Before revision:  The time-varying PHE feature allows us to apply such class of multifractional processes to model stock prices under various market conditions . An empirical study on modeling cross-listed stocks provides new evidence that equity's path roughness varies via time and  price informativeness properties from global  markets.
After revision:  The time-varying PHE feature allows us to apply such class of multifractional processes to model stock prices under various market conditions , that are both time-dependent and region-dependent. As an application to finance, an empirical study on modeling cross-listed stocks provides new evidence that equity's path roughness varies via time and price informativeness properties from global markets.