TY - JOUR
T1 - Long memory in finance and fractional brownian motion
AU - Kuroda, Koji
AU - Murai, Joshin
N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
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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U2 - 10.1143/PTPS.179.26
DO - 10.1143/PTPS.179.26
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 -