# Long memory in finance and fractional brownian motion

Koji Kuroda, Joshin Murai

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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 26-37 12 Progress of Theoretical Physics Supplement 179 Published - 2009

### Fingerprint

finance
scaling
mathematical models
time lag
theorems
exponents
decay

### ASJC Scopus subject areas

• Physics and Astronomy (miscellaneous)

### Cite this

In: Progress of Theoretical Physics Supplement, No. 179, 2009, p. 26-37.

Research output: Contribution to journalArticle

@article{64815c26767e411f91fbb154cb06bcf0,
title = "Long memory in finance and fractional brownian motion",
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.",
author = "Koji Kuroda and Joshin Murai",
year = "2009",
language = "English",
pages = "26--37",
journal = "Progress of Theoretical Physics Supplement",
issn = "0375-9687",
publisher = "Yukawa Institute for Theoretical Physics",
number = "179",

}

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 -