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)
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Long memory in finance and fractional brownian motion. / Kuroda, Koji; Murai, Joshin.
In: Progress of Theoretical Physics Supplement, No. 179, 2009, p. 26-37.Research output: Contribution to journal › Article
}
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.
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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 -