth- and sixth-order conditional moments.

For diffusive process, case (i), the function $Q$ has a linear relationship
with $\tau$ while it is constant in first order approximation in $\tau$ for
jump-diffusion process, case (ii). Additionally, the function $Q$ is also
constant for the Gaussian white noise, case (iii) which can be derived by
direct computations of conditional moments.

From the simulated data, we can determine $Q=Q(x=0,\tau=\tau_{\rm s})$ for
different $\tau_{\rm s}$. For jump-diffusion process theoretical prediction
Eq. (9) is valid for $\tau_{\rm s}<\frac{1}{\lambda}=T_{\rm J}$, which is the
average time scale between the jumps. If the integration time step $\tau_{\rm
s}>T_{\rm J}$, the jump-diffusion process behaves like diffusion process with
the apparent diffusion coefficient $\widetilde{D^{(2)}}$ which has the
relation, $\widetilde{D^{(2)}}\tau_{\rm s}=D^{(2)}\tau_{\rm
s}+\sigma_{\xi}^{2}$.

To prove this relation, the jump-diffusion equation can be written using
Euler’s method as

$x(t+\tau_{\rm s})-x(t)=D^{(1)}(x)~{}\tau_{\rm s}+\sqrt{D^{(2)}(x)}~{}\Delta
W_{t}+\xi~{}\Delta J_{t}.$ (49)

For $\tau_{\rm s}>\frac{1}{\lambda}$, $\Delta J_{t}=1$ due to zero-one jump
law, the Eq. (49) becomes

$x(t+\tau_{\rm s})-x(t)=D^{(1)}(x)~{}\tau_{\rm
s}+\sqrt{D^{(2)}(x)~{}\tau_{s}}~{}\eta+\xi.$ (50)

$\sqrt{D^{(2)}(x)~{}\tau_{s}}~{}\eta+\xi$ is the sum of two Gaussian random
variables which is equivalent to $\sqrt{\widetilde{D^{(2)}}(x)~{}\tau_{\rm
s}}~{}\widetilde{\eta}=\sqrt{D^{(2)}(x)~{}\tau_{s}+\sigma_{\xi}^{2}}~{}\widetilde{\eta}$.
$\widetilde{\eta}\sim N(0,1)$ is the standard normal Gaussian white noise.

Therefore, the jump-diffusion equation for $\lambda>\frac{1}{\tau_{\rm s}}$
becomes

$x(t+\tau_{\rm s})-x(t)=D^{(1)}(x)~{}\tau_{\rm
s}+\sqrt{\widetilde{D^{(2)}}(x)~{}\tau_{\rm s}}~{}\widetilde{\eta},$ (51)

respectively,

${\rm d}x=D^{(1)}(x)~{}{\rm d}t+\sqrt{\widetilde{D^{(2)}}(x)}~{}{\rm
d}\widetilde{W}_{t}~{},$ (52)

where $\widetilde{D^{(2)}}$ is the apparent diffusion coefficient and
$\widetilde{W}_{t}$ is the Wiener process.

###  A.4 $\Theta$\- and $Q$-criterion for General Diffusion and Jump-Diffusion
Processes

Now, the $\Theta$\- and $Q$\- criterion are determined for both diffusion and
jump-diffusion processes with non-linear drift and multiplicative diffusion
term. We numerically integrate Eq. (3) with $D^{(1)}(x)=-x^{3}$ and
$D^{(2)}(x)=1+x^{2}$, and Eq. (5) for the case with additional jump term with
$\lambda=100~{}{\rm s^{-1}}$ and $\sigma_{\xi}^{2}=1$ for different
integration time step $\tau_{\rm s}$. The result of the non-linear diffusion
process is shown in Fig. 5 and jump-diffusion process in Fig. 6.

Figure 5:  $\Theta$\- and $Q$-criterion of diffusion process at $x=0$ with
$D^{(1)}(x)=-x^{3}$ and $D^{(2)}(x)=1+x^{2}$, generated for different
integration time steps $\tau_{\rm s}$, are plotted in double logarithmic
scale. Theoretical predictions according to Eqs. (8) and (9) are presented
with the black solid lines. Standard errors are smaller than the symbol size.
Figure 6:  $\Theta$\- and $Q$-criterion of jump-diffusion process at $x=0$
with $D^{(1)}(x)=-x^{3}$, $D^{(2)}(x)=1+x^{2}$, $\lambda=100~{}{\rm s^{-1}}$
and $\sigma_{\xi}^{2}=1$, generated for different integration time steps
$\tau_{\rm s}$, are plotted in double logarithmic scale. The vertical blue
lines show the jump time scale $T_{\rm J}=\frac{1}{\lambda}$. Theoretical
predictions according to Eqs. (8) and (9) are presented with the black solid
lines. The blue dashed curve $Q=D^{(2)}(0)\tau_{\rm s}+\sigma_{\xi}^{2}$ and
line $\Theta=1$ show the behavior after $\tau_{\rm s}>T_{\rm
J}=\frac{1}{\lambda}$. Standard errors are smaller than the symbol size.

We can observe the similar behavior in the case with linear drift and additive
diffusion noise.

##  Appendix B Consequences of Downsampling of the Data

In real world, empirical data are given for a fixed sampling time $\tau_{\rm
s}$ due to experimental constraints. To handle this effect, we have to
downsample the numerical data in which a new time step $\tau_{\rm ds}$ is
defined for downsampling time step. For the estimation of the KM conditional
moments, it is essential to see in which relation $\tau_{\rm ds}$ is with the
correlation time $T_{\rm C}=\frac{1}{\gamma}$ and with the average time scale
between the jumps $T_{\rm J}=\frac{1}{\lambda}$.

Influence of downsampling on the diffusion coefficient is first examined. In
order to study the downsampled approximation of the diffusion term, it is
integrated over a finite downsampled time $\tau_{\rm ds}$. In the Itô sense,
the integral is generally interpreted as [6, 2]

$\displaystyle\int_{t}^{t+\tau_{\rm ds}}g(x(s),s)~{}{\rm d}W_{s}$
$\displaystyle=$ $\displaystyle g(x(t),t)~{}\int_{t}^{t+\tau_{\rm ds}}{\rm
d}W_{s}$ (53) $\displaystyle=$ $\displaystyle
g(x(t),t)\sum_{k=1}^{N_{\tau}}(W_{k}-W_{k-1})$ (54)

Therefore, the integral of diffusion part in (jump-)diffusion process becomes

$\displaystyle\int_{t}^{t+\tau_{\rm ds}}\sqrt{D^{(2)}(x)}~{}{\rm d}W_{s}$
$\displaystyle=$ $\displaystyle\sqrt{D^{(2)}(x)}\sum_{k=1}^{N_{\tau}}\Delta
W_{k}$ (55) $\displaystyle=$ $\displaystyle\sqrt{D^{(2)}(x)~{}\tau_{\rm
s}}\sum_{k=1}^{N_{\tau}}\eta_{k}~{}.$ (56)

where $\eta_{k}\sim N(0,\sigma_{\eta}^{2}=1)$ is the standard normal
distributed random variable. To estimate the summation
$\sum_{k=1}^{N_{\tau}}\eta_{k}$, $N_{\tau}=\frac{\tau_{ds}}{\tau_{s}}$ is the
number of time steps in the downsampled time step $\tau_{\rm ds}$. Let us
define the summation as a new effective noise $\eta_{\rm eff}$ for diffusion
term as,

$\eta_{\rm eff}=\sum_{k=1}^{N_{\tau}}\eta_{k}$ (57)

where variance of $\eta_{\rm eff}$ will be $\sigma_{\eta,{\rm
eff}}^{2}=N_{\tau}\sigma_{\eta}^{2}=N_{\tau}=\frac{\tau_{\rm ds}}{\tau_{\rm
s}}$, (as $\sigma_{\eta}^{2}=1$), and $\eta_{\rm eff}$ tends to a Gaussian
random variable. Then, the integral becomes

$\int_{t}^{t+\tau_{\rm ds}}\sqrt{D^{(2)}(x)}~{}{\rm
d}W_{s}\simeq\sqrt{D^{(2)}(x)~{}\tau_{\rm s}}~{}\eta_{\rm
eff}=\sqrt{D^{(2)}(x)~{}\tau_{\rm ds}}~{}\eta$ (58)

where $\eta\sim N(0,1)$ is the standard normal distributed. Therefore, we can
conclude that diffusion coefficient remains unchanged with downsampling and
summation of the noises approaches to a Gaussian white noise.

Influence of downsampling on the jump amplitude is then examined. The jump
term $\xi~{}{\rm d}J_{t}$ will provide the following stochastic integral

$\int_{t}^{t+\tau_{\rm ds}}\xi~{}{\rm d}J_{s}~{}.$ (59)

To deal with this integral, we use the integral relation [6, 5]

$\int_{t}^{t+\tau_{\rm ds}}h(x(s),s)~{}{\rm
d}J_{s}\stackrel{{\scriptstyle\text{ms}}}{{=}}\sum_{k=1}^{J_{t}}h(x(T^{-}_{k}),T^{-}_{k})~{},$
(60)

where $\stackrel{{\scriptstyle\text{ms}}}{{=}}$ is Itô mean square equals. The
$T_{k}^{-}$ denotes the limit from the left to jump time $T_{k}$. Therefore,
the jump integral becomes

$\int_{t}^{t+\tau_{\rm ds}}\xi~{}{\rm d}J_{s}=\sum_{k=1}^{J_{t}}\xi_{k}~{},$
(61)

where $\xi_{k}\sim N(0,\sigma_{\xi}^{2})$ is the Gaussian distributed random
variable with zero mean and variance $\sigma_{\xi}^{2}$. To estimate the
summation $\sum_{k=1}^{J_{t}}\xi_{k}$, we note that the average time step of
jumps in original time series is $\frac{1}{\lambda}$, then the average number
of jumps becomes $J_{t}\simeq\lambda\tau_{\rm ds}$ in the downsampled time
step $\tau_{\rm ds}$. Let us define the summation as a new noise $\xi_{\rm
eff}$ for jump amplitude as,

$\xi_{\rm eff}=\sum_{k=1}^{J_{t}}\xi_{k}~{},$ (62)

where variance of $\xi_{\rm eff}$ will be $\sigma_{\xi,{\rm
eff}}^{2}\simeq\lambda\tau_{\rm ds}\sigma_{\xi}^{2}$ and $\xi_{\rm eff}$ tends
to a Gaussian random variable. Then, we find,

$\int_{t}^{t+\tau_{\rm ds}}\xi~{}{\rm d}J_{s}=\xi_{\rm
eff}\simeq\sqrt{\lambda\tau_{\rm ds}}~{}\xi~{},$ (63)

where $\xi\sim N(0,\sigma_{\xi}^{2})$ is Gaussian distributed with variance of
jump amplitude $\sigma_{\xi}^{2})$. Therefore, if we downsample with
$\tau_{\rm ds}\gg\frac{1}{\lambda}$, $J_{t}\gg 1$, then zero-one jump law is
no longer fulfilled. However, it can be realized as a jump event with the
effective jump amplitude

$\sigma_{\xi,{\rm eff}}^{2}=\lambda\sigma_{\xi}^{2}\tau_{\rm ds}\;.$ (64)

In addition, the $\Theta$\- and $Q$-criterion are determined for both
downsampled diffusion and jump-diffusion processes with non-linear drift and
multiplicative diffusion terms. We numerically integrate Eq. (3) with
$D^{(1)}(x)=-x^{3}$ and $D^{(2)}(x)=1+x^{2}$, and Eq. (5) for the case with
additional jump term with $\lambda=100~{}{\rm s^{-1}}$ and
$\sigma_{\xi}^{2}=1$ with a fixed sampling time $\tau_{\rm
s}=10^{-4}~{}\mathrm{s}$. The results are shown in Fig. 7 for diffusion
process, and in Fig. 8 for jump-diffusion process.

Figure 7:  $\Theta$\- and $Q$-criterion of downsampled diffusion process at
$x=0$ with $D^{(1)}(x)=-x^{3}$ and $D^{(2)}(x)=1+x^{2}$ are plotted in double
logarithmic scale. Theoretical predictions according to Eqs. (8) and (9) are
presented with the black solid lines. The blue dashed line in (a) shows
$Q=\sigma_{x}^{2}$. Standard errors are smaller than the symbol size. Figure
8:  $\Theta$\- and $Q$-criterion of downsampled OU jump-diffusion process at
$x=0$ with $D^{(1)}(x)=-x^{3}$, $D^{(2)}(x)=1+x^{2}$, $\lambda=100~{}{\rm
s^{-1}}$ and $\sigma_{\xi}^{2}=1$ are plotted in double logarithmic scale. The
vertical blue lines show the jump time scale $T_{\rm J}=\frac{1}{\lambda}$.
Theoretical predictions according to Eqs. 8) and (9) are presented with the
black solid lines. The blue dashed lines show
$Q=(D^{(2)}(0)+\lambda\sigma_{\xi}^{2})\tau_{\rm ds}$ for $\tau_{\rm
ds}>T_{\rm J}$ and $Q=\sigma_{x}^{2}$ (horizontal line), and
$\Theta=\Lambda=1$. Standard errors are smaller than the symbol size.

We can observe the similar behavior in the case with linear drift and additive
diffusion noise.

So far for the jump diffusion processes, different cases of between the time
scales $T_{\rm J}$ and $T_{\rm C}$ are analyzed. It remains to study how the
magnitude of jump noise with respect to the diffusion noise affect the
stochastic process.

##  Appendix C Ornstein-Uhlenbeck Process in the Presence of Jumps

In the presence of compound Poisson jump process $Z_{t}=\xi J_{t}$ with
constant jump amplitude $\sigma_{\xi}^{2}$ and jump rate $\lambda$, Ornstein-
Uhlenbeck (OU) Process $x=x(t)$ can be written as Eq. (34).

By introducing the substituting variable $y=x\mathrm{e}^{\gamma t}$, applying
the Itô product rule,

$\mathrm{d}(f\cdot
g)=f\cdot\mathrm{d}g+g\cdot\mathrm{d}f+\mathrm{d}f\cdot\mathrm{d}g,$ (65)

and using the following properties,

$(\textrm{d}t)^{i}~{}(\textrm{d}W_{t})^{j}=0,\textrm{for}~{}i,j\geq 1,$ (66)

and

$(\textrm{d}t)^{i}~{}(\textrm{d}J_{t})^{j}=0,\textrm{for}~{}i,j\geq 1,$ (67)

we can solve Eq. (34). The solution in $\textrm{d}t$ precision is

$x=x_{0}\mathrm{e}^{-\gamma
t}+\sqrt{D}\int_{0}^{t}\mathrm{e}^{-\gamma(t-t^{\prime})}\mathrm{d}W_{t^{\prime}}+\int_{0}^{t}\mathrm{e}^{-\gamma(t-t^{\prime})}\mathrm{d}Z_{t^{\prime}},$
(68)

where $x_{0}=x(0)$.

The mean of the OU process can be calculated as

$\langle x\rangle=x_{0}\mathrm{e}^{-\gamma t}.$ (69)

and $\langle x\rangle=0$ for sufficiently long simulation time $t\to\infty$.

The covariance of OU process with jump can be calculated using the following
properties,

$(\textrm{d}W_{t})^{i}~{}(\textrm{d}J_{t})^{j}=0,\textrm{for}~{}i,j\geq 1,$
(70)

$\textrm{Cov}(\textrm{d}W_{s},\textrm{d}W_{t})=\langle\textrm{d}W_{s}~{}\textrm{d}W_{t}\rangle=\delta(s-t)~{}\textrm{d}s~{}\textrm{d}t,$
(71)

and

$\textrm{Cov}(\textrm{d}Z_{s},\textrm{d}Z_{t})=\langle\textrm{d}Z_{s}~{}\textrm{d}Z_{t}\rangle=\lambda\sigma_{\xi}^{2}~{}\delta(s-t)~{}\textrm{d}s~{}\textrm{d}t,$
(72)

such that

$C_{xx}(t,\tau)=\mathrm{Cov}(x(t)x(t+\tau))=\frac{D+\lambda\sigma_{\xi}^{2}}{2\gamma}\left[\mathrm{e}^{-\gamma\tau}-\mathrm{e}^{-\gamma(\tau+2t)}\right].$
(73)

We can calculate the variance of the OU process with jump from
$C_{xx}(t,\tau=0)$ which is

$\mathrm{Var}(x)=\langle x^{2}-\langle
x\rangle^{2}\rangle=\frac{D+\lambda\sigma_{\xi}^{2}}{2\gamma}~{}\mathrm{e}^{-2t}~{}.$
(74)

For sufficiently long simulation time $t\to\infty$, the covariance becomes

$C_{xx}(\tau)=\frac{D+\lambda\sigma_{\xi}^{2}}{2\gamma}~{}\mathrm{e}^{-\gamma\tau}~{},$
(75)

and the variance

$\mathrm{Var}(x)=\frac{D+\lambda\sigma_{\xi}^{2}}{2\gamma}~{}.$ (76)

The detailed discussions of the properties of stochastic processes that are
used in this derivation can be seen in Refs. [28, 5, 6, 4]. In order to verify
it, a synthetic time series of the OU jump-diffusion process is generated for
$\Delta t=10^{-3}~{}{\rm s}$ with $\gamma=100~{}{\rm s^{-1}}$, $D=10~{}{\rm
s^{-1}}$ and for the additional jump terms with $\lambda=100~{}{\rm s^{-1}}$
and $\sigma_{\xi}^{2}=1$. In Fig. 9 the timeseries of the OU jump-diffusion
processes is shown. Its covariance function $C_{xx}(\tau)$ is also evaluated
and plotted together with the theoretical result from Eq. (75). Here, we can
observed that the correlation time $T_{\rm C}$ depends solely on $\gamma$ but
not on the jump rate $\lambda$.

Figure 9: Timeseries of OU jump-diffusion processes, (a), and its covariance
function $C_{xx}(\tau)$, (b). The evaluated $C_{xx}(\tau)$ is plotted in open
circles and its theoretical result from Eq. (75) are plotted with solid line.

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