  We introduce a general class of stochastic processes driven by a
multifractional Brownian motion (mBm) and study the estimation problems of
their pointwise H\"older exponents (PHE) based on a new localized generalized
quadratic variation approach (LGQV). By comparing our suggested approach with
the other two existing benchmark estimation approaches (classic GQV and
oscillation approach) 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 mBm. Such multifractional processes, whose PHEs
are time-varying, can be used 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 the equity path's roughness varies via time and the
stock price informativeness properties from global stock markets.
