### Abstract

We present a mathematical model of the trade signs and trade volumes, and derive a fractional Brownian motion as a scaling limit of the signed volume process which describes a super-diffusive nature. In our model, we assume that traders place a market order at a single time or divide their order into two chunks and place orders at different times. When they divide their order into two chunks, the probability distribution of the time lag t of divided orders is assumed to decay as an inverse power law of t with exponent α. We obtain three types of scaling limit of the signed volume process according to the three cases of the value of α, (i) α <1, (ii) α = 1, and (iii) α > 1. (See Theorem 4.1.) We prove that a fractional Brownian motion having a super diffusive nature is obtained in a scaling limit of a signed volume process if and only if α <1.

Original language | English |
---|---|

Pages (from-to) | 26-37 |

Number of pages | 12 |

Journal | Progress of Theoretical Physics Supplement |

Issue number | 179 |

Publication status | Published - 2009 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Progress of Theoretical Physics Supplement*, (179), 26-37.

**Long memory in finance and fractional brownian motion.** / Kuroda, Koji; Murai, Joshin.

Research output: Contribution to journal › Article

*Progress of Theoretical Physics Supplement*, no. 179, pp. 26-37.

}

TY - JOUR

T1 - Long memory in finance and fractional brownian motion

AU - Kuroda, Koji

AU - Murai, Joshin

PY - 2009

Y1 - 2009

N2 - We present a mathematical model of the trade signs and trade volumes, and derive a fractional Brownian motion as a scaling limit of the signed volume process which describes a super-diffusive nature. In our model, we assume that traders place a market order at a single time or divide their order into two chunks and place orders at different times. When they divide their order into two chunks, the probability distribution of the time lag t of divided orders is assumed to decay as an inverse power law of t with exponent α. We obtain three types of scaling limit of the signed volume process according to the three cases of the value of α, (i) α <1, (ii) α = 1, and (iii) α > 1. (See Theorem 4.1.) We prove that a fractional Brownian motion having a super diffusive nature is obtained in a scaling limit of a signed volume process if and only if α <1.

AB - We present a mathematical model of the trade signs and trade volumes, and derive a fractional Brownian motion as a scaling limit of the signed volume process which describes a super-diffusive nature. In our model, we assume that traders place a market order at a single time or divide their order into two chunks and place orders at different times. When they divide their order into two chunks, the probability distribution of the time lag t of divided orders is assumed to decay as an inverse power law of t with exponent α. We obtain three types of scaling limit of the signed volume process according to the three cases of the value of α, (i) α <1, (ii) α = 1, and (iii) α > 1. (See Theorem 4.1.) We prove that a fractional Brownian motion having a super diffusive nature is obtained in a scaling limit of a signed volume process if and only if α <1.

UR - http://www.scopus.com/inward/record.url?scp=69549124460&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69549124460&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:69549124460

SP - 26

EP - 37

JO - Progress of Theoretical Physics Supplement

JF - Progress of Theoretical Physics Supplement

SN - 0375-9687

IS - 179

ER -