Article

Nonlinear Stochastic Equation within an Itô

Prescription for Modelling of Financial Market
Leonardo S. Lima ID

Departamento de Física, Centro Federal de Educação Tecnológica de Minas Gerais,

Belo Horizonte, MG 30510-000, Brazil; lslima7@yahoo.com.br




Received: 10 May 2019; Accepted: 23 May 2019; Published: 25 May 2019


Abstract: The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model
for price dynamics of financial markets. We study this model with relation to concrete stylised facts
about financial markets. We investigate the behavior of the long tail distribution of the volatilities
and verify the inverse power law behavior which is obeyed for some financial markets. Furthermore,
we obtain the behavior of the long range memory and obtain that it follows to a distinct behavior
of other stochastic models that are used as models for the finances. Furthermore, we have made
an analysis by using Fokker–Planck equation independent on time with the aim of obtaining the
cumulative probability distribution of volatilities P( g), however, the probability density found does
not exhibit the cubic inverse law.

Keywords: price dynamics; stochastic differential equation; Itô calculus

1. Introduction
In general, physics has long been a large source of ideas for economics. Every investor would
like to be able to predict the price of a stock in the same way as physicists predict the trajectory or
position in function of time of a particle. Since the prices of stocks of companies exhibit unpredictable
fluctuations, they can be modeled by stochastic differential equations which becomes very important in
the pricing of financial derivatives [1,2]. Since the celebre equation for price dynamics of the European
market derived by Black and Scholes [3] until nowadays, the modeling of financial has been focused
to simulate the behavior of market structure, trading mechanism and price dynamics [4]. Mike and
Farmer [5] have made an empirical behavioral model to simulate the dynamic of stock price formation.
In the following, Gu and Zhou [6] have modified the MF model by incorporating long memory into
aggressiveness of incoming orders [7].
A very important model in modelling of financial market is the Ising model, H = ∑hiji Jij σi σj .
It constitutes in a very general class of stochastic dynamical model developed to describe interacting
elements, particles, and agents in physics and biology. The tendency towards imitation is governed by
Jij , which is the “coupling strength”. This means the tendency towards noisy behavior is governed by
σi of the noise term. Hence, the value of J relative to σ determines the outcome of the battle between
order and disorder, and eventually the structure the market prices [8]. We consider the price pi as
analogous to the spins σiz of Ising chain and the magnetization interpreted as volatility, which is the
modulus of the return, g = |r (t)| [9–19]. The study of volatility is crucial to reveal the underlined
mechanism of markets dynamics, and it is also useful for traders because it helps them in estimating
risk and optimization of portfolio [20]. Another quantity of interest in price dynamics is the return r (t).
Its statistical analysis is well known as distribution of returns: r (t) ≈ S(t) = ln ( X (t + ∆t)) − ln ( X (t)),
where the long tail cumulative probabilities distribution of volatilities g obeys to an inverse cubic-law
P( g) ∼ g−γ , where γ ∼ 3 is the tail exponent [4,6,9,14,21–27].

Entropy 2019, 21, 530; doi:10.3390/e21050530 www.mdpi.com/journal/entropy

Entropy 2019, 21, 530 2 of 10

The modelling of stocks throughout stochastic differential equations is also often employed
in the modelling of market [28–34]. Furthermore, modelling through a set of linear equations [35]
and non-linear equations [36] is also often employed. Since exponentials and Gaussian functions
can emerge in economics theories [37–39], it is well known that such functions can be generalized
into a nonlinear one: f ( x ) = eqax , where eq is the q-exponential function defined in non-extensive
1/(1−q)
statistical mechanics [39] as eqx ≡ [1 + (1 − q) x ]+ , (e1x = e x ), with [1 + (1 − q) x ]+ = 1 + (1 − q) x
if z > 0 and zero otherwise. An important thing related to finances is the observation of scaling laws
exhibited by large price fluctuations, being corroborated for practically all types of financial data
and markets [40–43].
In this paper, we investigate an Itô diffusion model with additive noise and nonlinear terms as
a possible model for the financial market. The aim is to determine if the model obeys stylised facts
about financial markets such as the exponent of the long tail distribution of volatilities as well as the
long range memory or Hurst index. The stochastic differential equation with nonlinear terms until
quadratic terms has been already proposed as a model for stock market fluctuations and crashes,
being known as Bouchaud-Cont Langevin model [44]. Here, we study a more general model with
terms of higher order. The case of the cubic potential and higher order with additive white noise has
been treated in [28,31]. The case of multiplicative noise has been treated in [29]. The plan of this paper
is the following. In Section 2 we discuss the economic entropy. In Section 3, we describe the stochastic
model. In Section 4, we present the numerical results. In Section 5, we make a mathematical analysis
using the Chapman–Kolmogorov equation. In Section 5, we present our conclusions and final remarks.

2. Economic Entropy
When we consider the details of microstructure of an economic system, one can understand
the economic entropy. In the economy the entropy gives similarly a measure of the total number of
available ’economic’ states, whereas the energy measures the probability that any state in the ’economic
phase state’ will be realised. In a system of traders, it could be the total number of ways that the
money can be distributed across the agents [1]. In macroeconomics, the entropy function S takes the
role of a production function of the macroscopic economic system. In microeconomics, the entropy is
used to find the optimal number of different professionals in a company, the best choice of stocks in
a portfolio, and so on. By considering the total number of elements N = N1 + N2 of a binary system
with two different categories, containing N1 and N2 elements of two types. One has the binary entropy
or utility function
S( N1 , N2 ) = N ln N − N1 ln N1 − N2 ln N2 . (1)

One can rewrite it making x = N1 /N and hence 1 − x = N2 /N to obtain

S( x ) = − N [ x ln( x ) + (1 − x ) ln(1 − x )] . (2)

For a company with two different types of commodities, we have the Cobb-Douglas utility
function F = N1ε N21−ε , where the exponent ε ∈ [0.5, 0.7] The “elasticity” parameter ε allows for
adaptation of the function to data while S emerges from an entropy theoretical framework containing
no free parameter and thus has true predictive power.

3. Phenomenological Itô Equation

The model of interest is defined by
h i
dX (t) = αX (t) − µ( X (t))3 − δ( X (t))6 dt + β( X (t), t)dW (t), (3)

where W (t) is a Winner process. X (t) = p(t) is the price of a derivative. The equation above can be
used to describe the behavior of a particle in Brownian motion under action of an potential of type φ6
in which the prices of stocks ought obey. In the absence of sixth order term, the potential reduction
Entropy 2019, 21, 530 3 of 10

to one of double well separated by a barrier of potential of height V0 . When we include the sixth
order term, one have more wells. As in the neighboring of each minimum, the prices tend to have
an oscillatory behavior, the presence of a noise can cause the prices to move from a well to another well,
reflecting a crash in the market [44]. Moreover, we have a dissipative force given by −γ ẋ represented
by the friction term in Langevin’s equation and an environment stochastic white noise ζ (t), which if
relates with the Winner process W (t) by
Z t
W (t) = ζ (t0 )dt0 . (4)
t0

Although W (t) be the integral of ζ (t), the inverse is not true, i.e., ζ (t) 6= dW (t)/dt, since

dW (t) ∆W (∆t) 1
= lim ∼ √ → ∞. (5)
dt ∆t→0 ∆t ∆t

W (t) is a Markovian process with normal probability distribution. In a general way, non linear terms
in Equation (3) must model situations of instabilities with the appearance of crashes, where “panic” is self
reinforcing. They can also be responsible for the sudden collapse of speculative bubbles [44].

4. Numerical Results
We perform a simulation of the model Equation (3) with the√term βW (t) being the Winner
increment and an additive√white noise of standard deviation σw = ∆t. One can write the Winner
increment as βdW (t) ∼ dtβRG , where RG is an aleatory generator number with a Gaussian
distribution of mean zero and variance σw2 = 1. The dynamic behavior of the return r (t) ≈ S(t) =
ln( X (t + ∆t)) − ln( X (t)) is shown in Figure 1 for parameter values β = 1.0 × 103 , α = 1.0 and different
values of δ. We have the time series of changing of prices oscillating quickly within a range. It suffers a
large change with the increase of the coupling x6 as shown in the Figure 1.

Figure 1. Time evolution of the return r (t) ≈ S(t) = ln( X (t + ∆t)) − ln( X (t)) for values of parameters
β = 1.0 × 103 and α = 1.0 and different values of δ.

In Figure 2, we analyse the behavior of the long tail distribution of the volatilities g = |r (t)|,
P( g) which must obey the power law P( g) = g−γ , with γ ∼ 3 [21–24]. As we obtain the value of
exponent of long tail of the fitted curve near to 3 (γ ∼ 3.65), hence, we obtain that the model must
reproduce to well stylized facts of the financial market. However, it is necessary to analyse the Hurst
index, being the addition of more terms, such as φ8 terms, must generate an increase of it, making
the exponent of the long tail distribution nearer 3. However, the high order terms in Equation (3) are
Entropy 2019, 21, 530 4 of 10

not smaller than inferior order terms due to the fact the Taylor expansion is not valid in this case,
df r (h)
i.e., the usual approach f ( x + h) = f ( x ) + dx · h + r (h), with limh→0 |h| = 0 not if it applies in this
case. In a general way, there have been many models employing different mathematical methods to
describe the dynamical behavior of financial markets with the objective of seeking general evidences
for their behavior. However, few model are suitable. Moreover, there have been many kinds of assets
in the financial markets, such as stocks, futures, and other financial tools, each of them having different
price behaviors. Higher-order terms in general force to non Gaussian distributions in Equation (3)
since they still exhibit an exponential dependence that is not verified by most of the financial markets.
In Figure 3, we calculate the Hurst index using the Rescaled Range method (RS). We plot the graphic
of the volatility log( R/s) vs. t = log n for the time series, for values of parameters β = 1000.0, α = 1.0,
µ = 1.0 and δ = 50.0. In the RS method, we have that R is the range and s is the standard deviation.
We estimate the Hurst index using the RS method given by H ∼ 0.39. The RS method is an older
method that is not used very often nowadays. The Detrended Fluctuation Analysis (DFA) is recently
more employed to analyse the behavior of time series. As the value of Hurst index employing the RS
method is within the range H ∈ (0.0, 0.5] we have a time series with long-term switching between
high and low values in adjacent pairs, meaning that a single high value will be followed by a low
value [45]. Employing the DFA method, we have obtained the exponent α as α ∼ 0.33 for the returns
and α ∼ 0.12 for the volatilities (Figure 4). The exponent obtained using the DFA method is similar to
the Hurst exponent, except that DFA may also be applied to signals whose underlying statistics (such as
mean and variance) or dynamics are not stationary or if they relate to measure based upon spectral
techniques such as autocorrelation and Fourier transform. Thus, we have obtained that the inclusion
of higher order terms, such as sixth order terms on the potential φ4 , Equation (3), has improved the
model making that the model tends to a possible model of the market, since it has made the exponent
of the long-tail cumulative probability distribution nearer to 3 and thus, tends to obey the inverse
cubic law [28].

γ = 3.6464

0.8

0.6
P(|r|)

0.4

0.2
γ

0
0 1000 2000 3000 4000 5000
|r|

Figure 2. Long tail distribution of the volatilities, g = |r |, P(|r |) for values β = 1000.0, α = 1.0,
µ = 1.0 and δ = 1000.0. The least-squares fits of power laws varies as f (|r |) ∼ |r |−γ for the long tail
distribution, where we found the tail index given by γ ∼ 3.65.
Entropy 2019, 21, 530 5 of 10

-1.0
10

H = 0.3863
δ = 50.0

-2.0
10

R/s -3.0
10

-4.0
10

3.0 4.0 5.0

10 10 10
l

Figure 3. Log-Log graphic using the rescaled range method (RS) of the volatility log ( R/s) vs. t = log n
for a time series, determined for the value of β = 1000.0, α = 1.0 and δ = 50.0. The Hurst indexes is
obtained as H = 0.386(3).

-3.0
10

α = 0.3302(1)
F(n)

-4.0
10

-5.0
10 0.0 1.0 2.0 3.0
10 10 10 10
n
(a)

α = 0.1207(3)

-5.0
10
F(n)

-6.0
10

-7.0
10 0.0 1.0 2.0 3.0
10 10 10 10
n
(b)

Figure 4. Log-Log graphic using Detrended Fluctuation Analysis (DFA) method for log ( F (n)) vs. log n
for the returns (a) and volatilities (b) respectively, determined for the value of β = 1000.0, α = 1.0 and
δ = 50.0.
Entropy 2019, 21, 530 6 of 10

5. Analysis by Fokker–Planck Equation

Equation (3), which in the linear, deterministic case (that is, for a potential given by αX and
W = 0) reduces to the classical Fokker–Planck equation, describing the particle transport dynamics
(price dynamics) in disordered media driven by highly irregular or stochastic field forces [46].
When the transition probabilities are zero, the differential Chapman–Kolmogorov equation
reduces to Fokker–Planck equation [33]. From stochastic equation Equation (3), we obtain the time
development of an arbitrary f ( X (t)) using the Itô formula [33]

= f 0 [( X (t)][ αX (t0 ) − µ( X (t0 ))3 − δ( X (t0 ))6 dt + βdW ]

 
f [ X (t) + dX (t)] − f [ X (t)]
2 (6)
+ β2 f ”[ X (t)](dW )2 ,

where higher order terms have been discarded, and (dW (t))2 = dt. Taking the average of both sides in
the equation above and defining γ = β2 , we get

df
 
∂f h i γ ∂2 f  
= αX (t0 ) − µ( X (t0 ))3 − δ( X (t0 ))6 + . (7)
dt ∂x 2 ∂x2

Using
d
R∞ R∞
dt h f ( X ( t )i = dtd −∞ dx f ( x ) P( x, t) = −∞ dx f ( x ) ∂t∂ [ P( x, t)]
R ∞ ∂f 
= −∞ ∂x αX (t0 ) − µ( X (t0 ))3 − δ( X (t0 ))6 P( x, t)dx

(8)
R∞ 2
+ γ2 −∞ ∂∂x2f P( x, t)dx
we integrate by parts and discard surface terms to obtain
R∞ R∞
αX (t0 ) − µ( X (t0 ))3 − δ( X (t0 ))6 P( x, t) dx
∂ ∂
 
−∞ dx f ( x ) ∂t [ P( x, t)] = −∞ f ( x ) ∂x
R ∞ ∂2 (9)
+ γ2 −∞ f ( x ) ∂x 2 [ P ( x, t )] dx.

and hence

∂ ∂ nh i o γ ∂2
P( x, t) = − αX (t0 ) − µ( X (t0 ))3 − δ( X (t0 ))6 P( x, t) + P( x, t). (10)
∂t ∂x 2 ∂x2
The associated Fokker–Planck equation to the above model is given by

∂P( x, t) ∂[(−αx + µx3 + δx6 ) P( x, t)] γ ∂2 [ P( x, t)]

= + . (11)
∂t ∂x 2 ∂x2
Taking the Fourier transform of the Fokker–Planck equation, provided we can simultaneously
guarantee the normalization of the probability density and in which P( x ) is reasonably well behaved,
we take the boundaries at infinity for P( x, t) as limx→∞ P( x, t) = 0 and therefore ∂ x P( x ) being
reasonably well behaved. As limx→∞ ∂ x P( x, t) = 0 thus, a nonzero current of probability at infinity
will usually require that the terms in the equation above become infinite there [33]. We use the initial
condition P( x = x0 , t = 0) = P0 . The Fokker-Plank equation above, independent on time, can be
analytically solved making the power series expansion
P( x ) = ∑∞ n
n=−∞ an x generating the following recurrence relations

2αm 2( α + k )
(m + 2)(m + 1) am+2 − β am − γ am =0
6µ
µ ( m − 2 ) a m −2 − γ a m −2 = 0 (12)
δ(m − 5) am−5 − 12δ γ am−5 = 0.

We have for 0 ≤ m ≤ 2
β(m + 2)(m + 1)
am = a m +2 . (13)
2α(m + 1) + 2k
Entropy 2019, 21, 530 7 of 10

Hence, we obtain P( x ) as
   
γ 1 1
P( x ) = 1 + (α + k) x2 + 2 (α + k) (3α + k ) x4 + · · · a0 + x + (2α + k) x3 · ·· a1 . (14)
2 3β 3γ

For 2 ≤ m < 5, we have am−2 = 0 and for m ≥ 5, we also get am−5 = 0. Therefore,

P( x ) = a0 + a1 x, (15)

where the constants a0 and a1 are determined by the initial conditions, being a0 = P0 and k the
separation constant in Equation (11). For m ≤ −1, we have am = 0 for all m. From the normalization
condition, the second term in the density probability above must be zero and therefore, all coefficients
a1 must cancel. Therefore, we have
P( x, t) = P0 e−kt (16)

To ensure the normalization of the probability density, P0 must be non zero only within interval
−ε ≤ x ≤ ε and zero out it. Consequently, we have P( x, t) = (1/2ε)e−kt .
The cumulative probability density F ( x ) for the probability density above in the limit of x large is
given by
Z x
1 1
F ( x, t) = P( x 0 )dx 0 = ( x − ε)e−kt ' xe−kt (17)
−∞ 2ε 2ε
Despite the exponential decreasing of time behavior for the cumulative probability, we have the
cumulative probability distribution of the volatilities P(|r (t)|) that must exhibit the cubic inverse law
behavior, P(|r |) = 1/|r |3 , for the long tail, and not the cumulative probability density standard. For µ,
δ = 0, the Equation (11) can be easily solved giving ν(t) = x0 e−iαt and κ (t) = 2α γ
(1 − e−2αt ) and hence,
we have a Gaussian distribution for P( x, t) given by
2
1 − ( x−2κν((tt)))
P( x, t) = p e , (18)
2πκ (t)

were in t → ∞ limit, we have P( x, t) = P( x ) given by

α − αxγ2
r
P( x ) = e . (19)
πγ

Therefore, the probability density is non-Gaussian when terms of type φ4 and φ6 and higher order
are included in the model.

6. Conclusions
In summary, we have investigated a generalized nonlinear Itô’s stochastic model or generalized
Itô diffusion model as a mathematical model for the price dynamics of the financial market. As the
φn has local minimums for u = u0 and maximum for u = u∗, beyond which the potential plummets
to −∞, there are height barriers separating the stable from unstable regions. Near to local minimum,
the particle has a random harmonic-like until an activated event driven by noise term brings the
particle near u∗. In financial terms, the regime where the particle oscillates around u = u0 is the
random walk regime. This normal regime can be interrupted by “crashes”, where the derivative of
the price becomes large and negative due to the risk aversion given µ, δ terms which enhance the
drop in price. We have calculated the Hurst index using both the RS and DFA methods and verified
the behavior of the long tail distributions of the volatilities that must obey to the inverse cubic law
behavior [21–24]. Our results show that the inclusion of higher order terms in the Bouchaud-Cont
Langevin model [44] and on models of Refs. [28–31] still make it so that they are obeyed by the
financial markets. In a general way, there have been many methods, employing mathematical models,
Entropy 2019, 21, 530 8 of 10

to describe the price dynamics in finance, seeking general evidences. However, there are few models
suitable for real situations. Furthermore, there are many kinds of assets in the financial markets,
such as stocks, futures, and other financial tools, each of them with different price behaviors.
In a general way, interaction terms of the form λ4 φn /n! can be represented by connected graphs
corresponding to given initial and final states with n lines meeting at each vertex. Interaction
terms proportional φ6 introduced in the λ4 φ4 theory can give a further contribution to the six-point
amplitude and we could choose λ6 φ6 so that it cancels the divergence terms that are left. Therefore,
we could adjust nonlinear terms in the Equation (3) with the aim to give an empirical evidence on the
relationships between trading volume and return volatility of the Bitcoin market [47].

Funding: This research was funded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
grant number 1592874587824075.
Conflicts of Interest: The authors declare no conflict of interest.

References
1. Richmond, P.; Mimkes, J.; Hutzler, S. Econophysics and Physical Economics; Oxford University Press: Oxford,
UK, 2013.
2. Jacobs, K. Stochastic Processes for Physicists; Cambridge University Press: New York, NY, USA, 2013.
3. Black, F.; Scholes, M. The pricing of options and corporate liabilities. J. Political Econ. 1973, 81, 637–659.
[CrossRef]
4. Mantegna, R.S.; Stanley, H.E. An Introduction to Econophysics, Correlations and Complexity in Finance, 4th ed.;
Cambridge University Press: New York, NY, USA, 2007.
5. Mike, S.; Farmer, J.D. An empirical behavioral model of liquidity and volatility. J. Econ. Dyn. Control 2008, 32,
200–234. [CrossRef]
6. Gu, G.-F.; Zhou, W.-X. Emergence of long memory in stock volatility from a modified Mike-Farmer model.
Europhys. Lett. 2009, 86, 48002. [CrossRef]
7. Zhang, X.; Ping, J.; Zhu, T.; Li, Y.; Xiong, X. Are price limits effective? An examination of an artificial stock
market. PLoS ONE 2016, 11, e0160406. [CrossRef]
8. Sornette, D. Critical Phenomena in Natural Sicences—Chaos, Fractals, Self-Organization and Disorder: Concepts and
Tools, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2004.
9. Meng, H.; Ren, F.; Gu, G.-F.; Xiong, X.; Zhang, Y.-J.; Zhou, W.-X.; Zhang, W. Effects of long memory in the
order submission process on the properties of recurrence intervals of large price fluctuations. Europhys. Lett.
2012, 98, 38003. [CrossRef]
10. Assenza, T.; Gatti, D.D.; Grazzini, J. Emergent dynamics of a macroeconomic agent based model with capital
and credit. J. Econ. Dyn. Control 2015, 50, 5–28. [CrossRef]
11. Diks, C.; Wang, J. Can a stochastic cusp catastrophe model explain housing market crashes?
J. Econ. Dyn. Control 2016, 69, 68–88. [CrossRef]
12. In’t Veld, D. Adverse effects of leverage and short-selling constraints in a financial market model with
heterogeneous agents. J. Econ. Dyn. Control 2016, 69, 45–67. [CrossRef]
13. Lima, L. Modeling of the financial market using the two-dimensional anisotropic Ising model. Phys. A Stat.
Mech. Appl. 2017, 482, 544–551. [CrossRef]
14. Gu, G.-F.; Zhou, W.-X. On the probability distribution of stock returns in the Mike-Farmer model. Eur. Phys. J. B
2009, 67, 585–592. [CrossRef]
15. Zhou, W.-X.; Sornette, D. Self-organizing Ising model of financial markets. Eur. Phys. J. B 2007, 55, 175-181.
[CrossRef]
16. Sznajd-Weron, K.; Weron, R. A simple model of price formation. Int. J. Mod. Phys. C 2002, 13, 115–123.
[CrossRef]
17. Callen, E.; Shapero, D. A theory of social imitation. Phys. Today 1974, 27, 23. [CrossRef]
Entropy 2019, 21, 530 9 of 10

18. Montroll, E.W.; Badger, W.W. Introduction to Quantitative Aspects of Social Phenomena; Gordon and Breach:
New York, NY, USA, 1974.
19. Orléan, A. Bayesian interactions and collective dynamics of opinion: Herd behavior and mimetic contagion.
J. Econ. Behav. Organ. 1995, 28, 257–274. [CrossRef]
20. Wang, F.; Yamasaki, K.; Havlin, S.; Stanley, H.E. Indication of multiscaling in the volatility return intervals of
stock markets. Phys. Rev. E 2008, 77, 016109. [CrossRef]
21. Gopikrishnan, P.; Meyer, M.; Amaral, L.N.; Stanley, H.E. Inverse cubic law for the distribution of stock price
variations. Eur. Phys. J. B 1998, 3, 139–140. [CrossRef]
22. Gopikrishnan, P.; Plerou, V.; Amaral, L.A.N.; Meyer, M.; Stanley, H.E. Scaling of the distribution of fluctuations
of financial market indices. Phys. Rev. E 1999, 60, 5305. [CrossRef]
23. Plerou, V.; Gopikrishnan, P.; Amaral, L.A.N.; Meyer, M.; Stanley, H.E. Scaling of the distribution of price
fluctuations of individual companies. Phys. Rev. E 1999, 60, 6519. [CrossRef]
24. Botta, F.; Moat, H.S.; Stanley, H.E.; Preis, T. Quantifying stock return distributions in financial markets.
PLoS ONE 2015, 10, e0135600. [CrossRef]
25. Zhou, J.; Gu, G.-F.; Jiang, Z.-Q.; Xiong, X.; Chen, W.; Zhang, W.; Zhou, W.-X. Computational experiments
successfully predict the emergence of autocorrelations in ultra-high-frequency stock returns. Comput. Econ.
2017, 50, 579–594. [CrossRef]
26. Karamé, F. A new particle filtering approach to estimate stochastic volatility models with Markov-switching.
Econom. Stat. 2018, 8, 204–230. [CrossRef]
27. Anzarut, M.; Mena, R.H. Harris process to model stochastic volatility. Econom. Stat. 2019, 10, 151–169
28. Lima, L.S.; Miranda, L.L.B. Price dynamics of the financial markets using the stochastic differential equation
for a potential double well. Phys. A Stat. Mech. Appl. 2018, 490, 828–833. [CrossRef]
29. Lima, L.S.; Santos, G.K.C. Stochastic process with multiplicative structure for the dynamic behavior of the
financial market. Phys. A Stat. Mech. Appl. 2018, 512, 222–229. [CrossRef]
30. Lima, L.S.; Oliveira, S.C.; Abeilice, A.F. Modelling Based in Stochastic Non-Linear Differential Equation for
Price Dynamics. Pioneer J. Math. Math. Sci. 2018, 23, 93–103.
31. Lima, L.S.; Oliveira, S.C.; Abeilice, A.F.; Melgaço, J.H. Breaks down of the modeling of the financial market
with addition of non-linear terms in the Itô stochastic process. Phys. A Stat. Mech. Appl. 2019, 526, 120932.
[CrossRef]
32. Shreve, S.E. Stochastic Calculus for Finance II: Continuous-Time Models; Springer: New York, NY, USA, 2004.
33. Gardiner, C. Stochastic Methods, A Handbook for the Natural and Social Sciences, 4th ed.; Springer:
Berlin/Heidelberg, Germany, 2009.
34. Oksendal, B. Stochastic Differential Equations: An Introduction with Applications, 6th ed.; Springer:
Berlin/Heidelberg, Germany, 2013.
35. Xavier, P.O.; Atman, A.; de Magalhães, A.B. Equation-based model for the stock market. Phys. Rev. E 2017,
96, 032305. [CrossRef]
36. He, J.-H. Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 2006, 20, 1141–1199.
[CrossRef]
37. Tsallis, C. Economics and Finance: q-Statistical stylized features galore. Entropy 2017, 19, 457. [CrossRef]
38. Ribeiro, M.S.; Nobre, F.D.; Curado, E.M. Classes of N-dimensional nonlinear Fokker–Planck equations
associated to Tsallis entropy. Entropy 2011, 13, 1928–1944. [CrossRef]
39. Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 1988, 52, 479–487. [CrossRef]
40. Yu, W.-T.; Chen, H.-Y. Herding and zero-intelligence agents in the order book dynamics of an artificial double
auction market. Chin. J. Phys. 2018, 56, 1405–1414. [CrossRef]
41. Mandelbrot, B. The Variation of Certain Speculative Prices. J. Bus. 1963 35, 394 . [CrossRef]
42. Fama, E.F. Mandelbrot and the stable Paretian hypothesis. J. Bus. 1963, 36, 420–429. [CrossRef]
43. Lux, T.; Marchesi, M. Scaling and criticality in a stochastic multi-agent model of a financial market. Nature
1999, 397, 498. [CrossRef]
44. Bouchaud, J.-P.; Cont, R. A Langevin approach to stock market fluctuations and crashes. Eur. Phys. J. B 1998,
6, 543–550. [CrossRef]
Entropy 2019, 21, 530 10 of 10

45. Kleinow, T. Testing Continuous Time Models in Financial Markets. Ph.D. Thesis, Humboldt University of
Berlin, Berlin, Germany, 2002.
46. Barbu, V.; Röckner, M. Nonlinear Fokker–Planck equations driven by Gaussian linear multiplicative noise.
J. Differ. Equ. 2018, 265, 4993–5030. [CrossRef]
47. Wang, P.; Zhang, W.; Li, X.; Shen, D. Trading volume and return volatility of Bitcoin market: Evidence for the
sequential information arrival hypothesis. J. Econ. Interact. Coord. 2019, 14, 377–418. [CrossRef]

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).