# A Numerical Approach of Handling Fractional Stochastic Differential Equations

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## Abstract

**:**

## 1. Introduction

- ${W}_{0}=1$.
- ${W}_{t}\u2013{W}_{s}\sim \sqrt{t-s}N(0,1)$ for $0\le s<t$, where $N(0,1)$ indicates a standard normal distribution.
- The two increments ${W}_{t}\u2013{W}_{s}$ and ${W}_{\tau}\u2013{W}_{\upsilon}$ are independent on distinct time intervals for $0\le s<t<\tau <\upsilon $.

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1**

**.**Suppose that ${D}_{*}^{k\alpha}f\left(x\right)\in C(0,b]$ for $k=0,1,\dots ,n+1,$ where $0<\alpha \le 1$. Then the function f can be expanded about $x={x}_{0}$ as:

**Theorem**

**2**

**.**Let $d\xi \left(t\right)=adt+bdw\left(t\right)$ and let $f(x,t)$ be a continuous function in $(x,t)\in {\mathbb{R}}^{1}\times [0,\infty )$ with partial derivatives ${f}_{x},{f}_{xx},{f}_{t}$. Then the process $f\left(\xi \right(t),t)$ has a stochastic differential, given by:

**Definition**

**3**

**.**The definite fractional integral of the function f of order α is given by:

**Theorem**

**3**

**.**Suppose $f\in {C}^{3}[a,b]$ and ${x}_{0}$, ${x}_{1}$, ${x}_{2}$ are three distinct points in the interval $[a,b]$ such that $a={x}_{0}<{x}_{1}={x}_{0}+h<{x}_{2}={x}_{0}+2h=b$ with $h>0$. Then the modified three-point fractional formula for approximating the Caputo first derivative is given by:

## 3. Modified Three-Point Fractional Formula

**Corollary**

**1.**

**Theorem**

**4.**

**Proof.**

## 4. Handling FSDE Using the Modified Three-Point Fractional Formula

## 5. Applications

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Absolute error generated by the numerical solution (29) and Euler–Maruyama’s solution.

**Figure 7.**Absolute error generated by the numerical solution (31) and Euler–Maruyama’s solution.

**Figure 10.**Absolute error generated by the numerical solution (33) and Euler–Maruyama’s solution.

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**MDPI and ACS Style**

Batiha, I.M.; Abubaker, A.A.; Jebril, I.H.; Al-Shaikh, S.B.; Matarneh, K.
A Numerical Approach of Handling Fractional Stochastic Differential Equations. *Axioms* **2023**, *12*, 388.
https://doi.org/10.3390/axioms12040388

**AMA Style**

Batiha IM, Abubaker AA, Jebril IH, Al-Shaikh SB, Matarneh K.
A Numerical Approach of Handling Fractional Stochastic Differential Equations. *Axioms*. 2023; 12(4):388.
https://doi.org/10.3390/axioms12040388

**Chicago/Turabian Style**

Batiha, Iqbal M., Ahmad A. Abubaker, Iqbal H. Jebril, Suha B. Al-Shaikh, and Khaled Matarneh.
2023. "A Numerical Approach of Handling Fractional Stochastic Differential Equations" *Axioms* 12, no. 4: 388.
https://doi.org/10.3390/axioms12040388