We introduce a general class of multifractional stochastic processes driven by a multifractional Brownian motion and study estimation of their pointwise H\"older exponents (PHE) using the localized generalized quadratic variation approach. By comparing it with the other two benchmark estimation approaches through a simulation study, we show that our estimator has better performance in the case where the observed process is some unknown bivariate function of time and multifractional Brownian motion. 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.
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