# Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market

## Abstract

**:**

## 1. Introduction

## 2. Economic Entropy

## 3. Phenomenological Itô Equation

## 4. Numerical Results

## 5. Analysis by Fokker–Planck Equation

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of the return $r\left(t\right)\approx S\left(t\right)=ln\left(X\right(t+\Delta t\left)\right)-ln\left(X\right(t\left)\right)$ for values of parameters $\beta =1.0\times {10}^{3}$ and $\alpha =1.0$ and different values of $\delta $.

**Figure 2.**Long tail distribution of the volatilities, $g=\left|r\right|$, $P\left(\right|r\left|\right)$ for values $\beta =1000.0$, $\alpha =1.0$, $\mu =1.0$ and $\delta =1000.0$. The least-squares fits of power laws varies as $f\left(\right|r\left|\right)\sim {\left|r\right|}^{-\gamma}$ for the long tail distribution, where we found the tail index given by $\gamma \sim 3.65$.

**Figure 3.**Log-Log graphic using the rescaled range method (RS) of the volatility $log\left(R/s\right)$ vs. $t=logn$ for a time series, determined for the value of $\beta =1000.0$, $\alpha =1.0$ and $\delta =50.0$. The Hurst indexes is obtained as $H=0.386\left(3\right)$.

**Figure 4.**Log-Log graphic using Detrended Fluctuation Analysis (DFA) method for $log\left(F\left(n\right)\right)$ vs. $logn$ for the returns (

**a**) and volatilities (

**b**) respectively, determined for the value of $\beta =1000.0$, $\alpha =1.0$ and $\delta =50.0$.

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S. Lima, L.
Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market. *Entropy* **2019**, *21*, 530.
https://doi.org/10.3390/e21050530

**AMA Style**

S. Lima L.
Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market. *Entropy*. 2019; 21(5):530.
https://doi.org/10.3390/e21050530

**Chicago/Turabian Style**

S. Lima, Leonardo.
2019. "Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market" *Entropy* 21, no. 5: 530.
https://doi.org/10.3390/e21050530