# Noise and Financial Stylized Facts: A Stick Balancing Approach

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- For $\tau \le 5$, there is a critical threshold for ${R}_{0}$ (specifically ${R}_{0}^{*}=4.80545$ for $\tau =0$ and ${R}_{0}^{*}=4.83749$ for $\tau =5$), below which the sticks quickly falls, thus the point identified by the initial conditions $\theta \left(0\right)$ and $\dot{\theta}\left(0\right)$ in phase space results to be unstable, i.e., a repeller. Above the threshold, the system becomes stable, although the equilibrium point may not coincide with the initial one, and spiral trajectories can be observed approaching the node (in this case, of course, the stick never falls). In other words, in correspondence of the critical threshold ${R}_{0}^{*}$, there is a sudden transition from a completely disordered regime to a completely ordered one.
- For $5<\tau \le 10$, a third type of regime appears. Below a different critical threshold ${R}_{0}^{*}$ (which for $\tau =10$ becomes ${R}_{0}^{*}=4.88424$), we always observe a repeller in the phase space, and the sticks always falls; on the other hand, for ${R}_{0}^{*}<{R}_{0}<5$, the stick never falls, and we again find a spiral node; finally, for ${R}_{0}>5$, the stick falls again, but we now observe a spiral repeller.
- For $\tau >10$ we do not find anymore the regime where the initial point is a spiral node. The stick always falls and we pass from finding a repeller to find a spiral repeller in phase space in correspondence of a critical value starting from ${R}_{0}\sim 5$ and increasing as $\tau $ increases. For $\tau =50$ and for values of ${R}_{0}\le 10$, the transition to spiral repeller is no longer observed.

- For values of $\tau $ less than 5, and for ${R}_{0}$ less than 4, the stick immediately falls for any initial condition, thus showing repeller behavior. Within the small range $4<{R}_{0}<6$, more interesting dynamics start to be observed in the phase space. An example is shown in Figure 1 for $\tau =2$ and ${R}_{0}=5.2$. Finally, for larger values of ${R}_{0}$, the stick still falls but the representative point of the system barely moves from its initial position in the phase space.
- For $\tau >5$, the dynamics start to become very sensitive to the noise, for any ${R}_{0}$. Generally, as $\tau $ increases, a spiral-like repelling behavior emerges for values of ${R}_{0}>6$. Regardless, for $\tau \sim 10$ and for $4.5<{R}_{0}<6.5$, a window of complex behavior does appear, with longer trajectories more suitable for allowing statistical analysis.

## 3. Results

#### Empirical Data Collection and Comparison with Simulated Data

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Angular velocity returns as function of time for $\tau =10$, ${R}_{0}=5$, $\sigma =10$ and $L=49\%$.

**Figure 4.**Comparison of several distributions of angular velocity returns (cumulated over 20 runs) at $\sigma =10$ and for increasing values of the stick length L, expressed in percentage with respect to the width of the simulation environment. Gaussian distribution is also reported as a dashed line for comparison.

**Figure 5.**Comparison of several distributions of angular velocity returns (cumulated over 20 runs) at $L=49\%$ and for increasing values of noise $\sigma $. Gaussian distribution is also reported as a dashed line for comparison.

**Figure 6.**The angular velocity returns distribution of simulated stick balancing, with $L=49\%$ and $\sigma =10$, is compared with the returns distributions of real financial indexes (

**left**) and assets (

**right**). In both the panels, Gaussian distribution is also reported as a red dashed line, together with two fitting q-Gaussian curves, reported as blue and red full lines, respectively.

**Figure 7.**Comparison between the ACF of simulated data, with $L=49\%$ and $\sigma =10$, and the ACF of real financial indexes (

**left**) and assets (

**right**).

**Figure 8.**Comparison between the ACF of absolute value of simulated data, with $L=49\%$ and $\sigma =10$, and the ACF of absolute returns of real financial indexes (

**left**) and assets (

**right**).

Index | First Day | Last Day |
---|---|---|

AEX | 03/01/1983 | 06/07/2022 |

Dow Jones | 04/05/1950 | 06/07/2022 |

Euro stoxx 50 | 31/12/1986 | 06/07/2022 |

FTSE 100 | 30/12/1983 | 06/07/2022 |

FTSE MIB | 31/12/1997 | 06/07/2022 |

France CAC 40 | 09/07/1987 | 06/07/2022 |

IBEX 35 | 05/01/1987 | 06/07/2022 |

Nasdaq | 05/02/1971 | 06/07/2022 |

Nikkei 225 | 03/04/1950 | 06/07/2022 |

S&P 500 | 31/12/1963 | 06/07/2022 |

Asset | First Day | Last Day |
---|---|---|

American Express | 12/12/1972 | 06/07/2022 |

Amazon | 16/05/1997 | 06/07/2022 |

Apple | 15/12/1980 | 06/07/2022 |

BMW | 11/11/1996 | 06/07/2022 |

Colgate | 03/05/1973 | 06/07/2022 |

Ford | 02/06/1972 | 06/07/2022 |

General Electric | 03/01/1962 | 06/07/2022 |

JP Morgan | 18/03/1980 | 06/07/2022 |

Microsoft | 14/03/1986 | 06/07/2022 |

Pfizer | 02/06/1972 | 06/07/2022 |

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**MDPI and ACS Style**

Biondo, A.E.; Mazzarino, L.; Pluchino, A.
Noise and Financial Stylized Facts: A Stick Balancing Approach. *Entropy* **2023**, *25*, 557.
https://doi.org/10.3390/e25040557

**AMA Style**

Biondo AE, Mazzarino L, Pluchino A.
Noise and Financial Stylized Facts: A Stick Balancing Approach. *Entropy*. 2023; 25(4):557.
https://doi.org/10.3390/e25040557

**Chicago/Turabian Style**

Biondo, Alessio Emanuele, Laura Mazzarino, and Alessandro Pluchino.
2023. "Noise and Financial Stylized Facts: A Stick Balancing Approach" *Entropy* 25, no. 4: 557.
https://doi.org/10.3390/e25040557