### Abstract

Stochastic and fractional models are defined by applications of Liouville (and other) fractional operators. They underlie anomalous transport dynamical properties such as long range temporal correlations manifested in power laws. Prolific applications to finance and other domains have been published, based mostly on a randomness defined by the fractional Brownian Motion. Application to probability distributions (Tapiero and Vallois 2016, 2017, 2018), have indicated that fractional distributions are incomplete and their limit distributions (based on the Central LimitTheorem) depend on their fractional index. For example, for a fractional index 1∕2≤H≤1, we showed that a fractional Brownian Bridge defines a fractional randomness (rather than a Brownian Motion). In this paper we consider the case 0<H<1∕2 and prove that the underlying fractional distribution is a randomness defined by an α-stable distribution with α=1∕(1−H) to H∈1,2. Then, the smaller the fractional index, the greater the propensity for a randomness to be defined by a jump process rather than diffusions defining randomness. These properties are important in applications where risks, prices and their management are dependent of their definition of randomness.

Original language | English (US) |
---|---|

Pages (from-to) | 54-60 |

Number of pages | 7 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 511 |

DOIs | |

State | Published - Dec 1 2018 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*511*, 54-60. https://doi.org/10.1016/j.physa.2018.07.019

**Randomness and fractional stable distributions.** / Tapiero, Charles; Vallois, Pierre.

Research output: Contribution to journal › Article

*Physica A: Statistical Mechanics and its Applications*, vol. 511, pp. 54-60. https://doi.org/10.1016/j.physa.2018.07.019

}

TY - JOUR

T1 - Randomness and fractional stable distributions

AU - Tapiero, Charles

AU - Vallois, Pierre

PY - 2018/12/1

Y1 - 2018/12/1

N2 - Stochastic and fractional models are defined by applications of Liouville (and other) fractional operators. They underlie anomalous transport dynamical properties such as long range temporal correlations manifested in power laws. Prolific applications to finance and other domains have been published, based mostly on a randomness defined by the fractional Brownian Motion. Application to probability distributions (Tapiero and Vallois 2016, 2017, 2018), have indicated that fractional distributions are incomplete and their limit distributions (based on the Central LimitTheorem) depend on their fractional index. For example, for a fractional index 1∕2≤H≤1, we showed that a fractional Brownian Bridge defines a fractional randomness (rather than a Brownian Motion). In this paper we consider the case 0<H<1∕2 and prove that the underlying fractional distribution is a randomness defined by an α-stable distribution with α=1∕(1−H) to H∈1,2. Then, the smaller the fractional index, the greater the propensity for a randomness to be defined by a jump process rather than diffusions defining randomness. These properties are important in applications where risks, prices and their management are dependent of their definition of randomness.

AB - Stochastic and fractional models are defined by applications of Liouville (and other) fractional operators. They underlie anomalous transport dynamical properties such as long range temporal correlations manifested in power laws. Prolific applications to finance and other domains have been published, based mostly on a randomness defined by the fractional Brownian Motion. Application to probability distributions (Tapiero and Vallois 2016, 2017, 2018), have indicated that fractional distributions are incomplete and their limit distributions (based on the Central LimitTheorem) depend on their fractional index. For example, for a fractional index 1∕2≤H≤1, we showed that a fractional Brownian Bridge defines a fractional randomness (rather than a Brownian Motion). In this paper we consider the case 0<H<1∕2 and prove that the underlying fractional distribution is a randomness defined by an α-stable distribution with α=1∕(1−H) to H∈1,2. Then, the smaller the fractional index, the greater the propensity for a randomness to be defined by a jump process rather than diffusions defining randomness. These properties are important in applications where risks, prices and their management are dependent of their definition of randomness.

UR - http://www.scopus.com/inward/record.url?scp=85050652496&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050652496&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2018.07.019

DO - 10.1016/j.physa.2018.07.019

M3 - Article

AN - SCOPUS:85050652496

VL - 511

SP - 54

EP - 60

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

ER -