  This paper develops an It\^o-type fractional pathwise integration theory for
fractional Brownian motion with Hurst parameters \( H \in (\frac{1}{3},
\frac{1}{2}] \), using the Lyons' rough path framework. This approach is
designed to fill gaps in conventional stochastic calculus models that fail to
account for temporal persistence prevalent in dynamic systems such as those
found in economics, finance, and engineering. The pathwise-defined method not
only meets the zero expectation criterion but also addresses the challenges of
integrating non-semimartingale processes, which traditional It\^o calculus
cannot handle. We apply this theory to fractional Black-Scholes models and
high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the
advantages of this approach. Additionally, the paper discusses the
generalization of It\^o integrals to rough differential equations (RDE) driven
by fBM, emphasizing the necessity of integrand-specific adaptations in the
It\^o rough path lift for stochastic modeling.
