Investment time horizon and multifractality of stock price process

We construct a multifractal random walk for a stock trades model with inverse power law interaction. Consider a stock in a stock market and introduce a discrete time model for pair trades [(first trade →\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow$$\end{document} second (reverse) trade] transacted by various types of traders. The type of trader is characterized by the investment time horizon defined as a time difference of the pair trades. We assume that probability distributions of the investment time horizons are given by inverse power law interactions with different exponents depending on the types of traders, and define a discrete time log-volatility process of the stock. Using the method of abstract polymer expansion developed in the study of mathematical physics we obtain a continuous type log-volatility process as a scale limit, and define a log-return process from time 0 as a stochastic integral with respect to a Brownian motion. We then find a condition of the continuous cascade equation which derives a multifractality in the log-return process. Finally, we construct a multifractal random walk using the martingale convergence theorem.