# A Monte Carlo Approach to Bitcoin Price Prediction with Fractional Ornstein–Uhlenbeck Lévy Process

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- The process is likely to reverse the trend over the time frame considered.
- The process is random in which knowing one data point does not provide insight into predicting future data points.
- The process is persistent in the sense that a future data point is likely to be similar to a data point preceding it.

#### 2.1. Fractional Brownian Motion

#### 2.2. Lévy Process

#### 2.2.1. Infinitely Divisible Distributions

#### 2.2.2. Continuous-Time Stochastic Processes

- The random variables ${L}_{{t}_{0}},{L}_{{t}_{1}}-{L}_{{t}_{0}},\cdots ,{L}_{{t}_{n}}-{L}_{{t}_{n-1}}$ are independent for all $n\ge 1$ and $0\le {t}_{0}<{t}_{1}<\cdots <{t}_{n}$ (independent increments);
- ${L}_{t+s}-{L}_{t}$ has the same distribution as ${L}_{s}$ for all $s,t\ge 0$ (stationary increments);
- L is stochastically continuous; that is, for all $t\ge 0$ and $a>0$.$$\underset{s\to t}{lim}\mathbb{P}\left[\right|{L}_{s}-{L}_{t}|>a]=0.$$
- The paths $t\mapsto {L}_{t}$ are right-continuous with left limits (cadlag-continue á droite et limite á gauche).

#### 2.2.3. Normal Inverse Gaussian

- iv) $P({N}_{t}=k)={\displaystyle \frac{{\left(\lambda t\right)}^{k}}{k!}}exp(-\lambda t),\phantom{\rule{0.277778em}{0ex}}k\ge 0,\phantom{\rule{0.277778em}{0ex}}t\ge 0$ (Poisson distribution).

#### 2.3. Fractional Ornstein–Uhlenbeck Lévy Process

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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29 July | 8 August | 18 August | |
---|---|---|---|

Quantiles | 10 Day | 20 Day | 30 Day |

25% | 9812 | 9590 | 8915 |

50% | 10,360 | 10,283 | 9956 |

75% | 10,841 | 11,029 | 11,161 |

90% | 11,342 | 11,818 | 12,049 |

95% | 11,881 | 12,245 | 13,157 |

99% | 11,963 | 12,885 | 13,600 |

mean | 10,085.65 | 10,085.66 | 10,085.68 |

skweness | −0.0899 | −0.0891 | −0.0626 |

kurtosis | 1.1390 | 0.5511 | 0.3755 |

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**MDPI and ACS Style**

Mba, J.C.; Mwambi, S.M.; Pindza, E.
A Monte Carlo Approach to Bitcoin Price Prediction with Fractional Ornstein–Uhlenbeck Lévy Process. *Forecasting* **2022**, *4*, 409-419.
https://doi.org/10.3390/forecast4020023

**AMA Style**

Mba JC, Mwambi SM, Pindza E.
A Monte Carlo Approach to Bitcoin Price Prediction with Fractional Ornstein–Uhlenbeck Lévy Process. *Forecasting*. 2022; 4(2):409-419.
https://doi.org/10.3390/forecast4020023

**Chicago/Turabian Style**

Mba, Jules Clément, Sutene Mwambetania Mwambi, and Edson Pindza.
2022. "A Monte Carlo Approach to Bitcoin Price Prediction with Fractional Ornstein–Uhlenbeck Lévy Process" *Forecasting* 4, no. 2: 409-419.
https://doi.org/10.3390/forecast4020023