Randomness and fractional stable distributions

Charles Tapiero, Pierre Vallois

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)54-60
Number of pages7
JournalPhysica A: Statistical Mechanics and its Applications
Volume511
DOIs
StatePublished - Dec 1 2018

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Stable Distribution
Randomness
Fractional
finance
Brownian Bridge
Jump Process
Temporal Correlation
Long-range Correlations
operators
Limit Distribution
Fractional Brownian Motion
Finance
Anomalous
Brownian motion
Power Law
Probability Distribution
Dependent

ASJC Scopus subject areas

  • Statistics and Probability
  • Condensed Matter Physics

Cite this

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

In: Physica A: Statistical Mechanics and its Applications, Vol. 511, 01.12.2018, p. 54-60.

Research output: Contribution to journalArticle

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