Estimating the DJI Series by Multifractional Brownian Motion

This paper estimates the stock market and its price dynamics in terms of the multifractional Brownian motion. In our analysis, we use the financial dataset of the Dow Jones Industrial Average (DJI) time series from March 2009 to June 2015. First, we briefly introduce the definitions and properties of the Brownian motion (Bm), fractional Brownian motion (fBm) and multifractional Brownian motion (mBm). Then we model price processes as exponential of the sum of a regular process and a stochastic process. Finally, we estimate the Holder exponent and show how mBm can capture the fluctuations of the price dynamics