

Special Issue in honor of John W. Neuberger

Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 193–207.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

DETERMINING THE BACKGROUND DRIVING PROCESS OF

THE ORNSTEIN-UHLENBECK MODEL

MARIA C. MARIANI, PETER K. ASANTE, WILLIAM KUBIN, OSEI K. TWENEBOAH,

MARIA BECCAR-VARELA

Dedicated to the memory of Prof. John W. Neuberger

Abstract. In this work, we determine appropriate background driving pro-

cesses for the 3-component superposed Ornstein-Uhlenbeck model by analyz-

ing the fractal characteristics of the data sets using the rescaled range analysis
(R/S), the detrended fluctuation analysis (DFA), and the diffusion entropy

analysis (DEA).

1. Introduction

Ever since the Ornstein-Uhlenbeck model was proposed in 1930 by Leonard Orn-
stein and George Eugene Uhlenbeck, the technique has many areas of application
including health care [47], nanotechnology, thermodynamics [8], geophysics [30],
and finance [4, 25, 31, 41].

According to [1], the Ornstein-Uhlenbeck process is a natural model to consider
in a biological context since it stabilizes around some equilibrium point. When
Ornstein and Uhlenbeck proposed the model, it was an alternative to the Brownian
Motion, and thus in its original presentation, the background driving process (BDP)
was a standard Brownian Motion.

Unlike its original proposition which involved a Brownian motion as its back-
ground driving process, there have been many extensions or modifications to it to
truly capture the behavior of the data set, which otherwise could not be modeled
rightly with Brownian motions [27, 37]. Empirical results have shown evidence of
non-Brownian behavior in many real world complex systems [12, 31, 32]]. In fact,
according to [12] statistics of Lévy type is a ubiquitous phenomenon observed in a
wide variety of areas, including physics, seismology, and engineering, to mention a
few. Lévy motions constitute one of the essential and fundamental families of ran-
dom movements which have stationary and independent increments. This means
that the distribution of the increments is the same for any time interval, and that
the increments are statistically independent from each other.

2020 Mathematics Subject Classification. 34k50, 34F50, 60H10.
Key words and phrases. Stochastic differential equation; Itô Calculus; Lévy Process;
Ornstein-Uhlenbeck model; Superposed Ornstein-Uhlenbeck model; Gaussian process;
Background driving process (BDP); Diffusion entropy analysis (DEA); long-range correlations;

detrended fluctuation analysis (DFA); rescaled range analysis (R/S).
©2023 This work is licensed under a CC BY 4.0 license.
Published March 27, 2023.

193


194 M. C. MARIANI, P. K. ASANTE, W. KUBIN, . . . EJDE/SI/02

One major challenge in utilizing the Ornstein-Uhlenbeck model is the ability
to choose an appropriate BDP for the random process under consideration. As
[31] and [14] have shown, the choice of the BDP has a significant influence on the
performance of the model. In this work, our goal is to use the fractal characteristics
of some stochastic processes to characterize them as Gaussian, Lévy walks, or Lévy
flights.

If the characterization of the data is Gaussian, the appropriate BDP is the stan-
dard Brownian motion; if the data describes a Lévy walk or Lévy flight, the ap-
propriate BDP is a Lévy process. This work considers a Gamma(a, b) process and
an Inverse-Gaussian(a, b) process for our Lévy driven models. We compare results
when a different BDP is used from our proposed BDP to see how the performance
of the 3-component superposed Ornstein-Uhlenbeck model [33] is affected.

The outline of this article is as follows: In section 2 we introduce stochastic differ-
ential equations, the Ornstein-Uhlenbeck model, and the 3-component superposed
Ornstein-Uhlenbeck model. In section 3, we present the parameter estimation ap-
proach for the 3-component superposed Ornstein-Uhlenbeck model. We then char-
acterize the data in section 4. In section 5, we discuss the results obtained from
our simulation using Matlab software. Finally, section 6 offers our conclusion and
possible future work based on the results obtained.

2. Modeling with stochastic differential equations

2.1. Stochastic differential equations (SDEs). Stochastic differential equa-
tions (SDEs) have been used for modeling different phenomena in economics and
finance, physics, and population dynamics among others.

Definition 2.1. A stochastic differential equation (SDE) is a deterministic differ-
ential equation perturbed by random noise.

In general, an SDE can be formulated as

dXt = f(t,Xt)dt+ g(t,Xt)dBt, (2.1)

where xt is a stochastic process, and B denotes a Wiener process (standard Brow-
nian motion). For this stochastic differential equation, the random noise dBt is
also termed its background driving process. From the above equation we get the
corresponding integral equation

Xt+s −Xt =

∫ t+s

t

f(v,Xv)dv +

∫ t+s

t

g(v,Xv)dBv, (2.2)

we see that equation 2.2 defines the continuous time stochastic process Xt as the
sum of an ordinary Lebesgue integral and an Itô integral.

The theory of differential equations is the origin of classical calculus and moti-
vated the creation of differential and integral calculus. A differential equation is an
equation involving an unknown function and its derivative. Typically, a differential
equation is a functional relationship

f(t, x(t), x′(t), x′′(t), . . . ) = 0, 0 ≤ t ≤ T (2.3)

involving the time t, an unknown function x(t) and its derivative. The solution of
the differential equation is a function x(t) which satisfies (2.3).

Now, consider the deterministic differential equation

dx(t) = a(t, x(t))dt, x(0) = x0. (2.4)


EJDE-2023/SI/02 ORNSTEIN-UHLENBECK MODEL 195

The easiest way to introduce randomness in this equation is to randomize the initial
condition. The solution x(t) then becomes a stochastic process (Xt, t ∈ [0, T ])
defined as

dXt = a(t,Xt)dt, X0(t) = Y (t). (2.5)

Equation (2.5) is called a random differential equation. Random differential equa-
tions can be considered deterministic equations with a perturbed initial condition.
Note that this is not a full stochastic differential equation.

In general, a stochastic differential equation is a differential equation in which
one or more of the terms is a stochastic process; the solution will be also a stochastic
process.

2.2. Itô calculus. As we have observed above, in solving SDEs we encounter in-
tegrands and integrators that are stochastic processes. Thus, the classical calculus
methods cannot be used in solving SDEs. Kiyosi Itô, a Japanese mathematician,
advanced the solutions of SDEs by developing the theory of Itô calculus, which is
an extension of the methods of calculus to stochastic processes. See [24] for fur-
ther readings. Before stating the properties of the Itô integral, we first define some
important terms.

Definition 2.2. A probability space is a triplet (Ω,F , P ) where Ω is a sample
space, F is a σ-algebra on Ω and P is a probability measure P : F → [0, 1].

Definition 2.3. A continuous filtration of a set Ω is a collection Ft of σ-algebra
subsets of Ω such that Fs ⊂ Ft for all s < t. For n ∈ N, a discrete filtration Fn is
similarly defined.

Definition 2.4. An Ft-measurable random variable is a random variable whose
value is known at time t.

Definition 2.5. A sequence {Xt} of random variables is an adaptive process rel-
ative to {Ft}, if the random variable {Xt} is {Ft}-measurable for each t.

We recall that a stochastic process V = V (S, T ) is a collection of random vari-
ables V = Vt : t ∈ T , where each Vt is defined on a common probability space, and
takes values in a common set S, called the state space. The collection is indexed
or labeled by a set T , that in general is N or [0,∞). If T = N the process is called
discrete, and if T = [0,∞) it is called continuous.

Given a stochastic process V (S, T ) and a Brownian motion Bt, let f, g ∈ V (S, T )
and 0 ≤ S < U < T . Then the following properties hold for the Itô integral:

(i)
∫ T
S
fdBt =

∫ U
S
fdBt +

∫ T
U
fdBt;

(ii)
∫ T
S

(cf + g)dBt = c
∫ T
S
fdBt +

∫ T
S
gdBt, for c ∈ R;

(iii) E
[( ∫ T

s
fdBt

)]
= 0;

(iv)
∫ T
S
fdBt is {FT }-measurable.

Another essential property of the Itô integral is the fact that it is a martingale.
We recall that a martingale is a stochastic process or sequence of random variables
such that at a given time, the conditional expectation of the next value in the
sequence is equal to the present value, independently of all prior values.

2.3. Example. In this section, we present a classic application of SDEs in Finance.


196 M. C. MARIANI, P. K. ASANTE, W. KUBIN, . . . EJDE/SI/02

2.3.1. Modeling asset prices. The SDE that models the evolution of an asset price
is

dXt = µXtdt+ σXtdBt.

The above equation is in the class of SDEs formulated as

dXt = f(t,Xt)dt+ c(t)XtdBt.

Taking f(t,Xt) = µXt and c(t) = σ, we have that

Ft = exp(−
∫ t

0

σdBs +
1

2

∫ t

0

σ2ds) = exp(−σBt +
1

2
σ2t)

is an integrating factor for the SDE that transforms it into a deterministic differ-
ential equation. Setting Yt = FtXt we have

dYt = d(FtXt) = Ftf(t,Xt)dt = µFtXtdt = µYtdt.

Therefore,
dYt
Yt

= µdt.

By integrating both sides, we obtain log Yt = µt+ C, or equivalently,

Yt = eµt+C = Aeµt.

Furthermore, from this last equation, we can obtain Xt as

Xt = Ae
µt
Ft = AeµteσBt−

1
2σ

2t.

2.4. Ornstein-Uhlenbeck model (OU). Having introduced SDEs and some es-
sential tools used in solving them, let us discuss the Ornstein-Uhlenbeck model and
the 3-component superposition of the Ornstein-Uhlenbeck model (OU model). We
begin by defining a Markov process.

Definition 2.6. A Markov process, also known as a Markov chain, is a stochastic
process that has the Markov property, which means that the future state of the
process depends only on its present state and not on its past states. In other
words, given the present state, the probability of moving to any future state is
independent of how the process arrived at its present state.

The Gaussian OU process is the solution to the stochastic differential equation

dXt = λ(m−Xt)dt+ αdBt, t > 0, (2.6)

where λ, m, and α are real constants and Bt is a standard Brownian Motion on
R. The initial value X0 is a random variable independent of (Bt)t≥0. We can show
that the stochastic integral

Xt = m(1− e−λt) + α−λt
∫ t

0

eλsdBs +X0e
−λt, t ≥ 0, (2.7)

satisfies (2.6) for any λ, m, α and choice of X0. The solution X as defined in
equation (2.7) is the unique, strong Markov solution to (2.6) see [27].

As stated earlier, empirical results have shown that many financial stock indexes
deviate from normalcy. Hence, modeling with the ordinary Gaussian OU may result
in poor forecasts. A modification of the Gaussian OU model through replacing the
Weiner process in (2.6) with a Lévy process has been developed and applied in


EJDE-2023/SI/02 ORNSTEIN-UHLENBECK MODEL 197

various literature. To define the Lévy OU process, let (ε, η) be a bivariate Lévy
process and define

Xt = m(1− e−εt) + e−εt
∫ t

0

e−εsdηs +X0e
−εt , t ≥ 0, (2.8)

where X0 is independent of (εt, ηt)t≥0 and assumed F0 − measurable. Equation
(2.8) is the generalized OU model (GOU) and seems to have been first considered by
Carmona, Petit, and Yor (1997) as well as being implicit in Haan and Karandukar
(1989) [27, 9, 10].

Now, from equation 2.7, we set m = 0, α = 1 and replace Bt with a Lévy process
Zλt to get the SDE

dXt = −λXtdt+ dZλt, X0 > 0, λ ∈ R+. (2.9)

The OU process below is the solution for equation (2.9):

Xt = e−λtX0 +

∫ t

0

e−λ(t−s)dZλs. (2.10)

2.5. Superposed Ornstein-Uhlenbeck model. The OU process in (2.10) is re-
defined as a sum of m independent Ornstein-Uhlenbeck processes [33, 15, 36] as

Xt =

m∑
i=1

wie
−λitX0 +

∫ t

0

m∑
i=1

wie
−λi(t−s)dZλis, (2.11)

where
∑m
i=1 wi = 1.

For the purpose of this work, we consider a 3-component model of (2.11) which
results in

Xt = w1Xt1 + w2Xt2 + w3Xt3 ,

with
∑
i wi ≈ 1.

3. Parameter estimation

This section presents the estimation method for the λ parameters and the weights
for the 3-component superposed Ornstein-Uhlenbeck model. With our data, we
apply an iterative approach in estimating the λ values at different lags of the auto-
correlation function. We then derive a matrix system that uses the autocorrelation
function and estimated λi values to determine the weights (wi).

3.1. The 3-component model. We recall that the autocorrelation function ana-
lyzes the similarity between a time series and a lagged version of itself, see [5].

We consider the autocorrelation function at lags k, k + h1 and k + h2:

ρ(k) = w1e
−λ1|k| + w2e

−λ2|k| + w3e
−λ3|k|

ρ(k + h1) = w1e
−λ1|k+h1| + w2e

−λ2|k+h1| + w3e
−λ3|k+h1|

ρ(k + h2) = w1e
−λ1|k+h1| + w2e

−λ2|k+h1| + w3e
−λ3|k+h2|

(3.1)

Assume λ1 = λ2 = λ3, then from the first expression in equation (3.1)

ρ(k) = (w1 + w2 + w3)e−λ1|k| (3.2)

λ1 = − log(ρ(k))

|k|
. (3.3)


198 M. C. MARIANI, P. K. ASANTE, W. KUBIN, . . . EJDE/SI/02

From the second expression in equation (3.1),

ρ(k + h1) = (w1 + w2 + w3)e−λ2|k+h1|, (3.4)

λ2 = − log(ρ(k + h1))

|k + h1|
. (3.5)

From the third expression in equation (3.1),

ρ(k + h2) = (w1 + w2 + w3)e−λ3|k+h2|, (3.6)

λ3 = − log(ρ(k + h2))

|k + h2|
. (3.7)

For the 3-component OU model, h1 and h2 are shifts from lag k. To solve for the
weights, we construct the matrix equation from (3.1) as follows:

A =

 e−λ1|k| e−λ2|k| e−λ3|k|

e−λ1|k+h1| e−λ2|k+h1| e−λ3|k+h1|

e−λ1|k+h2| e−λ2|k+h2| e−λ3|k+h2|

 , (3.8)

b =

 ρ(k)
ρ(k + h1)
ρ(k + h2)

 , (3.9)

W =

w1

w2

w3

 . (3.10)

From equations (3.8), (3.9), and (3.10), we can solve for the weights as: W = A−1b
with

∑
W ≈ 1 and the inverse of A, where A−1 exists for an appropriately chosen

lag.

3.2. Ljung-Box Statistic. We state the null (H0) and alternate (HA) hypothesis
of the Ljung-Box test statistic as follows:

(H0) The residuals are independently distributed.
(HA) The residuals are not independently distributed; they exhibit serial corre-

lation.

Independence of residuals means that the error terms of the dependent variable
at different time points are not related to each other. This assumption is usually
checked by examining the autocorrelation function (ACF) of the residuals. If the
residuals are independent, the ACF will be close to zero for all lags. If the ACF
shows significant values at one or more lags, it indicates that the residuals are not
independent, and some kind of correlation exists. Hence, the goal of the Ljung-Box
test is to determine whether the residuals in a time series model exhibit autocor-
relation. A p-value greater than 0.05 indicates that there is not enough evidence
to reject the null hypothesis that the residuals are independently distributed. Not
rejecting the null hypothesis in the Ljung-Box test statistic is a desirable property
for a time series model. It indicates that our model has captured all the system-
atic patterns in the data and that the residuals are random fluctuations around
the mean. This is important for making accurate predictions and for conducting
statistical inference, as we want to ensure that any relationships we observe are not
spurious and are not driven by autocorrelated residuals.


EJDE-2023/SI/02 ORNSTEIN-UHLENBECK MODEL 199

3.3. Autoregressive integrated moving average (ARIMA). To ensure that
the matrix A is invertible, there is the need to choose appropriate lags for each λi.
Thus, we fit an ARIMA model to the data and perform an Ljung-box statistic of
the fitted ARIMA model to select the most significant lag based on the p-value at a
5% significant level. The ARIMA model assesses the significance of one dependent
variable in relation to other changing variables. We iterate the ARIMA model over
some lags and perform the Ljung-Box on each fit to examine the null hypothesis of
independence in our time series. We then select the lags that give a p-value > 0.05
as significant lags. The three lags chosen are then used in computing the λi’s. This
process is performed with the R-software. See [5, 17] for more information on the
ARIMA model and Ljung-Box statistic.

4. Data characterization

In this section, we present the data that we will analyze and we apply the
characterization approach in [32] to characterize them as Gaussian, Lévy walk, or
Lévy flight.

4.1. Data. We will use three different types of data in the analysis of our model,
namely the daily closing values of the Nasdaq, retrieved from Yahoo Finance, the
Japan earthquake data in 2011 after the magnitude nine occurrence (AfterM9),
and finally, a simulated fractional Brownian motion (FBM) using the function fbm
from the R package somebm. For each data set, we use two-thirds of the data for
estimating the parameters in our model. The remaining one-third is predicted with
our model, using the estimated parameters. Error estimates are then computed
between the true values and predicted values.

Figure 1. Time series plot of daily closing values of the Nasdaq
index. Here, time is measured in days, and the closing value is in
US dollars.

4.2. Variance scaling methods. This subsection briefly introduces the Rescaled
Range Analysis, Detrended Fluctuation Analysis, and Diffusion Entropy Analysis.


200 M. C. MARIANI, P. K. ASANTE, W. KUBIN, . . . EJDE/SI/02

Figure 2. Time series plot of recorded earthquake magnitudes
after magnitude nine event. Here, time is measured in minutes,
and the magnitude is measured on the Richter scale (ergs).

Figure 3. Time series plot of simulated fractional Brownian mo-
tion. Here, time is measured in days.

4.2.1. Rescaled range analysis. The Rescaled-Range analysis (R/S) was presented
by Hurst in his study on the long-run variations of the water level of the Nile river
[19].

Mandelbrot coined the name H for the parameter derived from this technique in
tribute to the hydrologist Hurst and the mathematician Holder. The parameter H,
also known as the index of dependence, represents the relative trend of a time series
(i.e. persistence, anti-persistence or randomness) and always lies between 0 and 1; it
is equal to 1/2 in the case of processes with independent increments. Of particular
interest for our work is the case in which 0.5 < H < 1 since it is an indicator of
long-range correlations [32]. However, because its sensitivity to abnormal values in
the series, the rescaled range analysis method is unsuitable for analyzing long-range
auto-correlation for non-stationary series [19].


EJDE-2023/SI/02 ORNSTEIN-UHLENBECK MODEL 201

4.2.2. Detrended fluctuation analysis. Peng et al. [38] proposed the Detrended Fluc-
tuation Analysis (DFA) while examining a sequence of DNA nucleotides to study
the self-similarity [35] and long-range dependence of time series. From the moment
it was submitted to date, DFA has become a widely used method for determining
fractal scaling properties and detecting long-range correlations in non-stationary
time series. There are known applications in biology, meteorology, geophysics, and
economics [34, 38, 46].

The principal advantage of the DFA lies in its ability to differentiate the intrinsic
autocorrelations of the time series from those imposed by non-stationary external
trends. The method focuses on the inherent structure of the correlations of market
fluctuations at different time scales, leaving aside non-stationary trends [20].

The application of the DFA method allows obtaining a scaling exponent α from
estimating the slope of the function F (s) that measures the mean square deviation
from an optimal linear approximation around the trend signal in segments of length
s. The fluctuation function vs. s behaves as a power law. Therefore, it is possible
to compute the value of the exponent α from the slope of the function in a log-log
scale plot of F (s) vs. s [35]. The DFA exponent α and the Hurst parameter H are
related by

H =

{
α if 0 < α < 1

α− 1 if α ≥ 1.
(4.1)

4.2.3. Diffusion entropy analysis. Based on the direct evaluation of the Shannon
entropy [16, 42, 44, 6], the Diffusion Entropy Analysis (DEA) is a probability density
function (PDF) scaling method that perceives the numbers in a time series as the
trajectory of a diffusion process [18]. The scaling property for the stationary time
series takes the form

p(x, t) =
1

tδ
F (

x

tδ
), (4.2)

where x denotes the diffusion variable, p(x, t) is its PDF at time t, and 0 < δ < 1 is
the scaling exponent. The scaling property for the non-stationary time series takes
the form

p(x, t) =
1

tδ(t)
F (

x

tδ(t)
). (4.3)

Note that for the stationary time series, the parameter δ in equation 4.2 is scalar,
whereas for the non-stationary time series, the parameter δ(t) in 4.3 is a function of
time [42]. We recall that a Lévy flight is a random walk in which the step-lengths
follow a Lévy distribution. The Lévy distribution is a probability distribution that
is heavy-tailed, see [29]. Lévy walks have been studied from 1937, see [26], they are
special forms of random walks. Both Lévy flights and Lévy walks are named after
the French mathematician Paul Lévy, see [40].

As derived in [42, 43], a diffusion process generated by Lévy walk characterization
follows the relation

δ =
1

3− 2(H,α)
. (4.4)

The variables (H,α) in (4.4) indicates either H or α will be used in equation (4.4).
Recall that H represents the scaling exponent from the rescaled range analysis,
while α represents the scaling exponent from the detrended fluctuation analysis.


202 M. C. MARIANI, P. K. ASANTE, W. KUBIN, . . . EJDE/SI/02

If δ = (H,α) = 0.5, the time series can be characterized by Fractional Brown-
ian Motion (FBM), since the variance methods are based subtly on the Gaussian
assumption [19, 20].

However, if δ 6= (H,α) and equation (4.4) holds, then the time series can be
characterized by Lévy statistics in particular a Lévy walk. Now, if δ 6= (H,α) and
equation (4.4) does not hold, then the time series can be characterized by a Lévy
flight.

4.2.4. Estimation procedure. In this subsection, we describe the estimation tech-
nique for the scaling exponent δ. We first present a brief background on the Shan-
non Entropy used for estimating δ.

4.2.5. Shannon entropy. Rudolph Clausius developed the concept of entropy in
1865, a few years after he stated the laws of thermodynamics [21, 22]. Entropy is an
indicator of the lack of information about the measure of an event that occurs with
probability p, [21]. Other types of entropies are the Kolmogorov-Sinai entropy, the
Renyi entropy and the Tsallis entropy [21, 42, 44]. The Shannon entropy measures
information of a probability distribution as

S(t) = −
N∑
1

pi log pi. (4.5)

The summation is replaced by the integral in the case of continuous probability
distributions. The above equation derives the log equation determining the DEA δ
scaling. We present below the process for estimating δ:

• First, we transform the time series into a diffusion process. See [42, 43].
• We compute Shannon’s Entropy of the diffusion process. A log-linear equa-

tion or log-quadratic equation is derived from the Shannon entropy by sub-
stituting equations (4.2))and(4.3)) respectively into equation (4.5). The
reader is invited to read [3, 32, 42, 43] for further information on the Shan-
non entropy, transformation of time series into diffusion processes and the
derivation of the shannon entropy for the stationary and non-stationary
series. Now, simplifying the result from the substitutions, we have the
relation for stationary time series given by

S(t) = A+ δ log(t). (4.6)

For the non-stationary series, the relation is

S(t) = A+ δ(t)τ, (4.7)

where δ(t) = δ0 + η log(t), and τ = log(t) with η log(t) < 1 − δ0 and δ0 and η are
constants. Substituting δ(t) and τ gives us

S(t) = A+ δ0 log(t) + η log2(t).

We assume there exists some constant K multiplying log(t) such that η = 1 − δ0
and equation (4.7) becomes

S(t) = A+ δ0 log(t)−K log(t) + (1− δ0) log2(t).

This simplifies to

S(t) = A+ (δ0 −K) log(t) + (1− δ0)(log(t))2. (4.8)


EJDE-2023/SI/02 ORNSTEIN-UHLENBECK MODEL 203

Thus, by fitting a log-quadratic model in the non-stationary series and a log-linear
model in the stationary series, we can determine the δ (or δ0) scaling. At t = 1, it
is clear that the constant A in both equations (4.6) and (4.7) is given by S(1).

We derive the δ (or δ0) by estimating the slope of the above linear-log equation
or the coefficients from the quadratic-log equation. For details of the algorithm
used when transforming the series into a diffusion process, we refer the reader to
[35, 42].

4.3. Scaling exponents. Now, we apply the variance scaling methods (Rescaled
Range Analysis and Detrended Fluctuation Analysis) and the PDF scaling method
(Diffusion Entropy Analysis) to compute the scaling exponents of the data under
consideration. We characterize the time series as Gaussian, Lévy walk, or Lévy
flight using the scaling methods and equation (4.4) discussed earlier. The charac-
terization approach of the time series has been extensively discussed in [32, 3].

Table 1. Scaling exponents for Financial time series obtained us-
ing the Rescaled range analysis, the Detrended fluctuation analysis
and the Diffusion entropy analysis. The δLevy(R/S)= 1

3−2H and

δLevy(DFA)= 1
3−2α are obtained from equation 4.4

Data R/S(H) DFA (α) DEA(δ) δLevy (R/S) δLevy (DFA)

AfterM9 0.3149 0.6518 0.77046 0.4219 0.5895

FBM 0.2928 0.4992 0.4986 0.4142 0.5

Nasdaq 0.5829 0.4298 0.4989 0.5452 0.4672

Remark 4.1. After running the characterization procedure developed in [32], we
obtain the characterization of the Nasdaq, earthquake data, and simulated FBM as
Lévy walk, Lévy flight, and Gaussian processes, respectively. Since all three scaling
methods are obtained using numerical approximations, we may not get two scaling
exponents to be the same. For example, for the simulated FBM, the α value ob-
tained in the DFA is 0.4992, while the δ value obtained in the DEA is 0.4986. Both
are approximately 0.5, but not equal; thus, we apply a (−0.06, 0.06) adjustment
to the scaling exponents obtained from our simulations. This adjustment interval
is chosen arbitrarily. Furthermore, despite the fact that the computed δLevy(R/S)
is approximately 0.5 after adjustments, the R/S method fails to compute the cor-
rect scaling exponent for the simulated FBM. The R/S method fails to correctly
estimate the scaling exponent due to the non-stationarity of the simulated FBM.

5. Results

This section presents the results from applying the proposed background driving
process (BDP) in modeling with the 3-component superposed Ornstein-Uhlenbeck
equation, i.e., the superposition of three Ornstein-Uhlenbeck processes (2.11) to
solve the stochastic differential equation in (2.9). We further compare results to
modeling the data with the two other processes discussed to ascertain the effect of
the background driving process (BDP) on the model performance.


204 M. C. MARIANI, P. K. ASANTE, W. KUBIN, . . . EJDE/SI/02

5.1. Model simulation. This section presents our simulation of the 3-component
superposed Ornstein-Uhlenbeck model [33]. The solution of the proposed Ornstein-
Uhlenbeck stochastic differential equation in 2.11 is used to simulate the process.
We simulate our Ornstein-Uhlenbeck stochastic differential equation model solu-
tions via the Brownian motion, Inverse-Gaussian(a, b) and the Gamma(a, b) pro-
cess. For the Inverse-Gaussian(a, b), a is the mean and b is the rate parameter,
while for the Gamma(a, b), a is the shape parameter and b is the rate parameter.
We determine the model’s performance by computing the root mean squared er-
rors. Table 2 summarizes the numerical results of simulating each data set. We
have utilized both R and Matlab software in the simulation of our model.

Table 2. Error estimates obtained from the model simulation
of the remaining one third of each data set using the Inverse-
Gaussian(a, b), the Gamma(a, b) and the Brownian motion as
Background processes.

Data Inverse-Gaussian(a, b) Gamma(a, b) Brownian Motion

Nasdaq 0.0286 0.25774 1.5939

AfterM9 0.0488 0.8037 1.3796

FBM 1.0525 1.4037 0.478

5.2. Discussion. The error estimates are compared in this section, with smaller
error values implying better performance. Recall that in this study, the Lévy driven
models use either a Gamma(a, b) process or an Inverse-Gaussian(a, b) process as the
background driving process. The table above records the error estimates between
the predicted values and true values. Each column represents the model being
simulated with the respective background driving process in the first row. Now ob-
serving the results above, table 2 shows that for both the financial and earthquake
data, the standard Brownian motion as a BDP leads to a poor performance (with
error estimate of 1.5939 and 1.3796 respectively) of the 3-component superposed
Ornstein-Uhlenbeck model, in comparison to the Lévy driven 3-component super-
posed Ornstein-Uhlenbeck model (with error estimates less than 1 for both IG(a, b)
and Gamma(a, b)). Similarly, for the simulated FBM characterized by a Gaussian
process, we see from table 2 that using the standard Brownian motion as a BDP
leads to a better model (with error estimate of 0.478) performance compared to
when a Lévy driven BDP (with error estimates greater than 1) is used. Finally, we
observe that for the Lévy driven BDP, the Inverse-Gaussian(a, b) performs better
than the Gamma(a, b). Thus, further investigation of the time series data is needed
to classify the performance when modeling with a Lévy driven 3-component super-
posed Ornstein-Uhlenbeck equation since we do not have enough data to conclude
that the IG(a, b) driven Ornstein-Uhlenbeck model will always perform better than
the Gamma(a, b) driven Ornstein-Uhlenbeck model.

6. Conclusion

This work shows the importance of selecting appropriate background driving
processes (BDP) when modeling time series with the 3-component superposed


EJDE-2023/SI/02 ORNSTEIN-UHLENBECK MODEL 205

Ornstein-Uhlenbeck stochastic differential equation (2.11)). We used the frac-
tal characteristics of the time series data to classify them as Gaussian processes,
Lévy walks, or Lévy flights. Next, we modeled each data set using the three
background driving processes presented in this work (Standard Brownian motion,
Inverse-Gaussian process, and Gamma process) which would help us determine
an appropriate BDP for better model performance. Our results show that when
the time series characterization is a Lévy process, a Lévy driven 3-component su-
perposed Ornstein-Uhlenbeck model performs better. In contrast, the standard
Brownian motion as a BDP performs better for a time series characterized by a
Gaussian process. By observing the performance of the Inverse-Gaussian(a, b) and
Gamma(a, b) Ornstein-Uhlenbeck processes, we conclude that further information
about the time series data is needed to classify the performance of the Lévy driven
3-component superposed Ornstein-Uhlenbeck model, which we will be investigating
in future works.

Acknowledgments. The authors are especially grateful to Prof. Alfonso Castro
and the reviewers for their careful reading of the manuscript and their fruitful
remarks.

This study was funded partially by the National Institute on Minority Health
and Health Disparities (NIMHD) grant U54MD007592, and by the Department of
Education (DoE) grant P120A220040-FY 2022 MSEIP.

References

[1] Aalen, O. O.; and Gjessing, H. K.; Survival models based on the Ornstein-Uhlenbeck process.
Lifetime data analysis, 10 (2004), 407-423.

[2] Abadie, L. M.; Current expectations and actual values for the clean spark spread: The case
of Spain in the Covid-19 crisis. Journal of Cleaner Production, 285 (2021), 124842.

[3] Asante, P. K.; Lévy Processes: Characterizing Volcanic and Financial Time Series(Doctoral

dissertation, The University of Texas at El Paso), 2020.
[4] Barndorff-Nielsen, O. E.; Shephard, N.; Non-Gaussian Ornstein–Uhlenbeck-based models and

some of their uses in financial economics. Journal of the Royal Statistical Society: Series B

(Statistical Methodology), 63 (2001) (2), 167-241.
[5] Box, G. E.; Jenkins, G. M.; Reinsel, G. C.; Time series analysis, forecasting and control.

Englewood Clifs, 1994.

[6] Brooks, C.; A measure of persistence in daily pound exchange rates. Applied economics
letters, 2 (1995) (11), 428-431.

[7] Calafiore, G. C.; Novara, C.; Possieri, C.; A time-varying SIRD model for the COVID-19

contagion in Italy. Annual reviews in control, 50 (2020), 361-372.
[8] Caprini, L.; Marconi, U. M. B.; Puglisi, A.; Vulpiani, A. ; The entropy production of Orn-

stein–Uhlenbeck active particles: a path integral method for correlations. Journal of Statistical
Mechanics: Theory and Experiment, 2019 (2019) (5), 053203.

[9] Carmona, P.; Petit, F.; Yor, M.; On the distribution and asymptotic results for exponen-
tial functionals of Lévy processes. Exponential functionals and principal values related to
Brownian motion, (1997), 73-121.

[10] De Haan, L.; and Karandikar, R. L.; Embedding a stochastic difference equation into a

continuous-time process. Stochastic processes and their applications, 32 (1989) (2), 225-235.
[11] Donado, F.; Moctezuma, R. E.; López-Flores, L.; Medina-Noyola, M.; and Arauz-Lara, J. L.;

Brownian motion in non-equilibrium systems and the Ornstein-Uhlenbeck stochastic process.
Scientific reports, 7 (2017) (1), 12614.

[12] Eliazar, I.; and Klafter, J.; Lévy, Ornstein–Uhlenbeck, and subordination: Spectral vs. jump
description. Journal of statistical physics, 119 (2005), 165-196.

[13] Endres, S.; and Stddotubinger, J.; Optimal trading strategies for Lévy-driven Orn-
stein–Uhlenbeck processes. Applied Economics, 51 (2019) (29), 3153-3169.


206 M. C. MARIANI, P. K. ASANTE, W. KUBIN, . . . EJDE/SI/02

[14] Habtemicael, S.; SenGupta, I.; Ornstein-Uhlenbeck processes for geophysical data analysis.

Physica A: Statistical Mechanics and its Applications, 399 (2014), 147-156.

[15] Habtemicael, S.; Ghebremichael, M.; and SenGupta, I.; Volatility and variance swap using
superposition of the Barndorff-Nielsen and Shephard type Lévy processes. Sankhya B, 81

(2019), 75-92.

[16] Haubold, H. J.; Mathai, A. M.; and Saxena, R. K.; Analysis of solar neutrino data from
Super-Kamiokande I and II. Entropy, 16 (2014) (3), 1414-1425.

[17] Hassani, H.; and Yeganegi, M. R.; Selecting optimal lag order in Ljung-Box test. Physica A:

Statistical Mechanics and its Applications, 541 (2020), 123700.
[18] Huang, J.; Shang, P.; Zhao, X.; Multifractal diffusion entropy analysis on stock volatility in

financial markets. Physica A: Statistical Mechanics and its Applications, 391 (2012) (22),

5739-5745.
[19] Hurst, H. E.; Long-term storage capacity of reservoirs. Transactions of the American society

of civil engineers, 116 (1951) (1), 770-799.
[20] https : //www.physionet.org/tutorials/fmnc/node5.html

[21] http : //www.ueltschi.org/teaching/chapShannon.pdf

[22] http : //galileo.phys.virginia.edu/classes/152.mf1i.spring02/Entropy.pdf
[23] https : //en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeckprocess

[24] https : //en.wikipedia.org/wiki/It%C3%B4 calculus#It%C3%B4 processes

[25] Janczura, J.; Orzel, S.; Wylomańska, A.; Subordinated α-stable Ornstein-Uhlenbeck process
as a tool for financial data description. Physica A: Statistical Mechanics and its Applications,

390 (2011) (23-24), 4379-4387.

[26] Lévy, P.; Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris., 1954.
[27] Maller, R. A.; Müller, G.; Szimayer, A.; Ornstein–Uhlenbeck processes and extensions. Hand-

book of financial time series, (2009), 421-437.

[28] Mandal, M.; Jana, S.; Nandi, S. K.; Khatua, A.; Adak, S.; Kar, T. K.; A model based study
on the dynamics of COVID-19: Prediction and control. Chaos, Solitons and Fractals, 136

(2020), 109889.
[29] Mantegna, R. N.; Stanley, H. E.; Introduction to econophysics: correlations and complexity

in finance. Cambridge university press, 1999.

[30] Mariani, M. C.; Tweneboah, O. K.; Stochastic differential equations applied to the study of
geophysical and financial time series. Physica A: Statistical Mechanics and its Applications,

443 (2016), 170-178.

[31] Mariani, M.C; Tweneboah, O. K.; Modeling high frequency stock market data by using sto-
chastic models. Stochastic Analysis and Applications, 40 (2022) (4), 573-588.

[32] Mariani, M. C.; Asante, P. K.; Bhuiyan, M. A. M.; Beccar-Varela, M. P.; Jaroszewicz, S.;

Tweneboah, O. K.; Long-range correlations and characterization of financial and volcanic
time series. Mathematics, 8 (2020) (3), 441.

[33] Mariani, M. C.; Asante, P. K.; Tweneboah, O. K.; Kubin, W.; A 3-component superposed

Ornstein-Uhlenbeck model applied to financial stock markets. Research in Mathematics, 9
(2022) (1), 2024339.

[34] Mariani, M. C.; Kubin, W.; Asante, P. K.; Guthrie, J. A.; Tweneboah, O. K.; Relation-
ship between Continuum of Hurst Exponents of Noise-like Time Series and the Cantor Set.

Entropy, 23 (2021) (11), 1505.

[35] Mariani, M. C.; Kubin, W.; Asante, P. K.; Tweneboah, O. K.; Beccar-Varela, M. P.;
Jaroszewicz, S.; Gonzalez-Huizar, H.; Self-Similar Models: Relationship between the Dif-

fusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models. Mathematics, 8
(2020) (7), 1046.

[36] Nicolato, E.; Venardos, E.; Option pricing in stochastic volatility models of the Ornstein-

Uhlenbeck type. Mathematical Finance: An International Journal of Mathematics, Statistics

and Financial Economics, 13 (2003) (4), 445-466.
[37] Obuchowski, J.; Wylomanska, A.; Ornstein-Uhlenbeck Process with Non-Gaussian Structure.

Acta Physica Polonica B, 44(5).
[38] Peng, C. K.; Buldyrev, S. V.; Havlin, S.; Simons, M.; Stanley, H. E.; Goldberger, A. L.;

Mosaic organization of DNA nucleotides. Physical review e, 49 (1994) (2), 1685.

[39] Punzo, A.; A new look at the inverse Gaussian distribution with applications to insurance
and economic data. Journal of Applied Statistics, 46 (2019) (7), 1260-1287.


EJDE-2023/SI/02 ORNSTEIN-UHLENBECK MODEL 207

[40] Reynolds, A. M.; Current status and future directions of Lévy walk research. Biology open,

7 (2018) (1), bio030106.

[41] Salmon, N.; SenGupta, I.; Fractional Barndorff-Nielsen and Shephard model: applications
in variance and volatility swaps, and hedging. Annals of Finance, 17 (2021) (4), 529-558.

[42] Scafetta, N.; An Entropic Approach to the Analysis of Time Series, University of North

Texas Libraries, 2003.
[43] Scafetta, N.; Grigolini, P.; Scaling detection in time series: diffusion entropy analysis, Phys-

ical Review E, 66 (2002) (3).

[44] Scafetta, N.; Latora, V.; Grigolini, P.; Lévy scaling: The diffusion entropy analysis applied
to DNA sequences, Physical review. E, Statistical, nonlinear, and soft matter physics, 66

(2002), 031906. 10.1103/PhysRevE.66.031906.

[45] Shu, Y.; Feng, Q.; Kao, E. P.; Liu, H.; Lévy-driven non-Gaussian Ornstein–Uhlenbeck pro-
cesses for degradation-based reliability analysis. IIE Transactions, 48 (2016) (11), 993-1003.

[46] Talkner, P.; Lutz, E.; Hddotanggi, P.; Fluctuation theorems: Work is not an observable,
Physical review. E, 75 (2007), 032903.

[47] Tian, M. Y.; Wang, C. J.; Yang, K. L.; Fu, P.; Xia, C. Y.; Zhuo, X. J.; Wang, L.; Estimating

the nonlinear effects of an ecological system driven by Ornstein-Uhlenbeck noise. Chaos,
Solitons and Fractals, 136 (2020), 109788.

[48] Valdivieso, L.; Schoutens, W.; Tuerlinckx, F.; Maximum likelihood estimation in processes of

Ornstein-Uhlenbeck type. Statistical Inference for Stochastic Processes, 12 (2009), 1-19.

Maria C. Mariani
Department of Mathematical Sciences, the University of Texas at El Paso, TX 79968,

USA

Email address: mcmariani@utep.edu

Peter K. Asante

Computational Science Program, the University of Texas at El Paso, TX 79968, USA
Email address: pkasante@miners.utep.edu

William Kubin
Computational Science Program, the University of Texas at El Paso, TX 79968, USA

Email address: wkubin@miners.utep.edu

Osei K. Tweneboah

Department of Data Science, Ramapo College of New Jersey, 505 Ramapo Valley Road,

Mahwah, NJ 07430, USA
Email address: otwenebo@ramapo.edu

Maria Beccar-Varela

Department of Mathematical Sciences, the University of Texas at El Paso, TX 79968,
USA

Email address: mpvarela@utep.edu


	1. Introduction
	2. Modeling with stochastic differential equations
	2.1. Stochastic differential equations (SDEs)
	2.2. Itô calculus
	2.3. Example
	2.4. Ornstein-Uhlenbeck model (OU)
	2.5. Superposed Ornstein-Uhlenbeck model

	3. Parameter estimation
	3.1. The 3-component model
	3.2. Ljung-Box Statistic
	3.3. Autoregressive integrated moving average (ARIMA)

	4. Data characterization
	4.1. Data
	4.2. Variance scaling methods
	4.3. Scaling exponents

	5. Results
	5.1. Model simulation
	5.2. Discussion

	6. Conclusion
	Acknowledgments

	References